Influence of Operating Parameters On the Retention of

However, upward and downward flow directions in the vertical channel lead to similar ... Accordingly, we speculate that a new transport phenomenon, ca...
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Anal. Chem. 2004, 76, 5787-5798

Influence of Operating Parameters On the Retention of Chromatographic Particles By Thermal Field-Flow Fractionation Anne Regazzetti, Mauricio Hoyos, and Michel Martin*

Ecole Supe´ rieure de Physique et de Chimie Industrielles, Laboratoire de Physique et Me´ canique des Milieux He´ te´ roge` nes (UMR CNRS 7636), 10 rue Vauquelin, 75231 Paris Cedex 05, France

We have investigated the retention behavior of chromatographic particles in thermal field-flow fractionation (FFF). Retention time is found to increase with increasing temperature drop across the channel thickness, as expected for species exhibiting a thermophoretic mobility. Experiments have been performed with a vertically oriented channel rather than by using the classical horizontal configuration as this leads to much more reproducible retention data. In acetonitrile, silica-based particles are more retained than octadecyl-bonded silica particles, which confirms our previous finding, by means of a different method, that the thermophoretic mobility of the latter is smaller than that of the former. Whatever the type of particles and the nature of the carrier liquid, the relative retention time is observed to decrease with increasing carrier flow rate. This indicates that a hydrodynamic lift force acts on particles to move them away from the accumulation wall, as is usually observed in all FFF experiments with micrometer-sized particles. However, upward and downward flow directions in the vertical channel lead to similar retention data, indicating that inertial lift forces have a minor influence on retention. In addition, the relative retention time steadily decreases with increasing sample concentration, suggesting that the hydrodynamic lift force increases significantly with sample concentration. Accordingly, we speculate that a new transport phenomenon, called shear-induced hydrodynamic diffusion, not previously accounted for in the modeling of retention in FFF, is controlling the migration of the particles in the FFF channel. Implications of the influence of this phenomenon in other FFF experiments are discussed. Thermal field-flow fractionation (thermal FFF) belongs to the family of the field-flow fractionation methods of separation of macromolecular, colloidal, and particulate materials conceived by Giddings.1 It is based on the application of a steady heat flux perpendicular to the flow axis of the thin separation channel. If the channel contains a binary mixture (for instance, a sample analyte and the carrier fluid), the resulting temperature gradient * Corresponding author. Phone: +33-1-4079-4707. Fax: +33-1-4079-4523. E-mail: [email protected]. (1) Giddings, J. C. Sep. Sci. 1966, 1, 123-125. 10.1021/ac040012t CCC: $27.50 Published on Web 08/31/2004

© 2004 American Chemical Society

induces concentration nonuniformities, one component being more concentrated near the cold wall and the other near the hot wall. This physicochemical process is called thermodiffusion (or thermal diffusion), or still, the Soret (or Ludwig-Soret) effect. It has successfully allowed the separation and characterization of lipophilic polymers according to their molar mass as well as to their chemical composition by thermal FFF.2 In FFF, the absence of porous packing as well as the tangential, rather than elongational, nature of the flow inside the separation channel allowed the analysis of polymeric materials with ultrahigh molar mass,3 and even of gels,4 without shear degradation of the macromolecules. In the gas phase, thermodiffusion is well understood from the kinetic theory of gases. However, there has been up to now no satisfying molecular theory of thermodiffusion in liquids. Therefore, one can neither predict a priori whether a given binary liquid system exhibits a Soret effect nor which of the two substances concentrates in the cold region and which one in the hot region. One has to rely on experimental observations to answer these questions. The word thermodiffusion used to describe the phenomenon implies that diffusivity is acting to disperse, although nonuniformly, the two components in the temperature gradient. In fact, the diffusive force is counteracting the thermodiffusion force resulting from the applied temperature gradient which induces a displacement of the molecules of one component in one direction and those of the other in the opposite direction. Such a displacement induced by a temperature gradient does not only arise in single-phase systems, it can also appear in biphasic systems. The migration of aerosol particles suspended in a gas is well documented.5 The phenomenon is called thermophoresis. The first demonstration that colloidal particles suspended in a liquid exhibit a thermophoretic migration appears to have been made by McNab and Meisen.6 By means of an optical visualization method, they observed that 0.79-µm and 1.01-µm polystyrene (PS) latex particles suspended in water or in n-hexane are driven toward cold regions. The existence of thermophoresis (2) Schimpf, M. E. J. Liq. Chromatogr. Relat. Technol. 2002, 25, 2101-2134. (3) Gao, Y. S.; Caldwell, K. D.; Myers, M. N.; Giddings, J. C. Macromolecules 1985, 18, 1272-1277. (4) Lee, S. In Chromatography of Polymers. Characterization by SEC and FFF; Provder, T., Ed.; ACS Symposium 521, American Chemical Society: Washington, DC, 1993; pp 77-88. (5) Zheng, F. Adv. Colloid Interface Sci. 2002, 97, 255-278. (6) McNab, G.; Meisen, A. J. Colloid Interface Sci. 1973, 44, 339-346.

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of colloidal particles in liquids was later confirmed by various thermal FFF investigations, performed since slightly more than a decade ago, on different kinds of particles, such as polystyrene,7-21 modified polystyrene,12,13 polybutadiene,7,9,10,21 poly(methyl methacrylate),7,8,10,21 core-shell and copolymer10-12,15,21,22 latex particles, rubber particles,22 silica7-10,12,17,18,21 and modified silica10,12,21 particles, other organic or inorganic particles,7 and, even, metal particles.10,17,21 In some of these studies, the thermophoretic mobility, DT (also called the thermodiffusion coefficient or thermal diffusion coefficient), defined as the thermophoretic velocity per unit value of the temperature gradient, and the Soret coefficient, ST, equal to DT/D, where D is the mass diffusion coefficient, were determined from retention measurements. It was found that DT depends on the chemical nature of the particles, especially the chemical nature of their outer part, on the particle size, and on the nature and composition of the carrier liquid (ionic strength and pH). Besides these thermal FFF studies, thermophoresis was also observed for ferrofluids, which are magnetic particles with a size of about 10-30 nm that are suspended in liquids, and investigated by other instrumental methods, such as forced Rayleigh scattering,23-25 the Clusius-Dickel thermogravitational column,26-28 and a nonlinear optical method based on concentration rings.29 The values of ST and DT for polyorganosiloxane nanoparticles doped with light-absorbing gold were also determined by dynamic light scattering30 and forced Rayleigh scattering.31 A Soret-driven convective flow instability was found to take place in a cell (7) Liu, G.; Giddings, J. C. Anal. Chem. 1991, 63, 296-299. (8) Liu, G.; Giddings, J. C. Chromatographia 1992, 34, 483-492. (9) Shiundu, P. M.; Liu, G.; Giddings, J. C. Anal. Chem. 1995, 67, 2705-2713. (10) Shiundu, P. M.; Giddings, J. C. J. Chromatogr., A 1995, 715, 117-126. (11) Ratanathanawongs, S. K.; Shiundu, P. M.; Giddings, J. C. Colloids Surf., A 1995, 105, 243-250. (12) Jeon, S. J.; Schimpf, M. E.; Nyborg, A. Anal. Chem. 1997, 69, 3442-3450. (13) Jeon, S. J.; Schimpf, M. E. In Particle Size Distribution III; Provder, T., Ed.; ACS Symposium 693; American Chemical Society: Washington, DC, 1998; pp 182-195. (14) Mes, E. P. C.; Tijssen, R.; Kok, W. Th. J. Chromatogr., A 2001, 907, 201209. (15) Mes, E. P. C.; Kok, W. Th.; Tijssen, R. Chromatographia 2001, 53, 697703. (16) Janca, J. J. Liq. Chromatogr. Relat. Technol. 2002, 25, 2173-2191. (17) Shiundu, P. M.: Munguti, S. M.; Williams, S. K. R. J. Chromatogr., A 2003, 983, 163-176. (18) Shiundu, P. M.; Munguti, S. M.; Williams, S. K. R. J. Chromatogr., A 2003, 984, 67-79. (19) Jancˇa, J.; Berneron, J.-F.; Boutin, R. J. Colloid Interface Sci. 2003, 260, 317323. (20) Janca, J. J. Liq. Chromatogr. Relat. Technol. 2003, 26, 849-869. (21) Shiundu, P. M.; Williams, P. S.; Giddings, J. C. J. Colloid Interface Sci. 2003, 266, 366. (22) Shiundu, P. M.; Remsen, E. E.; Giddings, J. C. J. Appl. Polym. Sci. 1996, 60, 1695-1707. (23) Bacri, J. C.; Cebers, A.; Bourdon, A.; Demouchy, G.; Heegaard, B. M.; Perzynski, R. Phys. Rev. Lett. 1995, 74, 5032-5035. (24) Lenglet, J.; Bourdon, A.; Bacri, J. C.; Demouchy, G. Phys. Rev. E 2002, 65, 031408-1-14. (25) Alves, S.; Demouchy, G.; Bee, A.; Talbot, D.; Bourdon, A.; Figueiredo Neto, A. M. Philos. Mag. 2003, 83, 2059-2066. (26) Blums, E.; Mezulis, A.; Maiorov, M.; Kronhalns, G. J. Magn. Magn. Mater. 1997, 169, 220-228. (27) Blums, E.; Odenbach, S.; Mezulis, A.; Maiorov, M. Phys. Fluids 1998, 10, 2155-2163. (28) Vo ¨lker, T.; Odenbach, S. Phys. Fluids 2003, 15, 2198-2207. (29) Du, T.; Luo, W. Appl. Phys. Lett. 1998, 72, 272-275. (30) Schaertl, W.; Ross, C. Phys. Rev. E 1999, 60, 2020-2028. (31) Spill, R.; Ko¨hler, W.; Lindenblatt, G.; Schaertl, W. Phys. Rev. E 2000, 62, 8361-8368.

