Effects of dissolved polymer on the transport of colloidal particles through a microcapillary

Journal of Colloid and Interface Science 311 (2007) 77–88 www.elsevier.com/locate/jcis

Effects of dissolved polymer on the transport of colloidal particles through a microcapillary S. Amnuaypanich a,b , M.S. El-Aasser b , E.S. Daniels b , C.A. Silebi b,∗ a Department of Chemistry, KhonKaen University, KhonKaen 40002, Thailand b Emulsion Polymers Institute and Department of Chemical Engineering, Lehigh University, Bethlehem, PA 18015, USA

Received 14 September 2006; accepted 25 February 2007 Available online 2 March 2007

Abstract The effect of water-soluble polymer on the transport of latex particles through a microcapillary was investigated. Capillary hydrodynamic fractionation (CHDF) experiments were performed using polystyrene (PS) particles and poly(ethylene oxide) (PEO) solutions as the eluant. Generally, the average particle velocities were greater than those corresponding to a polymer-free eluant. A decrease in the sample axial dispersion was also observed using the PEO solutions. In addition, increasing the polymer molecular weight resulted in lower particle residence times in the capillary tube. The enhanced particle transport arises primarily from an increase in the particle diameter resulting from the adsorption of PEO onto the PS surfaces, and, more importantly, from the migration of particles toward the capillary axis due to the normal stress of the PEO solution. © 2007 Elsevier Inc. All rights reserved. Keywords: Capillary hydrodynamic fractionation; Poly(ethylene oxide); Polystyrene latex particle; Separation factor; Axial dispersion; Particle migration; Normal stress

1. Introduction Under laminar flow conditions for a fluid moving through a microcapillary, the fluid develops a parabolic velocity profile whereby the local fluid velocity attains a maximum value at the center line of the capillary and decreases toward the capillary wall. Therefore, colloidal particles dispersed in this moving fluid would experience different velocities within the microcapillary according to their radial positions. Transport of colloidal particles through a microcapillary can be described as a convective-diffusion process [1]. The fluid velocity distribution inside the capillary causes a convective effect that distorts the particle slug when it enters the flow field. This elongated shape induces a concentration gradient, which causes the diffusion of the particles primarily in the radial direction. For elution times that are long compared to the time needed for the particles to sample all of the radial positions in the Poiseuille flow; i.e., when the condition D∞ t/R02  1/λsm is satisfied, where D∞ is the particle diffusivity, t is the elution time, λsm is the * Corresponding author.

E-mail address: [email protected] (C.A. Silebi). 0021-9797/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2007.02.068

smallest eigenvalue of the diffusion equation and R0 is the tube radius, the convective effect will be balanced by particle diffusion resulting in a fully-developed radial concentration profile of the particle within the microcapillary [2,3]. For dilute particle suspensions, interactions between particles are weak, and several types of particle–wall interactions become important. Due to a particle’s finite size, its center of gravity is unable to approach the tubular boundary more closely than a distance equal to the particle radius. This inaccessible volume increases with increasing particle size; hence, larger particles will tend to sample faster streamlines and elute from the capillary sooner than smaller ones. Another effect is the inertial force of the fluid which takes place at relatively high Reynolds numbers of the moving fluid [4,5]. The experimental works by Segre and Silberberg [4,5] show that the inertial force causes the particles to migrate across the streamlines to attain an equilibrium non-central radial position roughly at a distance of 0.6 of the tube radius. Theoretically, the inertial effect depends on the eluant average velocity, particle size and capillary inner diameter [6]. More complex effects include colloidal interactions such as the van der Waal’s attractive forces as well as colloidal re-

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pulsive forces. Generally, for colloidal particles moving in a Newtonian fluid through the capillary, two primary repulsive forces dominate, i.e., electrostatic repulsions and electrokinetic lift. Electrostatic repulsions arise from the overlap of the electrical double layers (EDL) surrounding the colloidal particles as well as the capillary wall. The thickness of this layer is indicated by the Debye length (1/κD ). The presence of electrolyte ions in the fluid phase has a strong influence on the interplay between the electrostatic repulsion and the van der Waal’s attraction. As the ionic strength of the fluid is raised, the EDL’s surrounding the particles and the capillary wall are compressed and van der Waal’s attractions dominate. Consequently, the particles will have a greater probability to sample slower velocity streamlines near the wall resulting in an increase of particle axial dispersion and a decrease in the average particle velocity. In contrast, under low ionic strength conditions, the particles move more rapidly on average resulting from enhanced electrostatic repulsions compared to the van der Waal’s attractions [7,8]. The electrokinetic lift force occurs when a charged surface on the capillary wall and the diffuse part of the electrical double layer surrounding the particles are made to move relative to each other. When a small electrically-charged sphere translates through a polar fluid in close proximity and parallel to a charged surface, the convection of charge within the diffuse layer of the electrical double layer surrounding the particle and wall surfaces induces a streaming potential profile between the surfaces. The streaming potential imposes an electrical stress on the charged interface and integration of the normal component of the Maxwell stress yields the electrokinetic lift [9,10]. Unlike the electrostatic repulsion, the electrokinetic lift force is less well described theoretically. All extant theories are based on the thin double layer approximation which strictly speaking does not apply to the CHDF experiments. Nonetheless, experimental results by Hollingsworth and Silebi [11] showed qualitative agreement with the low Peclet number theory [12,13] and are believed to be of an electrokinetic origin. Generally, this lift force predominates only under certain conditions in CHDF experiments, i.e., low ionic strength conditions and high eluant flow rate. The presence of a high molecular weight polymer dissolved in the eluant can significantly affect particle transport through the microcapillary. The polymer molecules could absorb on the particle surfaces and give rise to an increase in the effective particle diameter depending upon the chain length of the polymer as well as the solvency of the medium. The thickness of adsorbed polymer was found to be twice as large as the radius of gyration of the polymer molecule in solution [14,15] for adsorption of poly(ethylene oxide) on polystyrene latex particles. More importantly, the viscoelastic property of polymer solutions can play a major role in inducing the radial migration of particles when moving through the microcapillary under Poiseuille flow. A polymer solution is known to possess nonzero normal stresses and it is believed to be responsible for the migration of particles toward the region of lower velocity gradient in low Reynolds number Couette and Poiseuille flows [16– 19]. In plane-Poiseuille flow, Jefri and Zahed [20] investigated particle migration in three different types of fluids: Newtonian