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containing a suspension of colloidal silica in water and heated from above.32 All these non-FFF studies were performed on submicrometer, colloidal particles, apart from the McNab and Meisen experiments with 1.01-µm PS latex. Still, these latter particles can well be considered as colloidal, i.e., Brownian, as their Pe´clet number, Pe, defined as the ratio of the time of diffusion of the particles over their diameter to that of convection by sedimentation over that distance, is much lower than 1 (Pe ∼ 0.06). There has been a long debate in the community of theoretical physicists about the existence or nonexistence of thermophoresis for non-Brownian particles in liquids. Answering this question by means of various modern methods of investigation of thermodiffusion and thermophoresis, such as forced Rayleigh scattering, the nonlinear optical method, the optical beam deflection method, or the thermogravitational column, has not yet been possible, apparently because of the too strong perturbations caused by the gravitational sedimentation of micrometer-sized particles. A few studies demonstrated that micrometer-sized polystyrene particles are retained in thermal FFF, giving apparent credence to the fact that they have a thermophoretic mobility.7-9,18 However, the retention mechanism of micrometer-sized particles in FFF is entirely different from that of submicrometer particles. Indeed, it is not simply controlled by the balance between field-induced and diffusive forces, as it is for submicrometer particles, but it is dependent on the magnitude of hydrodynamic lift forces which strongly increase with increasing particle size.33 These lift forces are dominating the FFF behavior of particles with size larger than about 1 µm. Accordingly, except for vertically oriented channels, the balance between the component of the gravitational force perpendicular to the flow axis with the hydrodynamic lift force can lead to the retention of particles, i.e., to their elution at a time larger than the void time. This mechanism is efficient and was used to separate and characterize, in a simple gravitational FFF channel, chromatographic particles34 and even PS particles for which the density difference with the aqueous carrier was much smaller.35,36 Therefore, from the sole observation of retention of micrometer-sized particles in a horizontal thermal FFF channel, one cannot conclude that they do exhibit a thermophoretic effect. To reach this goal, it must, at least, be demonstrated that the levels of retention of the particles in the horizontal thermal FFF channel with and without an applied temperature gradient are clearly different. In order to provide evidence of the thermophoresis of nonBrownian particles, or of its absence, and, eventually, to determine DT for these particles, we have run samples of 3-µm spherical chromatographic particles in a thermal FFF channel.37 The perturbing gravitational effect was eliminated by orienting the channel vertically and the temperature gradient horizontally. Part of the sample particles eluted as nonretained while the remaining (32) Cerbino, R.; Vailati, A.; Giglio, M. Philos. Mag. 2003, 83, 2023-2031. (33) Martin, M. In Advances in Chromatography; Brown, P. R., Grushka, E., Eds.; Marcel Dekker: New York, 1998; Vol. 39, pp 1-138. (34) Giddings, J. C.; Myers, M. N.; Caldwell, K. D.; Pav, J. W. J. Chromatogr. 1979, 185, 261-271. (35) Petersen, R. E., II; Myers, M. N.; Giddings, J. C. Sep. Sci. Technol. 1984, 19, 307-319. (36) Martin, M. In Particle Size Analysis 1985; Lloyd, P. J., Ed.; Wiley: New York, 1987; pp 65-85. (37) Regazzetti, A.; Hoyos, M.; Martin, M. J. Phys. Chem. B, in press.

part was well retained, which in itself indicates that there is a thermophoretic effect for these particles. Furthermore, it was observed that the relative amount of the retained particles increases with increasing durations of the period of interruption of the carrier flow following sample introduction in the channel, until a threshold duration is reached. Above this threshold value, the relative amount of the retained peak was approximately constant. The value of this threshold stop-flow duration was interpreted as the time required to transport the particles by thermophoresis across the whole channel thickness, which provided a method of determination of the thermophoretic mobility, DT. The consistency of the DT values obtained with different temperature gradients gave confidence in the correctness of this interpretation. This method was used for the DT determination of silica and octadecyl-bonded silica porous particles suspended in various pure liquids.37 If the evidence of thermophoresis for non-Brownian particles in liquids has been clearly established, the mechanism of retention of the particles in the thermal FFF channel is not well understood. In order to get some insight into this mechanism, the influence of operating parameters on the retention of the particles is reported in the present study. THEORY Flow Velocity Profile in the Thermal FFF Channel. In the laminar flow conditions in which FFF channels are operated, the flow velocity profile is generally parabolic. However, it is not so in thermal FFF for two reasons. First, because the viscosity of the carrier is not uniform across the channel thickness, due to its temperature dependence, the flow profile is distorted, and, at an equal distance from the walls, the velocity is lower near the cold wall than near the hot wall. This effect appears whatever the orientation of the channel. Second, when the channel is not horizontal, even in absence of an imposed flow rate, a unicellular flow arising from the temperature dependence of the carrier density takes place, the denser liquid near the cold wall moving down while the lighter liquid moves up near the hot wall. The overall velocity profile, u(x), where x is the distance from the accumulation wall, can be described as

u(s) ) 6[(1 + ν)s - (1 + 3ν)s2 + 2νs3] 〈u〉

(1)

where s is the reduced coordinate, x/w, 〈u〉, the mean flow velocity, and ν, a dimensionless parameter accounting for flow distortion from the Poiseuille parabolic profile. As shown in the appendix, ν can be expressed as38,39

(| |

)

Fγgw2∆T ν ) δacc(|νhor| + δfl|νvert|) ) δacc νhor + δfl 72η〈u〉

channel, +1 for vertical channel with upward flow, and -1 for vertical channel with downward flow. F and η are the mean values of the carrier density and viscosity in the channel, γ the carrier thermal expansion coefficient, g the gravitational acceleration, and ∆T the temperature difference between the hot and cold walls. |νhor| depends on ∆T, on the accumulation wall temperature, and on the nature of the carrier liquid. Empirical expressions are available for its determination.40 Note that polymeric as well as particulate species generally migrate toward cold regions,6,21,26,28-31,38,39,41,42 although opposite situations have been encountered.24,25,32 Taking into account the downward sedimentation of the chromatographic particles, the particle velocity profile, up(s), is given by (see eq A-5 in the Appendix)

up(s) 〈u〉

|US| ) 6[(1 + ν)s - (1 + 3ν)s2 + 2νs3] - δfl (3) 〈u〉

where US is the particle sedimentation velocity. Note that in this equation, it was assumed that US does not depend on the position in the channel and that the flow drags the particle at a velocity equal to the unperturbed flow velocity at the particle center, i.e., retardation effects due to the wall are assumed negligible. Thermophoretic Mobility and Thermophoretic Force. In a vertical thermal FFF, let tthres be the threshold value of the stop-flow duration required to transport all particles, of diameter d, across the channel thickness, w, by thermophoresis. Assuming, as a first-order approximation, that the temperature gradient is constant across the channel thickness and equal to ∆T/w, the thermophoretic mobility, DT, defined as the particle thermophoretic velocity per unit temperature gradient, is obtained as

DT )

(w - d)/tthres w(w - d) ) ∆T/w ∆Ttthres

(4)

Here, to express the transverse migration distance, the channel thickness is corrected by the diameter of the particles to account for their steric exclusion from the walls. With the use of the Stokes-Einstein relation for computing the mass diffusion coefficient, D, the Soret coefficient is derived as

ST ≡

DT 3πηdDT ) D kBT

(5)

(2)

δacc is the accumulation wall indicator, equal to +1 if the analyte accumulates at the hot wall and to -1 if it accumulates at the cold wall. δfl is the flow direction indicator, equal to 0 for horizontal (38) Giddings, J. C.; Martin, M.; Myers, M. N. Sep. Sci. Technol. 1979, 14, 611643. (39) Martin, M.; Min, B. R.; Moon, M. H. J. Chromatogr., A 1997, 788, 121130.

where η is the liquid viscosity, kB the Boltzmann constant, and T the average absolute temperature in the channel. The dimensionless thermodiffusion factor, RT, is equal to T ST. (40) Belgaied, J. E.; Hoyos, M.; Martin, M. J. Chromatogr., A 1994, 678, 8596. (41) Martin, M.; Reynaud, R. Anal. Chem. 1980, 52, 2293-2298. (42) Vastama¨ki, P.; Johannes, G.; Jussila, M.; Martin, M.; Riekkola, M.-L. Presented at the 11th International Symposium on Field-Flow Fractionation, Cleveland, OH, October 7-10, 2003; L-25.

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The thermophoretic force, FT, is derived from the thermophoretic mobility and the particle friction factor (3π η d) as

FL1, FL2, and FL3, respectively, on the relative position, s, of the particle center in the channel:

[

g1(s) ) 52.62(0.19 - s)(0.5 - s)(0.81 - s) 1 + ∆T FT ) 3πηdDT w

(6)

16 s(1 - s) 25 (8)

5 g2(s) ) -9.72s(0.5 - s)(1 - s) 1 - s(1 - s) 2

[

Hydrodynamic Lift Forces. Hydrodynamic forces are exerted on particles by the carrier flow. The main force is obviously the drag force which makes the particle move along the channel toward the outlet. However, forces perpendicular to the flow axis are also acting on particles along the x-direction. In principle, these lift forces can be calculated by solving the Navier-Stokes flow equation. However, this calculation represents a formidable fluid mechanical task which has been attempted only in some specific flow situations by making assumptions which limit the range of application of the resulting expressions. Anyway, it is easily shown, by means of the absurdity principle, that such forces can only arise from the inertial, nonlinear part of the Navier-Stokes equation. Indeed, if such a force were to exist in the Stokes condition, i.e., neglecting this inertial, nonlinear contribution, it would be linearly proportional to the flow rate. Hence, doubling the flow rate would lead to a doubling of the transverse hydrodynamic force. But also, multiplying the flow rate by -1 would lead to a transverse force also multiplied by -1. In other words, if the flow from channel inlet to outlet gives rise to a transverse force moving the particles away from the channel walls, reversing the flow direction, now from outlet to inlet, would lead to a reversal of the transverse force which would now push the particles toward the channel walls. This is obviously absurd on physical grounds, which proves that the assumption based on the linear Stokes equation is invalid. Hence, the hydrodynamic lift forces are arising from the nonlinear inertial term of the NavierStokes equation. The inertial lift force on a neutrally buoyant particle, FL1, has been computed assuming that the particle is relatively far away from the walls and that its size is much smaller than the channel thickness. This applies for both horizontal and vertical channels. Additional lift force contributions, FL2 and FL3, are found in vertical channels when the particles are buoyant and have a sedimentation velocity, US. The expressions of these forces have been reviewed and modified by Williams to account for the presence of two parallel walls in the FFF channel.43 The resulting expression of the inertial lift force, FLi, is given as the sum of these three forces: 43

[( )

FLi ) F〈u〉2d2

( )

( ) ]

|US| US 2 d 2 g1(s) + δflδsed g2(s) + g (s) w 〈u〉 〈u〉 3

(7)

g3(s) ) 0.884(0.5 - s)(1 + s - s2)2

]

]

(9) (10)

In a horizontal channel, g2(s) ) g3(s) ) 0. The lift force FL1 drives the particles away from the walls as well as from the channel center. The lift force FL2 drives the particles away from the walls if they sediment against the flow and toward the walls in the opposite case. The lift force FL3 drives the particles away from the walls whatever the flow direction. Attempts to estimate the lift forces from retention data in sedimentation FFF in various operating conditions (particle size, field strength, flow rate, carrier ionic strength, and viscosity) have led Williams et al. to propose an additional, empirical, lift force, which they called near-wall lift force, FLnw, given by44-47

η〈u〉d3 FLnw ) C 2 w (s - R)

(11)

with R ) d/(2w). However, depending on the procedure followed to fit experimental data and on the operating conditions, the value of the dimensionless coefficient C was found to change by nearly 3 orders of magnitude from 0.129 in the initial study44 to a few ten-thousandths in the latest one.47 It should be noted that the coefficient C in eq 11 is equal to 0.75 times the corresponding coefficient in publications by Williams et al. in which the nearwall lift force is expressed in terms of particle radius and shear rate at the accumulation wall. Equations 7 and 11, together with eq 13 below, indicate that the lift forces increase very fast with increasing particle sizes. This explains that they are dominating the retention behavior for large particles while they are usually negligible for smaller, submicrometer particles. Sedimentation Velocity. If Fp is the particle density, the force due to gravitational sedimentation exerted on a particle is expressed as

π FS ) d3(Fp - F)g 6

(12)

The sedimentation velocity, US, is given as

d2(Fp - F)g US ) 18η

(13)

δsed is the sedimentation direction indicator, equal to +1 or to -1 if particles sediment upward or downward, respectively. g1, g2, and g3 are dimensionless functions representing the dependences of

For porous siliceous particles, the apparent particle density, Fp, can be expressed in terms of the specific porous volume, Vg, the

(43) Williams, P. S. Sep. Sci. Technol. 1994, 29, 11-45.

(44) Williams, P. S.; Koch, T.; Giddings, J. C. Chem. Eng. Commun. 1992, 111, 121-147.