fluid, constant-viscosity viscoelastic fluid, and shear-thinning viscoelastic fluid. They observed that no migration took place for particles in Newtonian fluid, whereas migration toward the upper and lower plates was seen for the shear-thinning fluid. For particles present in a viscoelastic fluid having a constant viscosity, the particles were found to migrate toward the centerline of the capillary. The study of particle migration in a cone-andplate viscometer by Highgate and Whorlow [21,22] showed that rigid spheres present in a viscoelastic fluid moved radially toward the outer edge of the viscometer. Recently, Ponche and Dupuis [23] studied the migration of glass spheres in a solution comprised of polyisobutylene dissolved in decalin under shear flows at constant stress in a cone and plate geometry. They observed a depletion region in the central zone of the viscometer surrounded by concentric rings containing most of the glass spheres. The objective of the current work was to demonstrate the pronounced effect of dissolved polymer on particle transport through a microcapillary. Specifically, we report the enhanced particle separation efficiency and reduced axial dispersion in the CHDF process. The paper is organized in two parts. First, we present experimental results demonstrating the effect of polymer adsorbed on the latex particles. Second, PEO-free particles are used to probe radial particle migration due to the normal stress of the polymer solutions. 2. Experimental: materials and methods 2.1. Capillary hydrodynamic fractionation (CHDF) Capillary hydrodynamic fractionation is an analytical technique for the characterization of particle size and particle size distribution of submicron colloidal dispersions [7]. The separation of colloidal particles according to size is made possible by utilizing the parabolic velocity of the fluid flow field through a small open-bore capillary tube. Fig. 1 displays the schematic diagram of the custom-built CHDF instrument used in the experiments. Eluant is delivered into the system using a positive displacement pump (Milton Roy Minipump Model 396) equipped with a pulse dampener. The eluant is pumped through a Rheodyne sample injection valve (Model 8125), which is used to introduce colloidal dispersions into the microcapillary. Sample volumes injected into the CHDF are standardized using a 20 µL sample loop. The eluant flow from the pump is divided into two streams at a T-joint before entering the capillary; one stream continues through the microcapillary, while the other is discarded. The split flow is required to allow the use of a regular HPLC pump to operate at a relatively high flow rate which will minimize dead-volume sample-mixing issues at the injection valve, in addition, the sample volume introduced into the smallID capillary will be reduced. The fused-silica microcapillary (Polymicro Technologies Inc.), which has an approximate internal diameter of 25 µm, is connected one end to the T-connector while the other end is plugged into the 15 µL detector cell of a multi-wavelength UV detector. The actual average internal diameter of the microcapillary is calculated using the Hagen–

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Fig. 1. Schematic diagram of experimental capillary hydrodynamic fractionation (CHDF).

Poiseuille law for Newtonian fluid flowing through a cylindrical tube under laminar flow conditions [24]. The flow-through UV detector is used to determine the particle elution times as well as relative particle concentrations by measuring the optical densities of the emerging species. The detector output is recorded every 0.46 s at four different UV wavelengths simultaneously and interfaced to a computer for data acquisition and analysis. A constant flow of deionized water (make-up stream) is passed through the detector cell and is combined with the small flowstream from the microcapillary in order to decrease the mixing of fractionated particles, thus enhancing the fractionation resolution. 2.2. Quantification of particle separation efficiency in CHDF separation factor (Rf ) The effectiveness of particle separation using the microcapillary can be evaluated in terms of a separation factor, Rf . The separation factor is defined by the ratio of the average particle velocity to the average fluid velocity; that is, Rf =

vpz  , vz 

(1)

where vpz  and vz  are the average velocities of the particles and the fluid, respectively. In fact, for a constant length of the microcapillary, the separation factor can be calculated by taking the ratio of the mean elution time of the fluid to that of the particles. The mean elution time of the eluant is denoted by the average elution time of a molecular-size marker species, sodium benzoate. For a symmetrical fractogram, the mean elution time corresponds to the peak elution time. For an asymmetrical fractogram, the mean elution time is measured at the mean retention time at the peak’s center of mass. In this paper, a fractogram is defined as the plot of the detector response against the elution time. This statistical moment can be calculated using the following formula [25]:  ti S(ti )ti ts  = i (2) , i S(ti )ti