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density of silica, FSi and that of the suspending liquid, F, which gives

F FSi Fp - F ) 1 Vg + FSi 1-

Table 1. Thermophoretic Mobility, DT, Soret Coefficient, ST, and Thermodiffusion Ratio, rT, for Various Particle/Liquid Systems and Relative Error on the Determination of These Values at the 90% Probability Level (( δ %)a

(14)

EXPERIMENTAL SECTION The thermal FFF channel used in this study is the same as that used for the determination of DT.37 It has been described previously.48 It had a tip-to-tip length of 46 cm, a breadth of 1.8 cm, and a thickness of 0.1 mm. The channel walls were nickelplated. The temperatures of the hot and cold plates were selected by controlling the voltage of cartridge heaters in the hot plate and the flow of tap water through the cold plate, respectively. The temperature drop between the plates varied from 15 to 40 K, and the cold wall temperature, Tc, was 293 K. The carrier liquid was introduced by means of a syringe pump (Model A-99, Razel Scientific Instruments, Stamford, CT) at the upstream end of the channel. Sample injection was made by means of a six-port injection valve equipped with a 20-µL sample loop (Model 7125, Rheodyne, Cotati, CA). The effluent was fed to a flow-through UV photometer (Model 2138 Uvicord S, LKB, Bromma, Sweden) operating at a wavelength of 254 nm by means of a Teflon connection tube at the downstream end of the channel. The detector signal was recorded on a computer using a data acquisition board (Model DAS-801, Keithley, Taunton, MA). Two kinds of chromatographic porous particles were used in this study, Hypersil 3 µm, and Hypersil-ODS 3 µm, with a nominal pore size of 120 Å, a specific surface area of 170 m2 g-1, and a specific pore volume, Vg, of 0.69 cm3 g-1. They were kindly gifted by Thierry Domenger (Thermo-Electron, Courtabœuf, France). The Hypersil particles are silica particles for normal-phase LC, and the Hypersil-ODS particles are silica-based particles bonded with octadecylsilane groups for reversed-phase LC. Both types of particles were suspended in acetonitrile (SDS, Peypin, France). The sample concentration in the suspending liquid was generally 1% (volume fraction), a value found to be a good compromise between detector sensitivity and absence of overloading and aggregation effects. For the study of the influence of sample concentration on retention, the concentration was varied from 0.5% to 3%. No additive (salt or surfactant) was added to the carrier, which was degassed before use in thermal FFF to prevent bubble formation. We have previously observed that retention times in horizontally oriented channels were not highly reproducible.37 Accordingly, experiments described in this study were performed with the thermal FFF channel oriented vertically, which provided much more reproducible results (variations of about 1% in retention times). Unless otherwise specified, the carrier flow was stopped (45) Williams, P. S.; Lee, S.; Giddings, J. C. Chem. Eng. Commun. 1994, 130, 143-166. (46) Williams, P. S.; Moon, M. H.; Xu, Y.; Giddings, J. C. Chem. Eng. Sci. 1996, 19, 4477-4488. (47) Williams, P. S.; Moon, M. H.; Giddings, J. C. Colloids Surf., A 1996, 113, 215-228. (48) Van Batten, C.; Hoyos, M.; Martin, M. Chromatographia 1997, 45, 121126.

Si-OH/ water DT (m2 s-1 K-1) ST (K-1) RT (-) δ% a

2.2 × 10-11 130 4.0 × 104 14

Si-OH/ Si-ODS/ acetonitrile acetonitrile

Si-ODS/ n-heptane

1.6 × 10-11 37 1.1 × 104 14

5.7 × 10-12 16 4.7 × 103 9

5.2 × 10-12 12 3.7 × 103 14

DT values are reported from ref 37.

for a duration, tsf1, of 300 s just after sample introduction in the channel, while the temperature gradient was applied, to allow transverse thermophoretic migration of the particles. Whatever value of ∆T and kind of particles were used, this duration was found to largely exceed the threshold time required for the completion of this migration.37 The fractograms of the chromatographic particles most often exhibit two peaks, one “unretained” peak near the void time and one well-retained peak. These two peaks are sometimes not entirely separated, and the detector signal does not go back to the baseline at the valley between these two peaks. The retention time of the retained peak was computed as the first moment of the fractogram in the time interval from the elution time of the valley between the two peaks to the end of the retained peak. RESULTS AND DISCUSSION The Pe´clet numbers, Pe, for the 3-µm chromatographic particles used in this study, defined as US d/D, were equal to 46 when using water and to 56 when using acetonitrile or n-heptane, respectively, as suspending liquids. Because Pe increases as d4 and because the density difference between the particles and liquid is about 10 times larger for the silica-based porous particles than for the PS latex, these Pe values are about 3 orders of magnitude larger than those for the particles used by McNab and Meisen and are much larger than 1. The 3-µm Hypersil particles can therefore well be considered as non-Brownian particles. Thermophoretic Mobility. The DT values previously obtained for 3-µm silica and octadecyl-bonded silica particles suspended in acetonitrile, as well as for Si-OH particles in water and for Si-ODS particles in n-heptane,37 are reported in Table 1, together with the values of the Soret coefficient, ST, and of the thermodiffusion factor, RT. The values of the ST and of RT reported in Table 1 are much larger than any value ever reported in the literature. This is because the mass diffusion coefficient, D, of the 3-µm particles used in this study is much smaller than that of ordinary liquids and, even, of polymers. It is more interesting to compare the values of DT obtained for the chromatographic particles with those reported for other systems. For polymer solutions, DT was found nearly independent49 or only slightly dependent50 on the molar mass and to lie approximately in the (49) Schimpf, M. E.; Giddings, J. C. J. Polym. Sci., Part B: Polym. Phys. 1989, 27, 1317-1332. (50) Martin, M.; Van Batten, C.; Hoyos, M. In Thermal Nonequilibrium Phenomena in Fluid Mixtures; Ko ¨hler, W.; Wiegand, S., Eds.; Lecture Notes in Physics 574; Springer: Berlin, 2002; pp 250-284.

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range (4-16) × 10-12 m2 s-1 K-1,49 for various polymer-organic solvent systems. The few values reported for colloidal silica and modified silica particles are somewhat lower. They are not strictly comparable with the data in Table 1 because the suspending carriers were not pure liquids but ionic and/or surfactant solutions. Nevertheless, the values reported for 0.1-, 0.15-, and 0.25-µm silica particles suspended in 0.1% FL-70 aqueous solution with sodium azide vary little with particle size around 10-12 m2 s-1 K-1.7,8 Still, for the same system, a significant 3-fold change in DT from 0.29 to 0.98 × 10-12 m2 s-1 K-1 was recently reported for an increase in particle size from 0.25 to 0.3 µm.21 In sodium azide aqueous solutions, somewhat larger DT values have been obtained, for similar size Si-OH particles, increasing from about 2.7 to 4.0 × 10-12 m2 s-1 K-1 when the sodium azide concentration increases from 3 to 9 mM.12 In comparison, the value reported in Table 1 for the 3-µm Si-OH in pure water is about 1 order of magnitude larger. It is not clear whether this comes from the larger particle size or from the fact that we used pure water instead of an ionic solution. For Si-OH particles, we obtained a DT value about 30% smaller in acetonitrile than in water. Shiundu et al. observed that, in a 1.0 mM solution of tetrabutyammonium phosphate (TBAP) in acetonitrile, DT decreases from 1.4 to 0.6 × 10-12 m2 s-1 K-1 when the silica particle size increases from 0.05 to 0.25 µm.9 In a 10 times less concentrated solution, Jeon et al. observed a similar decrease of DT from about 4.7 to 3.4 × 10-12 m2 s-1 K-1 when d increases from 0.15 to 0.3 µm.12 More recently, Shiundu et al. reported, for the same 0.1 mM TBAP solution in acetonitrile, similar values decreasing from 5.0 to 4.84 to 3.69 × 10-12 m2 s-1 K-1 when d increases from 0.15 to 0.25 to 0.5 µm.21 Again our value for the 3-µm Si-OH particles in pure acetonitrile is several times larger than the values obtained for colloidal Si-OH particles. Only one DT value is reported for colloidal Si-ODS particles; it is equal to 2.06 × 10-12 m2 s-1 K-1 for 0.3-µm particles suspended in an aqueous solution of 0.1% FL-70 and 0.02% sodium azide.21 It is only about 2 times smaller than our value for the 3-µm SiODS particles in pure acetonitrile, but the carrier was not the same. However, the striking difference between the thermophoretic behavior of colloidal particles with that of non-Brownian particles arises from the fact that colloidal Si-ODS particles were more retained than Si-OH particles of the same size in the same aqueous carrier, and hence have a larger DT value than the latter,10,12,21 while for 3-µm particles in acetonitrile, we observed the opposite trend, i.e., that DT is larger for Si-OH than for SiODS particles. Again, one does not know whether this change in trend comes from the change of particle size, from the change of the suspending liquid, or from the fact that it is pure instead of being a solution. It must be noted that the method developed for the determination of DT for non-Brownian particles37 provides the absolute value of this parameter, not its sign. One does not know toward which of the two walls the particles are driven by thermophoresis. Velocity Profile. As discussed in the Theory, the flow distortion parameter, which expresses the deviation of the flow profile from the classical parabolic profile, is made of two parts, one, νhor, due to the temperature dependence of the carrier viscosity, the other, νvert, due to the thermogravitational effect in the vertical configuration. Their values depend on the nature of 5792 Analytical Chemistry, Vol. 76, No. 19, October 1, 2004

Figure 1. Analyte relative velocity, R, vs ∆T for Si-OH particles (lower curve) and Si-ODS particles (upper curve) suspended in acetonitrile. Tc ) 293 K. Upward flow. Q ) 0.5 mL/min. tsf1 ) 300 s.

the carrier, on ∆T, and, for νvert, on 〈u〉. In the operating conditions used in this study, νhor is found to be nearly proportional to ∆T. For ∆T ) 40 K and for samples accumulating at the cold wall, νhor is equal to about -0.12 for acetonitrile as the carrier liquid. νvert is inversely proportional to 〈u〉. Whatever ∆T, for a typical flow rate of 0.5 mL/min, νvert represents about 30% of νhor for acetonitrile. Therefore, when ∆T ) 40 K and for an upward flow rate of 0.5 mL/min, the retention time in the high-retention domain, i.e., when the particle cloud is located in the region where the velocity profile is almost linear, near the accumulation wall, is, for upward flow, 16% larger for acetonitrile than it would be if the flow profile was parabolic. Influence of the Temperature Difference on Retention. The relative analyte velocity (also called retention ratio), R, equal to the ratio of the void time to the retention time of the retained part of the Si-OH and Si-ODS particles suspended in acetonitrile, is plotted in Figure 1 as a function of ∆T, for an upward flow rate, Q, of 0.5 mL/min. It is seen that, for both kinds of particles, R is much smaller than 1, hence that they are well retained, which indicates that their migration in the channel occurs in close vicinity to one channel wall. The lift force FL3 being negligible in the experimental conditions of Figure 1 (see below) and in absence of a transverse gravitational force for a vertically oriented channel, this clearly demonstrates that these non-Brownian particles exhibit a thermophoretic effect, because the thermophoretic force is the only force which can drive them toward a plate. Increasing ∆T leads to an increase of the absolute value of the flow distortion parameter, ν. The effect of the increase of ∆T on ν should lead to an increase in R for particles accumulating at the hot wall and to a decrease in R for those accumulating at the cold wall. Clearly, if the particles are driven to the hot wall, this effect cannot explain the results of Figure 1. If they are driven to the cold wall, it can be shown that, in acetonitrile, the resulting decrease in R is slightly less than 11%. However, the decrease in R when increasing ∆T from 15 to 40 K observed in Figure 1 amounts to 25% for Si-ODS and 33% for Si-OH. This indicates that the change in R is not solely due to the change of the velocity profile and that the particles are more compressed near the accumulation wall when ∆T is larger. Again, this demonstrates that the thermophoretic force acts on particles, this force being proportional to ∆T, as shown in eq 6.

Figure 2. Analyte relative velocity, R, vs carrier liquid flow rate for (a) Si-OH particles and (b) Si-ODS particles in acetonitrile. From upper to lower curves: ∆T ) 15, 25, and 40 K. Tc ) 293 K. tsf1 ) 300 s.

Retention of Si-OH versus Si-ODS Particles. Figure 1 clearly shows that Si-OH particles are more retained than SiODS particles, when suspended in acetonitrile, hence that the thermophoretic mobility of the former is larger than that of the latter. This trend constitutes an independent confirmation of the results shown in Table 1. As noted above, this trend is opposite to that found for Brownian particles in aqueous suspensions. The higher retention of Si-OH particles implies that, on average, they migrate in closer vicinity to the accumulation wall than Si-ODS particles. It could be argued that, apart from the thermophoretic effect, this could arise from a stronger electrostatic repulsion of the latter particles, due to their higher hydrophobicity. However, the influence of the electrostatic interactions is most likely negligible for several reasons. First, the average gap between the particles and the wall is, in typical experiments, of the order of 3 to 4 µm, a distance quite larger than the range found for electrostatic interactions in water.33,51 Second, the dielectric constant of acetonitrile is less than half that of water, which implies that electrostatic interactions are weaker and decay faster with distance in acetonitrile than in water.33 Influence of Flow Rate on Retention. The variations of R for the retained peaks as a function of the upward carrier flow rate, Q, at different ∆T, are shown in parts a and b of Figure 2, for Si-OH and Si-ODS particles in acetonitrile, respectively. Whatever the particle type and the flow rate, it is seen that R decreases when ∆T increases. This is in agreement with the (51) Hansen, M. E.; Giddings, J. C. Anal. Chem. 1989, 61, 811-819.

observations of Figure 1 and reflects the fact that the stronger the thermophoretic force is, the larger the retention is. Furthermore, a comparison of parts a and b of Figure 2 shows that, whatever ∆T and Q, R is larger for Si-ODS particles than for Si-OH particles, which again confirms and generalizes the observations from Figure 1. Parts a and b of Figure 2 show that, whatever the type of particle and ∆T, R increases steadily with increasing carrier flow rate. This kind of behavior is typical of that observed for retention of micrometer-sized particles in sedimentation FFF, either gravitational or centrifugal,34,35,52-59 and implicitly in symmetrical flow FFF channels.60 This behavior has been interpreted as a result of the lift mechanism of retention in FFF. In this mechanism, particles having a low or negligible diffusivity are believed to focus near a transversal position where the hydrodynamic lift force driving particles away from the walls exactly balances the fieldinduced force. Such a focusing effect occurs because the lift force depends on transverse position, as seen in eqs 7-11, while the field force is constant or approximately constant across the channel. As, with increasing flow rates, or mean flow velocities, all lift forces increase, the transverse position at which this balance occurs is located at increasing distances from the accumulation wall. Accordingly, the carrier velocity at the focusing position increases faster than the flow rate, which can explain that the ratio, R, of mean particle migration velocity along the channel to the mean flow velocity increases with increasing flow rates. Following this interpretation, and using the appropriate value of the flow distortion parameter according to eq 2, one can estimate the focusing position, sfoc, from R, such that R ) up(sfoc)/ 〈u〉 in eq 3. Then, the values of the lift forces at this focusing position can be computed using eqs 7-11. For ∆T ) 25 K, and an upward flow rate of acetonitrile of 0.5 mL/min, one gets FLi values of 4.1 × 10-16 N and 3.8 × 10-16 N for Si-OH and SiODS particles, respectively, if they accumulate at the cold wall. If the accumulation occurs at the hot wall, the inertial lift forces are 4.3 × 10-16 N and 4.1 × 10-16 N, respectively. In any case, these values are much smaller than those of the thermophoretic forces computed from eq 6, equal to 3.8 × 10-14 N and 1.3 × 10-14 N, respectively.37 Therefore the inertial lift forces are too small to account for the retention of the particles and explain their relatively large R values despite the relatively large field-induced forces. The assumption made above of neglecting, in the particle axial velocity, the retardation arising from the presence of the nearby accumulation wall is justified by the fact that, in the selected (52) Caldwell, K. D.; Nguyen, T. T.; Myers, M. N.; Giddings, J. C. Sep. Sci. Technol. 1979, 14, 935-946. (53) Caldwell, K. D.; Cheng, Z.-Q.; Hradecky, P.; Giddings, J. C. Cell Biophys. 1984, 6, 233-251. (54) Martin, M.; Williams, P. S. In Theoretical Advancement in Chromatography and Related Separation Techniques; Dondi, F., Guiochon, G., Eds.; Kluwer Academic Publishers: Dordrecht, 1992; pp 513-580. (55) Merino-Dugay, A.; Cardot, Ph. J. P.; Czok, M.; Guernet, M.; Andreux, J. P. J. Chromatogr. 1992, 579, 73-83. (56) Urba´nkova, E.; Vacek, A.; Nova´kova´, N.; Matulı´k, F.; Chmelı´k, J. J. Chromatogr. 1992, 583, 27-34. (57) Pazourek, J.; Filip, P.; Matulı´k, F.; Chmelı´k, J. Sep. Sci. Technol. 1993, 28, 1859-1873. (58) Tong, X.; Caldwell, K. D. J. Chromatogr., B 1995, 674, 39-47. (59) Metreau, J. M.; Gallet, S.; Cardot, P. J. P.; Le Maire, V.; Dumas, F.; Hernvann, A.; Loric, S. Anal. Biochem. 1997, 251, 181-186. (60) Ratanathanawongs, S. K.; Giddings, J. C. Anal. Chem. 1992, 64, 6-15.