where ts  is the mean elution time of the sample, S(ti ) is the detector response at time ti and ti is the measuring time difference. For particles and molecules that do not interact with the CHDF capillary wall, the separation factor is always greater than 1 indicating that on average, the particles move through the microcapillary faster than the eluant. Theoretically, the maximum value of Rf is 2 for particles dispersed in Newtonian fluids, which occurs when the particles are traveling exclusively along the center streamline. Nonetheless, this situation never exists because of the retarding hydrodynamic effects of the close proximity of the particles to the wall. The theoretical particle velocity and diffusivity include the appropriate corrections which, when incorporated in the transport model [8], predict separation factors of less than 2. 2.2.1. Axial dispersion Axial dispersion of the particles takes place when a slug of colloidal particles flowing through a capillary is stretched axially while it is transported by the carrier phase. Axial dispersion causes fractogram broadening. Fully-developed radial concentration of particles produces Gaussian-shaped fractograms. In the absence of axial dispersion, the fractogram peak appears as a sharp spike rather than a bell-like shape. The particle separation efficiency in the microcapillary is influenced not only by the separation factor, but also by the extent to which the peak broadens during the separation process. Axial dispersion is quantified by the distance-based variance of the peak width per unit length of axial displacement which is denoted as a theoretical plate height of chromatography, HTP . An increase in dispersion of a solute is characterized by an increase in the effective dispersion coefficient, and hence, an increase in HTP . The axial dispersion of a solute in a microcapillary can be determined directly from the CHDF fractogram. The broadening of the fractogram is characterized by the axial dispersion coefficient which can be evaluated from the time-based variance σt2 of the fractogram [25]:  (ti − ts )2 S(ti )ti 2 σt = i  (3) . i S(ti )ti

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Table 2 Particle diameter and standard deviation of polystyrene latex standardsa

Table 1 Molecular weights and polydispersity index (PDI) of PEOs Polymer PEO1 PEO2 PEO3 PEO4 a b c d

Mn a (g/mol)

Mw b (g/mol)

Mv c (g/mol)

PDId

Particle diameter (nm)

Standard deviation (nm)

9700 27 800 70 800 248 400

11 500 122 200 368 600 1 217 500

11 300 99 200 308 400 1 061 600

1.2 4.4 5.2 4.9

91 109 176 234 357 610 794

5.8 2.7 2.3 2.6 5.6 9.0 4.4

Mn : number-average molecular weight. Mw : weight-average molecular weight. Mv : viscosity-average molecular weight. PDI: polydispersity index = Mw /Mn .

a Data obtained from the manufacturer.

To express the variance in terms of a distance-based variance, σz2 , the time-based variance is multiplied by the square of the average axial velocity; that is, σz2 = σt2 vpz 2 .

(4)

And thus, for the capillary of length L, the plate height HTP is defined as HTP =

σz2 . L

(5)

2.3. Eluant compositions and latex standards 2.3.1. Eluant Poly(ethylene oxide) (PEO; Sigma–Aldrich), with different molecular weights, were used as received. The molecular weights and molecular weight distributions of the PEOs determined by aqueous GPC are shown in Table 1. The PEOs were dissolved in deionized (DI) water to prepare the eluants. All eluants were cleaned by passing them through an ion-exchange resin column, followed by filtering using a 2 µm pore size membrane. The ion-exchange cleaning process was repeated until the conductivity of the aqueous PEO solutions fell below 3 µS/cm. The conductivity was measured using a YSI model 32 conductance meter with a YSI model 3403 electrode (cell constant = 1.0 cm−1 ) at room temperature. For some conditions, the conductivity of the eluant was adjusted by adding a certain amount of NaCl solution. NaCl was added as eluant because it has a low tendency to form complexes with PEO [26]. In addition, the viscosities of PEO solutions were measured at room temperature for shear rates up to 10 000 s−1 using a coneand-plate viscometer (Boehlin). 2.3.2. Latex standards A series of cleaned monodisperse polystyrene (PS) latex standards, manufactured by the Dow Chemical Co., of various particle diameters were used as the colloidal samples. The latex standards were cleaned by stirring with an ion-exchange resin to remove ionic residues in the dispersed phase. Average particle sizes and the standard deviations as determined by transmission electron microscopy (TEM) are shown in Table 2. To prepare the samples, these latexes were dispersed in the eluant to a concentration that provides a reasonable signal to noise ratio, while minimizing the particle–particle interactions [27]. Before injection into the CHDF instrument, the dis-

persion samples were sonicated for 1 min in order to break up any aggregates that might have formed during the sample preparation. 2.4. Determination of the amount of PEO adsorbed on PS particles To obtain good stability for a dispersion of PS particles in PEO solution, the particle surfaces should be fully covered by absorbed PEO to reduce the free surface area of the particles to eliminate bridging of PEO molecular chains from nearby particles. In addition, the densely adsorbed polymer layer provides an additional repulsive force to the particles preventing particle–particle attachments. The saturated surface concentrations indicating the full coverage of PEO on the PS particle surfaces for each PEO molecular weight were determined by adsorption isotherm experiments. In these experiments PS latex particles with a diameter of 234 nm were vigorously mixed into PEO solutions with known concentrations at a 0.10% w/w particle concentration. The mixtures were left undisturbed to equilibrate for 12 h to ensure adsorption equilibrium. After that, the dispersions were centrifuged at 20 000 rpm for 6 h at 25 ◦ C. The supernatants were carefully removed with a syringe and filtered through a 5 µm pore size filter paper. Calibration curves of the refractive index as a function of PEO concentrations were constructed and the concentrations of free polymer in the supernatant could be determined. The amount of PEO adsorbed onto the PS particle surface was calculated by difference from a mass balance [28]. 2.5. Determination of the apparent diameter of PS particles with PEO adsorbed The apparent diameter of the PEO-adsorbed PS latex particles was determined experimentally using dynamic light scattering (Nicomp Model 370). The thickness of the adsorbed PEO is obtained by subtracting the diameter of the bare PS particles from the diameter of the PEO-adsorbed PS particles. Standard PS latex particles were mixed with PEO solutions where the concentrations were selected to fall in the plateau regions of the adsorption isotherms to assure complete coverage of PEO molecules on the particle surfaces. The mixtures were vigorously shaken and then left undisturbed overnight before performing the measurement. Since all samples must be diluted by deionized (DI) water before the measurement, detachment

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of PEO molecules from particle surfaces may occur. However, all measurements were conducted over a 30-min period and the results showed that the diameter of the particles remained constant and the particle size distribution was consistent, indicating that the rate at which PEO molecules become desorbed from the latex particle surfaces is much slower than the time period of the measurement. 2.6. Transport of colloidal particles through a microcapillary 2.6.1. Transport of PEO-absorbed particles using eluant without PEO present CHDF experiments were performed using 234 nm PS latex particles dispersed in PEO4 solution. To minimize the amount of free PEO present in the sample dispersion medium, the concentration of PEO4 was limited to values in the vicinity of the saturation point based on the adsorption isotherm. The mixtures of latex particles and PEO4 solution were left undisturbed for at least 12 h in order to ensure that an equilibrium state of adsorption was attained. The eluant in this experiment was 4 mM NaCl aqueous solution in order to diminish the long range interactions such as electrostatic and electrokinetic repulsions. In regards to the results obtained from dynamic light scattering, the desorption of PEO from the latex particle surfaces was assumed to be slow in comparison to the transport time of particles through the microcapillary. High shear inside the microcapillary may promote the detachment of polymer chains from the particle surfaces; however, as the molecular weight increased, the polymer chains form multiple attachments on the particle surfaces, which make the disengagement of the contact points difficult [29]. 2.6.2. Transport of PS particles using eluant with PEO present Another set of experiments were performed to probe the effect of dissolved PEO in the eluant on particle transport in the microcapillary. Here, precautions were taken to avoid PEO adsorption on the particles during the transport process of particles. The particle surfaces were passivated by pre-adsorbing a UV-transparent nonionic surfactant. The steric barrier created by the adsorbed surfactant can deter the PEO molecules from coming in contact with the particle surfaces [30]. Brij35SP (polyoxyethylene lauryl ether with Mw = 1.198 g/mol manufactured by ICI America Inc.) was used as the pre-adsorbed surfactant. The ability of Brij35SP to prevent PEO adsorption was tested by performing dynamic light scattering (DLS) measurements (NICOMP Model C370). Latex particles were added into the Brij35SP solution, before dispersing the mixture in the various PEO solutions. The concentration of Brij35SP was above its critical micelle concentration (cmc), which is about 0.0072% w/w [27]. Then, the mixture of Brij35SP pre-adsorbed latex particles and PEO was diluted in deionized water before performing DLS measurements. Latex particles used in CHDF experiments were mixed with Brij35SP before being injected into the system. Moreover, Brij35SP was also dissolved in the PEO solutions that were used as the eluants to assure a constant supply of Brij35SP during the particle transport process. Also, the concentration

Fig. 2. Adsorption isotherms of various molecular weight PEOs on 234 nm PS latex particles. The surface saturation concentrations are: PEO1, 0.39 mg PEO/m2 PS; PEO2, 0.55 mg PEO/m2 PS; PEO3, 0.86 mg PEO/m2 PS; and PEO4, 1.06 mg PEO/m2 PS.

of Brij35SP in the eluant was selected to be slightly above its cmc, since at this concentration, full-coverage of surfactant on particle surface is assured. 3. Results and discussion 3.1. Adsorption of PEOs on PS particles and CHDF fractograms Fig. 2 shows the adsorption isotherms for various molecular weight PEOs adsorbed on 234 nm PS particles plotted against the concentrations of added PEO. The saturation surface concentrations (see the caption of Fig. 2 for numerical values), were estimated from the plot where the adsorption amount (mg PEO/m2 PS) reached its plateau value. By supplying PEO in the dispersion samples above the saturation point, particle aggregation should be greatly reduced. Since aggregates become larger than singlet particles, they travel faster than do wellseparated particles and then elute first from the capillary. Hence, the formation of particle aggregates can be observed in the CHDF fractograms in the form of the extra peaks that appear before the singlet particle peak. Moreover, the ratio of particle optical densities between two UV wavelengths (the so-called turbidity ratio) can be used to follow the extent of aggregation because the turbidity ratio has a unique value for each particle size [31]. Fig. 3a shows particle aggregates that appear as several extra peaks in the CHDF fractogram at earlier elution times. Also, the decreasing value of the turbidity ratio for these extra peaks confirms that larger aggregates were formed. The fractogram shown in Fig. 3a represents a dispersion of 91 nm PS particles in 0.01% w/w PEO4 at a particle concentration of 0.5% w/w. This condition corresponds to a surface concentration of 0.32 mg PEO4/m2 PS which is below the saturated surface concentration (1.06 mg PEO/m2 PS from Fig. 2). These aggregated latex particles that were formed at low PEO4 concentrations probably occurred because the long protruding regions of the PEO