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Figure 3. Analyte relative velocity, R, of Si-ODS particles in acetonitrile vs flow rate, for upward flow (squares) and downward flow (diamonds). ∆T ) 25 K. Tc ) 293 K. tsf1 ) 300 s.

operating conditions, the focusing position is more than twice as large as the particle diameter.61 One can note that, in the present experimental conditions, the contribution of sedimentation to the particle axial velocity, which has been taken into account in the above calculation of the focusing positions, is quite small. Indeed, according to eqs 13 and 14, this contribution amounts to 0.76% and 0.63%, respectively, for Si-OH and Si-ODS particles. One could wonder if the empirical near-wall lift force, FLnw, may be strong enough to compensate the thermophoretic force. However, the wide variation found for the dimensionless coefficient C in eq 11 is disturbing and the empirical lift force does not appear to have any physical basis. Influence of Flow Direction. The variations of R versus flow rate for Si-ODS particles in acetonitrile and ∆T ) 25 K are reported in Figure 3 for both upward and downward carrier flow. One can imagine that two effects may affect the retention as one reverses the direction of the flow: first, a change in the focusing position, and second, a change in the particle axial velocity at the focusing position. A reversal of the flow direction does not affect the lift forces FL1 and FL3, according to eq 7, which may lead to a change in the focusing position. However, for Q ) 0.5 mL/min and ∆T ) 25 K, the lift force FL1 accounts for about 90% of FLi. FL2 represents about 10% of FLi while FL3 is negligible. It was seen above that FLi is too small to counterbalance the thermophoretic force; hence, one does not expect that a change, even by 20% of its value, will have a significant effect on the focusing position. As there is apparently no effect of change in flow direction on the analyte concentration profile, the difference in R associated with this change might come from a change in the particle axial velocity, because of the resulting changes in the contribution of the free convection to the flow profile and in the relative sedimentation velocity. This difference in R can be obtained from eq A-11 in the Appendix, in the high-retention limit. For particles which sediment downward, this gives

|US|

Rup - Rdown ) 12δacc|νvert|scg - 2

〈u〉

(15)

where Rup and Rdown are the R values for upward flow and 5794 Analytical Chemistry, Vol. 76, No. 19, October 1, 2004

Figure 4. Analyte relative velocity, R, of Si-ODS particles in acetonitrile vs sample concentration (particle volume fraction in percent). ∆T ) 25 K. Tc ) 293 K. Upward flow, Q ) 0.5 mL/min. tsf1 ) 300 s.

Figure 5. Superimposed fractograms for various sample concentrations of Si-ODS particles in acetonitrile. From lower to upper curves at 250 s, particle volume fraction: 0.5%, 1%, 1.5%, 2%, 3%. ∆T ) 25 K. Tc ) 293 K. Upward flow, Q ) 0.5 mL/min. tsf1 ) 300 s.

downward flow, respectively. In principle, measuring Rup - Rdown + 2|US|/〈u〉 should allow one to determine the accumulation wall, a negative value indicating that particles accumulate at the cold wall. In the experimental conditions, at a flow rate of 0.5 mL/ min, and assuming that scg corresponds to sfoc, the first term in the RHS member of eq 15 dominates the second one. According to eq 15, the difference in R should be equal -0.015 and -0.017 if the Si-OH and Si-ODS particles, respectively, accumulate at the cold wall, and to 0.006 and 0.008 if they accumulate at the hot wall. These differences are small and of the order of magnitude of the experimental uncertainties. Accordingly, no meaningful conclusion can be drawn about the direction of thermophoresis from Figure 3. However, this figure supports the conclusion that the transverse particle distribution is not significantly affected by the direction of flow. Influence of Sample Concentration. The variations of R for Si-ODS particles in acetonitrile for an upward flow rate of 0.5 mL/min at ∆T ) 25 K versus sample concentration are plotted in Figure 4 and the corresponding fractograms are shown in Figure 5. In Figure 4, R is seen to increase with increasing particle concentration in the sample. The fractograms in Figure 5 illustrate the shift of the peak maximum toward lower retention times as the sample concentration increases. Such a behavior has been experimentally observed,62 and theoretically predicted,63 in the

case of colloidal particles. However, these latter particles were eluted in the Brownian retention mode, for which particles, in diluted suspensions, are exponentially, or nearly exponentially, distributed near the accumulation wall as a result of the balance between the field-induced and diffusive forces. This is understood essentially by the fact that an increase of the sample concentration leads to a displacement of the center-of-gravity of the particle distribution toward larger distances from the accumulation wall, in flow streamlines with a larger velocity. However, the theoretical description of concentration effects for colloidal particles does not apply to micrometer-sized particles for which the retention mechanism is entirely different and dominated by hydrodynamic lift effects. An increase in R with increasing concentrations of 5-µm porous silica particles was also observed in gravitational FFF and attributed to the variation of the gradient of the lift force FL1 near the focusing position.64 However, this cannot explain the concentration effects shown in Figures 4 and 5 because, as discussed above, FL1 is not significant in our experiments. A mechanism has to be found to account for the variations for the influence of concentration on retention shown in Figure 4. This influence is quite strong if one considers that the retained fraction of the sample injected in a 20-µl sample loop is eluted in a time interval of about 200-300 s (see Figure 5) at a flow rate of 0.5 mL/min, i.e., in an elution volume interval of 1.7-2.5 mL. It is therefore diluted, on average, by a factor of the order of 100. Hence, with R around 0.33, in average along the channel, the cross-sectional concentration is around 30 times more diluted than in the sample loop. Still, the particles might be concentrated in a small layer within the channel thickness, the more so as they have a very low diffusivity. Hence their concentration in this layer can be important, especially near the channel inlet where axial dispersion has not significantly occurred. Effect of the Reduction of the Temperature Gradient during a Run. In order to get further insight into the particle behavior in the thermal FFF channel, the following experiments were conducted: the classical stop-flow procedure was applied immediately after sample introduction in the channel under the steady temperature gradient during a period, tsf1, of 300 s, then, at this time, the heating of the hot plate was stopped and the stopflow procedure was pursued during an additional period, tsf2, before resuming the flow. The mean elution time of the unretained and retained Si-ODS particle peaks in acetonitrile are plotted as a function of tsf2 in Figure 6, for an upward flow rate of 0.5 mL/min and ∆T ) 25 K during the period of duration tsf1, and the corresponding fractograms are shown in Figure 7. The retention time obtained for the retained peak when the temperature gradient was applied during the whole sojourn of the particles in the channel is indicated by a large × on the y-axis of Figure 6. In the selected experimental conditions, this corresponds to the R value of 0.27 noted in Figures 1-4. Due to the thermal inertia of the thermal FFF channel, ∆T does not drop to zero as soon as the heating of the plate is stopped but decays progressively, as shown in Figure 8. This corresponds essentially to a (61) Chaoui, M.; Feuillebois, F. Q. J. Mech. Appl. Math. 2003, 56, 381-410. (62) Hansen, M. E.; Giddings, J. C.; Beckett, R. J. Colloid Interface Sci. 1989, 132, 300-312. (63) Hoyos, M.; Martin, M. Anal. Chem. 1994, 66, 1718-1730. (64) Pazourek, J.; Chmelı´k, J. J. Chromatogr., A 1995, 715, 259-265.