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(a)

(b) Fig. 3. CHDF fractograms of 0.5% w/w 109 nm PS particles dispersed in: (a) 0.01% w/w PEO4 and (b) 0.1% w/w PEO4.

polymer chains can overcome the distance of closest approach between adjacent PS particles due to the high particle concentration used and attach themselves to the bare surface of other particles. Well-stabilized particles can be obtained when the concentration of PEO4 is increased above the saturation point. At this stage, there are sufficient numbers of PEO4 chains in the solution to generate full surface coverage of PEO4 on the PS particles, thus imparting colloidal stability to the particles provided by the steric barrier of the densely adsorbed polymer layer. This is evident in Fig. 3b where the amount of PEO4 was equivalent to an adsorbed amount of 3.8 mg PEO/m2 PS, and where the CHDF fractogram only exhibited a single symmetrical peak with a constant turbidity ratio indicating that the particles were well separated from each other and no aggregates were formed.

3.2. Dynamic light scattering of PS particles with adsorbed PEO Table 3 reports the volume-average particle diameter (Dv ) and the mean adsorbed layer thickness (δ) for the adsorption of PEO with different molecular weights on 234 nm PS latex particles as determined by dynamic light scattering. An increase of the particle diameter with increasing molecular weight of PEO (see Table 1 for the molecular weights of the various PEOs that were used in this study) was observed which is primarily due to the adsorbed PEO layer surrounding the particles. The dependency of molecular weight of PEO on δ was obtained by plotting log(Mw ) versus log(δ). A linear relationship was found as illustrated in Fig. 4 suggesting a power-law type dependency of δ on the molecular weight of PEO; i.e.,δ ∝ (Mw )0.63 . This experimental molecular weight dependence of Mw0.63 was among

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Table 3 Volume-average diameter (Dv ) and mean adsorbed layer thickness (δ) for adsorption of PEO on PS latex particles as determined by dynamic light scattering Dispersed phase

Dv (nm)

δ (nm)

DI water 0.1% w/w PEO1 0.1% w/w PEO2 0.1% w/w PEO3 0.1% w/w PEO4

256.6 ±25.4 261.6 ± 35.1 284.1 ± 39.3 298.6 ± 62.9 353.5 ± 106.3

– 2.5 13.7 21.0 48.4

(a)

Fig. 4. Plot of log δ vs log Mw for PEO-adsorbed PS particles.

the values previously found for PEO adsorption studies which ranged from Mw0.27 to Mw0.8 [14,15,32–34]. A large standard deviation in the particle size measurements for the PS particles in PEO3 and PEO4 solutions raised the uncertainty of the results. This deviation perhaps results from the free PEO chains and aggregation of PEO in aqueous solution. If the molecular weight of PEO is high, the coil size of PEO chains in water is comparable to the particle size and can interfere with the light scattering measurement. If one imagines a polymer coil as a rigid sphere with a hydrodynamic radius of RD , the size of a polymer chain is given by  RD =

3π 128

1/2

 2 1/2 r ,

(6)

where r 2 1/2 is the root-mean-square end-to-end distance of a polymer molecule [35]. RD of PEO2 calculated from Eq. (6) is equal to 9.7 nm, while it is equal to 87.8 nm for PEO4. Evidently, the coil size of the PEO4 polymer chain is comparable to the probe particle size and can be detected by light scattering. Moreover, PEO can form aggregates in aqueous solution, which can have a molecular weight as high as 50 times the nominal molecular weight of a single molecule [36]. Boils and Hair [37] have shown that PEO aggregates can be formed even for very monodisperse PEO samples. Using dynamic light scattering and gel permeation chromatography, Polverari and van de Ven [38] found that PEO clusters diameters were in the range of 0.45–0.90 µm and were independent of the molecular weight.

(b) Fig. 5. (a) CHDF separation factor (Rf ) and (b) theoretical plate height (HTP ) of 234 nm PS latex particles dispersed in 0.0098% w/w PEO4 (Q) compared to PS particles dispersed in DI water (P); capillary ID = 24.0 µm and length = 655.0 cm, eluant = 4 mM NaCl and particle concentration = 0.25% w/w.

3.3. Transport of PEO-absorbed particles through a microcapillary using eluant without PEO present Fig. 5 shows the dependence of Rf and HTP values for 234 nm PS particles as a function of the average fluid velocity in the presence and absence of PEO4. The PEO4 concentration of 0.0098% w/w was calculated based on the saturated surface concentration assuming that the particle surfaces would be totally covered by adsorbed PEO4 molecules and the free PEO4 concentration in the latex sample would be minimized. PS particles in the presence of PEO4 exhibited higher Rf values compared to those of PS particles without PEO4 present especially at relatively high velocities of the fluid. The enhancement of Rf most likely resulted from the increased effective diameter of the PS particles due to the adsorbed layer of PEO4 surrounding the latex particles. Under a high ionic strength condition (4 mM NaCl), the effects of the electrokinetic and electrostatic interactions are weakened allowing the particles to sample all moving fluid streamlines. Thus, on approaching the

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Table 4 Volume-average particle diameter (Dv ) of 234 nm PS latex particles with preadsorbed Brij35SP dispersed in PEO solutions as determined by dynamic light scattering Dispersed phase

Dv (nm)

Std. deviation (nm)

Brij35SP PEO2 PEO3 PEO4

253 257 256 255

25.0 14.8 39.0 26.7

capillary wall, the particle with a PEO-adsorbed layer present will be excluded from the wall at a longer distance than the radius of the bare PS particle. Consequently, PEO-adsorbed particles will experience the faster fluid velocities. More clearly, the increase in the PS particle diameter resulting from the adsorbed PEO layer was indicated by an increase in HTP with the average fluid velocity. If the PEO-adsorbed particles are pictured as composite spheres comprised of a rigid PS particle core surrounded by a permeable adsorbed PEO shell layer, then the diffusivity of the composite sphere Dcs is given as [39] Dcs =

aD , a + LH

(a)

(7)

where D is the diffusivity of a rigid sphere in a medium and is calculated from the well-known Stokes–Einstein relationship, D=

kB T , 6πμa

in which kB is the Boltzmann constant, T is the absolute temperature, μ is the viscosity of the medium and a is the particle radius. LH represents the impermeable polymer adsorbed layer which provides an equivalent drag force as the adsorbed layer thickness δ [40]. For high permeability of the polymer adsorbed layer, the flow retardation of the adsorbed layer is low; that is LH → 0, and thus the diffusivity of the composite sphere will be close to that of a rigid sphere. On the other hand, the composite sphere will behave like a hard sphere with an increased radius of a + δ for a low permeable adsorbed layer. Regarding the diffusivity of the composite sphere, the adsorption of PEO on PS particles can cause a reduction in particle diffusivities depending upon the thickness and permeability of the PEO adsorbed layer. Because of this diminishing particle diffusivity, the convective effect from the moving fluid will be more prominent, resulting in an increase in the particle axial dispersion with increasing fluid velocity. As a result, PEO4-adsorbed PS particles yielded a higher HTP than did the non-adsorbed particles. 3.4. Transport of PS particles using eluant with PEO present 3.4.1. Transport of colloidal particles in the presence of PEO The diameters of Brij35SP surfactant-adsorbed particles as measured by dynamic light scattering are reported in Table 4. The results show that the particle diameter was almost the same for all PEO solutions; this indicated the effectiveness of Brij35SP in shielding the PS surface from PEO.

(b) Fig. 6. (a) CHDF separation factor (Rf ) and (b) theoretical plate height (HTP ) of various size PS latex particles in the presence of 0.1% w/w PEO4 solution (Q), DI water (P) and 4 mM NaCl ("); capillary ID = 24.1 µm and length = 655.0 cm; eluants = 0.1% w/w PEO4 with 4 mM NaCl and DI water with estimated ionic strength = 1.5 × 10−6 M; average fluid velocity = 3.4 cm/s.

Fig. 6 shows the CHDF separation factor (Rf ) and theoretical plate height (HTP ) of various size PS latex particles using 0.1% w/w PEO4 solution with 4 mM NaCl as the eluant compared to those particles dispersed in DI water with a low ionic strength of 1.5 × 10−6 M (DI water) and a high ionic strength of 4 mM (NaCl solution). The separation factors increased considerably for all particle sizes in the presence of PEO in the eluant. This implies that the particles moving with PEO4 solution through the capillary tube were subjected to the stronger repulsions toward the tube axis than the repulsions that existed in the low ionic strength DI water. In PEO-free eluants, it is believed that the particle radial migration taking place in low ionic strength DI water is most likely influenced by the electrokinetic lift force because as the ionic strength of PEO-free eluant increased, the Rf curve shifted markedly to lower values. This force is predicted to increase with decreasing fluid conductivity (ionic strength) and increasing particle velocity [13]. The strong radial migration of submicrometer particles in a micro-