Figure 6. Mean elution times of the unretained peak (squares) and of the retained peak (diamonds) for Si-ODS particles in acetonitrile vs duration, tsf2, of the period of flow interruption after stopping the heating of the hot plate, which follows the initial period of duration, tsf1, of stop-flow under application of the temperature drop ∆T ) 25 K. Tc ) 293 K when ∆T ) 25 K and decreases to 287 K after 300 s of stopping the heating of the hot plate. Upward flow, Q ) 0.5 mL/ min. tsf1 ) 300 s. The “×” on the ordinate axis indicates the retention time obtained when ∆T is maintained during the whole migration of the sample in the channel.

Figure 7. Fractograms corresponding to Figure 6. At a time of 220 s, the lowest curve corresponds to the fractogram obtained when ∆T is maintained during the whole migration of the sample in the channel. Other curves from lower to upper, at a time of 220 s: tsf2 ) 300, 100, 50, and 0 s.

decrease of the hot wall temperature. But the cold wall temperature decreases also slightly, from 293 to 287 K after 300 s. It is seen in Figure 6 that the mean elution time of the unretained peak does not significantly change with tsf2. However, the retention time of the retained peak decreases with increasing values of tsf2 until it fuses with the unretained peak. When tsf2 equals 0, particles start their downstream migration under ∆T ) 25 K. As time elapses, ∆T is decreasing. Hence the particles migrate in fieldprogramming conditions, which explains that their retention time is shorter than when a constant ∆T is maintained during their migration. When tsf2 increases, the effective ∆T at the beginning of the downstream migration decreases as seen from Figure 8; hence, their migration along the channel occurs under a lower length- or time-average ∆T. Their retention time is then seen to decrease. This is clearly seen in the fractograms of Figure 7. When ∆T is maintained during the whole run, the unretained and retained peaks are well separated. Their resolution decreases as tsf2 increases. Analytical Chemistry, Vol. 76, No. 19, October 1, 2004

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Figure 8. Temperature difference between the hot and cold plates, ∆T, vs time after stopping the heating of the hot plate. ∆T at time 0 ) 25 K. Tc: 293 K at time 0; 287 K at time 300 s.

It can be considered that, at the end of the initial stop-flow period tsf1, all particles have been driven to contact with the accumulation plate by thermophoresis since tsf1 is larger than the time tthres required to complete this process. The fact that, on increasing tsf2, their retention time decreases indicates that their average migration velocity along the channel increases, hence that their average distance to the accumulation wall during their migration increases. The accumulation process at the wall is thus observed to be reversible, which excludes the hypothesis that some kind of chromatographic mechanism related to particlewall interactions is controlling the retention. As the particles are in contact with the accumulation wall, or very close to this wall, when they start their migration, a lift mechanism is required to make them moving away from the wall in order to elute at a moderate retention time. The corresponding lift force is counteracted by the thermophoretic force, which is proportional to ∆T. Hence the larger ∆T is at the beginning of the downstream migration, the stronger this counteraction is, and the larger the retention time is. It is noticeable that the time scale over which ∆T drops to zero after stopping the heating of the hot plate is quite similar to the time scale over which the mean elution time of the retained peak decreases to reach that of the unretained peak. Again, this provides a strong evidence that the thermophoretic effect exists for non-Brownian particles and that, together with a lift mechanism, it is controlling the behavior of these particles in thermal FFF. New Lift Mechanism in FFF. It was noted above that the expressions of the hydrodynamic inertial lift forces given by eqs 7-10 give values of the overall inertial lift force, FLi, which are too small to counteract the relatively strong thermophoretic force acting on particles at the mean distance from the accumulation wall at which they migrate along the channel. A similar observation in the case of sedimentation FFF of micrometer-sized particles has led Williams et al.44 to suggest the existence of an additional lift force, called near-wall lift force. We have noted above that questions are raised about the consistency of the expression of this force, which is empirical and lacks a physical background. The experiments performed at various sample concentrations, showing that R increases with the concentration, lead us to believe that another lift mechanism, depending on concentration, is acting on particles. Such a phenomenon, called viscous resuspension, was observed in the process of resuspension of a settled layer of 5796 Analytical Chemistry, Vol. 76, No. 19, October 1, 2004

particles when a shear flow was applied.65 The equilibrium distribution of the particles results from the balance between the flux of particles toward the wall due to an external field (usually gravity) and a flux away from the wall arising from shear-induced diffusion of the particles. This diffusive flux is not connected to the ordinary mass diffusion or Brownian motion but has a hydrodynamic origin. Various expressions of the shear-induced diffusion coefficient have been proposed. They show that this coefficient increases with increasing particle size, increasing shear rate (hence increasing flow rate), and increasing particle concentration.66 The shear-induced diffusion mechanism was considered by Williams et al.44 They, however, dismissed it as a possible lift mechanism explaining retention of micrometer-sized particles in sedimentation FFF for two reasons. First, they did not observe an influence of concentration on retention. Second, the narrowness of the peaks suggested that the particle clouds were confined in a very thin layer between the plane walls and eluted as a coherent set. In our experimental conditions, however, we observe a clear concentration effect on retention, as seen in Figure 4, and the peaks are rather broad, as seen in Figure 5. This leads us to reconsider the possible role of shear-induced diffusion in FFF. Such a process can explain that, in thermal FFF of non-Brownian particles, the mean distance of the particle cloud from the accumulation increases, and thus that R increases, when the shearinduced diffusion coefficient increases, hence when increasing the flow rate, as seen in Figure 2, and increasing the sample concentration, as seen in Figure 4. A further discussion about this coefficient and a quantitative comparison of its effect with our experimental data is behind the scope of the present study. CONCLUSION The retention experiments reported in this study confirm clearly our previous finding37 that the 3-µm Si-OH and Si-ODS chromatographic particles, which do not have a Brownian behavior, exhibit a thermophoretic effect when suspended in a liquid, here acetonitrile. Although the previous and present studies were both performed with the same thermal FFF instrument, their positive conclusions about the existence of this effect are based on independent processes. The previous one relies on the influence of the duration of the stop-flow period tsf1 on the extent of the thermophoretic migration during this period. This extent affects the relative areas of the unretained and retained particle peaks. It does not rely on the mechanism of migration of these peaks. Instead, the evidence of thermophoresis provided in the present study is based on the influence of the temperature gradient on the retention and thus on the migration process of the retained peak in the thermal FFF channel. This provides an answer to a question which has been vividly debated in the community of theoretical physicists. It is seen that the thermophoretic force depends on the chemical nature of the outer part of the particles, as this was observed for colloidal particles as well as for block copolymers. Still, the order of retention that we observed between Si-OH and Si-ODS micrometer-sized particles is opposite to that found for Si-OH and Si-ODS submicrometer particles. Further (65) Leighton, D.; Acrivos, A. Chem. Eng. Sci. 1986, 41, 1377-1384. (66) Leighton, D.; Acrivos, A. J. Fluid Mech. 1987, 177, 109-131.

work is needed to elucidate whether the origin of the difference lies in the particle size or in the nature of the suspending liquid. Anyway, this shows that the thermophoretic mobility is dependent on the chemical nature of the outer part of silica-based particles and that thermal FFF is potentially a method of characterization of the surface properties of such particles. It should be noted that the demonstration of the thermophoretic effect for the non-Brownian particles and the determination of their thermophoretic mobilities have been made feasible because of the possibility of setting the thermal FFF channel vertically. Running the channel in the horizontal configuration led to a failure to reach these objectives. We have found that the values of the lift forces given by models invoked to account for FFF retention of micrometer-sized particles are too small to counteract the rather intense thermophoretic force. This leads us to postulate that another lift mechanism, shearinduced hydrodynamic diffusion, is playing a significant role on the migration behavior of the chromatographic particles in the channel. This mechanism is most likely acting not only in the present thermal FFF experiments but also in most experiments performed with micrometer-sized particles with other FFF methods (gravitational FFF, sedimentation FFF, flow FFF, magnetic FFF, acoustic FFF, ...). In fact, the expressions of the inertial lift forces given in eqs 7-10 have been obtained by integration of the stress exerted on a particle surface by the flow, the flow field being obtained from the Navier-Stokes equation. In these calculations, a single particle was considered. Hence the resulting lift force equations correspond, in their domain of validity, to infinitely diluted suspensions. In practice, in FFF, even if the particle concentration in the sample is low, the concentration in the vicinity of the accumulation wall may be too high to allow neglecting the hydrodynamic interactions between particles, which are long-range interactions, especially in high field strength conditions. It might be for this reason that the inertial lift forces on individual particles are too small to account for the lift effect observed, and that an additional, empirical lift force, called nearwall lift force had to be invoked to make up for the missing force. The fact that the influence of particle concentration on this nearwall lift force has not been considered may explain, at least in part, the large variations of the dimensionless coefficient C of eq 11 obtained from different sets of experimental data. In fact, it is likely that, in most FFF experiments on micrometer-sized particles, the particle concentration near the accumulation wall is sufficiently large that the effects of inertial lift forces and of shear-induced hydrodynamic diffusion, both of which are strongly dependent on particle size and on flow rate, are simultaneously influencing the particle migration. Further theoretical as well as experimental investigations are needed to unravel the respective influences of these two effects in FFF and to better characterize each of them. Because of its great sensitivity to phenomena occurring in close vicinity to a wall, FFF can be an instrumental tool well suited to the experimental investigations of these hydrodynamic effects. In the FFF literature, the retention of micrometer-sized particles in FFF is described as occurring according to either the steric mode or the lift-hyperlayer mode. However, the mechanism is the same in both cases: particles are thought as migrating at or toward a transverse focusing position, sfoc, where the field-