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capillary due to the electrokinetic lift also have been observed in the previous CHDF study when using the eluant with ionic strengths of less than 10−3 M [11]. For particles dispersed in high ionic strength PEO4 solution, the electrokinetic lift is less pronounced and hence, the particle migration is mainly affected by the viscoelastic properties of PEO4 solution. Many studies in different flow geometries [16–23] have shown that spherical particles dispersed in viscoelastic fluids would migrate to a region of lower velocity gradient, i.e., toward the inner cylinder of the couette apparatus and towards the center of the capillary. Theoretical studies of the moving particles in the viscoelastic fluids [41–44] showed that the migration of particles in the direction of decreasing shear rate is indeed generated by the normal stress of viscoelastic fluids. The stronger radial migration effect of particles in the PEO4 solution is also evident in the plot of HTP . As the particle size increased, the axial dispersion of particles in PEO4 solution decreased more dramatically than that of particles in DI water. This is because the rate of particle migration resulting from the normal stress effect of PEO4 solution increases more rapidly with particle size than the migration arising from the electrokinetic lift. However, for particle diameters less than 200 nm, particles dispersed in PEO4 exhibited higher HTP compared to those dispersed in DI water. According to the Stokes–Einstein equation, the particle diffusivity is reduced as the viscosity of the medium increased for a given particle size. Since PEO4 solutions have a higher viscosity, this is expected to produce more peak broadening. For particles larger than 200 nm in PEO4 solution, Rf and HTP almost reached a constant value implying that the particles sampled the region where the fluid velocity difference is small and thus, they experienced similar velocities despite their size differences. PEO4 solutions at 1.0% w/w and 0.5% w/w exhibited shearthinning behavior, but Newtonian behavior (constant viscosity) was observed for 0.1% w/w and 0.2% w/w PEO4, the concentrations used in these experiments. If shear-thinning fluid flows through a capillary tube, the fluid velocity profile is blunter. Expectedly, the CHDF peak for particles dispersed in the shear thinning fluid will appear narrower. For 0.1% w/w PEO4, shear thinning was not observed for a shear rate up to 10 000 s−1 however, the shear rate at the capillary wall appeared to be twice at large under this experimental condition. If PEO4 solutions behave like a power-law fluid, the power-law exponent for 1.0% w/w and 0.5% w/w PEO4 are estimated as 0.76 and 0.86, respectively. For 0.1% w/w PEO4, if shear thinning exists, the power law exponent should be close to 1.0. Thus, even though shear thinning in 0.1% w/w PEO4 may take place at a shear rate larger than 10 000 s−1 , the effect of reduced viscosity and more precisely the effect of reduced diffusivity should be relatively small. For example, the diffusivity of particles in a power-law fluid with a power-law exponent equal to 0.9 drops 5% compared to the diffusivity of particles in a Newtonian fluid [45]. 3.4.2. Effect of molecular weight of PEO Fig. 7 shows the plot of the separation factor, Rf , and the degree of axial dispersion, HTP , as a function of particle size for various molecular weights of PEO. These results show that the

85

(a)

(b) Fig. 7. CHDF separation factor (Rf ) and theoretical plate height (HTP ) of various size PS latex particles in the presence of different molecular weight PEO solutions; capillary ID = 24.1 µm and length = 655.0 cm; eluants = 0.1% w/w PEO1 ("), 0.1% w/w PEO2 (2), 0.1% w/w PEO3 (Q) and 0.1% w/w PEO4 (a), all PEO solutions contain 4 mM NaCl; average fluid velocity = 3.4 cm/s.

separation factors for the latex particles were improved, and the degree of axial dispersion was lower, as the molecular weight of the PEO that was used in the experiment was increased. This implies that the radial migration of particles toward the capillary axis is dependent on the molecular weight of PEO. Considering the analysis of Ho and Leal [41] for a single sphere migration in a second-order fluid under two-dimensional capillary flow, the lateral migration velocity of the particle, vpnr , in a viscoelastic fluid can be estimated from the force balance on a particle between the lateral force due to the normal stress gradient and the viscous drag, giving [19] vpr = k

a2 ∂ γ˙ ψ1 γ˙ , μ ∂r

(8)

where in the unidirectional capillary flow, γ˙ can be represented by the velocity gradient of the unperturbed velocity field, a is the particle radius, ψ1 is the primary normal stress coefficient, r is the radial position, and k is the constant arising from the para-

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meters appear in the second-order fluid model. The viscoelastic effect is reflected through the primary normal stress coefficient which depends on the molecular weight and concentration of polymer molecules in medium. If a polymer molecule is modeled as a linear elastic dumbbell, the kinetic theory of polymer molecules suggests that the primary normal stress coefficient is related to the intrinsic relaxation time, λH , and number of polymer molecules by the following expression [46]: ψ1 = 2nkB T λ2H .

Table 5 1/2 Root-mean-square end-to-end distance of unperturbed PEO molecule r 2 0 and estimated intrinsic relaxation time λH 1/2

PEO

r 2 0

PEO1 PEO2 PEO3 PEO4

14.3 37.5 63.4 116.4

(nm)

λH (s) 3.77 × 10−7 6.71 × 10−6 3.23 × 10−5 1.99 × 10−4

(9)

The intrinsic relaxation time is a function of the root-meansquare end-to-end distance of unperturbed polymer molecule 1/2 r 2 0 which is proportional to the molecular weight of polymer [47] as illustrated in Table 5. Therefore, a larger molecular weight of PEO possesses a longer λH which resulted in a higher ψ1 . As a result, the normal stress is increased with molecular weight causing the faster migration of the particle toward the center of the microcapillary. For low molecular weight PEO1, the normal stress effect is insignificant, especially at smaller particle sizes; therefore, the particles will experience more pronounced effects from convection and diffusion. The increase in HTP for particle sizes smaller

than 357 nm results from a decrease in the particle diffusivity with particle size; hence, the convection from fluid velocity will be prominent, thus causing the peak broadening. 3.4.3. Effect of ionic strength of PEO solution The effect of ionic strength of PEO solutions on the particle separation is presented in Fig. 8. For all PEO molecular weights, the separation factors of PS particles in the presence of high-ionic-strength PEO solutions (conductivity ∼ 470 µS/cm and estimated ionic strength = 4 × 10−3 M) exhibits lower values than those of particles dispersed in low ionic strength (conductivity ∼1 µS/cm and estimated ionic strength = 1.5 ×