induced force is exactly counteracted by the hydrodynamic lift force.67 The historical development of FFF theory has led to an arbitrary distinction between the two modes: steric mode when sfoc - R < 1 and lift-hyperlayer mode when sfoc - R > 1.67 The physical representation of the particle behavior according to this mechanism is deterministic: each particle follows a trajectory entirely determined by the dependence of the lift force and the axial drag force on the transverse position. Accordingly, the distribution of the residence time of the particles in the channel is determined by the one-dimensional distribution of their positions across the channel thickness at the beginning of their migration. When the shear-induced hydrodynamic diffusion mechanism is playing a significant role in the particle migration, the particle behavior is, in principle, also deterministic. But the hydrodynamic interaction process between particles is so complex that the particle trajectories strongly depend on the initial distribution of the particles at the channel inlet, which is almost impossible to reproduce from one run to another. Hence, the resulting particle behavior is somewhat similar to that observed in the presence of a randomizing diffusive process. Such a process is described by an effective diffusivity. Still, in the case of the shear-induced diffusion process, this diffusivity depends on local values of shear rate and concentration. In FFF, the values of these parameters depend on the distance from accumulation wall as well as, at least for concentration, on the position along the channel. A full characterization of shear-induced diffusion coefficient in these conditions is complex and has not yet been attempted. The FFF method, and the related SPLITT fractionation method, might reveal themselves as adequate instrumental tools for the investigation of this hydrodynamic process. ACKNOWLEDGMENT Antoine-Michel Siouffi (University of Aix-Marseille 3, France) is gratefully acknowledged for his strong scientific support in this work and for allowing one of us (A.R.) to perform experiments at ESPCI, Paris. We are indebted to Thierry Domenger (ThermoElectron, Courtabœuf, France) for his generous donation of the Hypersil samples. APPENDIX Flow Velocity Profile and Particle Velocity Profile in a Vertical Thermal FFF Channel. Because of the viscosity dependence on temperature, the carrier flow velocity profile in a thermal FFF channel is not parabolic but slightly distorted. In a horizontal channel, this profile, u(x), can be approximated by a third-degree polynomial in s33,38

u(s) ) 6[(1 + νhor)s - (1 + 3νhor)s2 + 2νhors3] (A-1) 〈u〉 with s ) x/w, x is the distance from the accumulation wall and νhor is an adjustable parameter accounting for the distortion of the flow profile. If the profile was expressed in terms of the relative (67) Giddings, J. C. In Field-Flow Fractionation Handbook; Schimpf, M., Caldwell, K., Giddings, J. C., Eds.; Wiley-Interscience: New York, 2000; pp 3-30.

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distance from the depletion wall, r ) (w - x)/w ) 1 - s, one would get, by replacing s by 1 - r in eq A-1

u(r) ) 6[(1 - νhor)r - (1 - 3νhor)r2 - 2νhorr3] (A-2) 〈u〉

Retention in the High-Retention Limit. Whatever the analyte concentration profile, c(s), in the channel thickness, and whatever the mechanisms giving rise to this concentration profile, the analyte relative velocity, R, is given by33

R) This expression is identical to that of eq A-1, except that νhor has been replaced by -νhor. In practice, because the flow profile in the FFF channel is not monotonic, one cannot know at which wall the analyte accumulates from the measurement of its retention time. When the analyte accumulates at the cold wall, νhor is negative as the flow velocity is reduced near the cold wall because of the increased viscosity in this region. νhor is positive if accumulation occurs at the hot wall. Hence, whatever the accumulation wall, one can write

νhor ) δacc|νhor|

(A-3)

where δacc is an accumulation wall indicator, equal to -1 if this wall is the cold one and to +1 if it is the hot wall. If the channel is oriented vertically, a free convection profile develops due to the horizontal gradient of the carrier density. This gives a downward flow near the cold wall and an upward flow near the hot wall. Assuming that this gradient is constant, this free convection velocity profile is also expressed as a third degree polynomial in s with zero mean velocity. Its effect on the resulting carrier flow velocity profile in a vertical thermal FFF channel depends on the direction of the forced carrier flow. Hence, this resulting profile, which is the sum of the distorted profile expressed by eq A-1 in the horizontal configuration and of the free convection profile, can be expressed as39

u(s) ) 6[(1 + ν)s - (1 + 3ν)s2 + 2νs3] 〈u〉

ν ) δacc(|νhor| + δfl|νvert|)

(A-4)

where δfl is a flow direction indicator, equal to +1 for upward forced flow and to -1 for downward flow. When the particles are buoyant, they sediment along the direction of the flow. If Fp > F, they sediment downward, while the opposite is true if Fp < F. Assuming that the sedimentation velocity US does not depend on the particle position in the channel, one can write the particle velocity profile, up(s), as

up(s) 〈u〉

)

|US| u(s) + δflδsed ) 6[(1 + ν)s - (1 + 3ν)s2 + 〈u〉 〈u〉 |US| 2νs3] + δflδsed (A-5) 〈u〉

where δsed is a sedimentation direction indicator, equal to -1 for Fp > F and to +1 for Fp < F, and where ν is given by eq A-4. 5798

Analytical Chemistry, Vol. 76, No. 19, October 1, 2004

up(s)

1c(s)

〈c〉 〈u〉

0

ds

(A-6)

where 〈c〉 is the cross-sectional average concentration, defined as

〈c〉 ≡

∫ c(s) ds 1

(A-7)

0

In the high-retention limit, when the analyte is concentrated near the accumulation wall, the flow velocity profile can be considered as linear in the region where c differs from 0. Then the particle velocity profile is given by eq A-5 in which the terms in s2 and s3 are set to 0. In these conditions R becomes

R ) 6(1 + ν)

1c(s)

|US|

〈c〉

〈u〉

∫

0

s ds + δflδsed

(A-8)

The integral represents the mean distance of the particle distribution from the accumulation wall, scg ) xcg/w, where the subscript cg stands for center-of-gravity. Hence R becomes in the highretention limit

|US| R ) 6(1 + ν)scg + δflδsed 〈u〉

(1)

with

∫

(A-9)

Let Rup, Rdown, scg,up, and scg,down be the values of R and scg for upward flow and downward flow, respectively. With the use of eqs A-4 and A-9, the difference in R when reversing the flow direction from downward to upward is given by

Rup - Rdown ) 6(1 + δacc|νhor|)(scg,up - scg,down) + |US| 6δacc|νvert|(scg,up + scg,down) + 2δsed (A-10) 〈u〉

If the position of the center-of-gravity is not affected by the flow direction, i.e., scg,up ) scg,down ) scg, then this difference becomes

Rup - Rdown ) 12δacc|νvert|scg + 2δsed

|US| 〈u〉

(A-11)

Received for review January 15, 2004. Accepted July 5, 2004. AC040012T