Fig. 8. CHDF separation factor (Rf ) of various size PS latex particles in the presence of different molecular weight PEO solutions for high ionic strength PEO solutions with estimated ionic strength = 4 × 10−3 M (2) and low ionic strength PEO solutions with estimated ionic strength = 1.5 × 10−6 M (1); capillary ID = 24.1 µm and length = 655.0 cm; eluants = 0.1% w/w PEO1, 0.1% w/w PEO2, 0.1% w/w PEO3 and 0.1% w/w PEO4; average fluid velocity = 3.4 cm/s.

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87

Fig. 9. Comparison of the experimental Rf (") with the predicted results taking into account the effect of the normal stress (—) and the electrokinetic effect (- - -) for transport of latex particles in 0.1% w/w PEO1 and PEO4; capillary ID = 24.1 µm and length = 655.0 cm; eluant = low ionic strength 0.1% w/w PEO1 and PEO4; average velocity of fluid = 3.4 cm/s.

10−6 M) PEO solutions. These results suggest that under low ionic strength conditions, the electrokinetic lift force may also be significant in addition to the normal stress effect that repelled the particles to move closer to the center of the microcapillary. Furthermore, the competing effects between the normal stress and the electrokinetic lift can be evident as the difference of the particle separation factor between low and high ionic strength PEO solutions became larger with decreases in the PEO molecular weight. This is probably because the electrokinetic lift is the major repulsive force for particles in the low molecular weight PEO. As the molecular weight of PEO is increased, the normal stress effect becomes more significant and takes part in the particle migration. The dominant role of the normal stress effect and the electrokinetic lift in the low and high molecular weight PEO is demonstrated in Fig. 9 where the experimental separation factors were compared with the ones predicted from the CHDF dynamic model for particles dispersed in low ionic strength PEO1 and PEO4 solutions. The CHDF dynamic model was employed since it does not require a condition of fully-developed radial concentration. The details of the CHDF dynamic model can be found elsewhere [48]. The migration velocity of the particle generated by the normal stress of PEO solution was estimated using Eq. (8) and the migration velocity from the electrokinetic lift vpek was calculated by [27]  vpek =

ε0 εr 4π

3 

9a 32η



2   −4 r 1− 2 R0 KR0 vpz

× ζ2 (ζ2 + 2ζ1 ), where εr and ε0 are the relative permittivity of the medium and the permittivity of a vacuum, vpz is the local axial particle velocity, R0 is the capillary radius, ζ1 and ζ2 are the zeta potentials of the wall and particle which are assumed to be −100 and −50 mV, respectively [27], and K is the fluid conductivity. For the low ionic strength PEO1 solution, when vpek was included in the CHDF dynamic model, the calculated Rf provided a better agreement with the experimental Rf indicating the su-

perior effect of the electrokinetic lift compared to the normal stress. On the contrary, for particles dispersed in the low ionic strength PEO4, better agreement between the calculated and the experimental Rf values was obtained when vpnr is considered, revealing the dominant effect of the normal stress for the high molecular weight PEO. 4. Conclusions The transport of PS latex particles through a microcapillary in the presence of PEO solutions was investigated. The effect of PEO that was present in the dispersion medium containing PS latex particles and in the eluant was examined. It was observed experimentally using a dynamic light scattering technique that PEO present in the dispersed medium apparently increases the effective PS particle size because of the presence of an adsorbed layer of PEO molecules on the particle surfaces. The thickness of the adsorbed PEO layer (δ) increased with increasing molecular weight of the PEO. A log–log plot between the molecular weight of PEO and the layer thickness of the adsorbed PEO exhibited a linear dependence with a slope of 0.63, indicating a power law relationship. In addition, the CHDF experimental results carried out using 4 mM NaCl as the eluant showed a higher particle separation factor Rf and degree of axial dispersion HTP of PS particles dispersed in PEO4 solution compared with PS particles dispersed only in DI water. This ensures the occurrence of PEO adsorption on the PS latex samples. The CHDF experiment carried out with high ionic strength PEO solution with molecular weight ca. 1 000 000 g/mol as the eluant produced a substantial increase in Rf and a rapid decrease in HTP as a function of particle size when compared to the CHDF experiment in which DI water was used as the eluant. Therefore, a stronger migration effect was generated for particles traveling through PEO4 solution that resulted from the normal stress of PEO4 solution under Poiseuille flow. The radial migration of PS particles was found to depend upon the molecular weight of PEO such that the higher molecular weight PEO provides a larger normal stress resulting in an increase of Rf and de-

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crease of HTP . Under low ionic strength conditions of PEO solutions, the particle–wall repulsion from the electrokinetic lift is also imposed on the particles. This electrokinetic lift is more significant for low molecular weight PEO since the normal stress effect is weakened as the molecular weight of PEO was decreased. For larger molecular weight PEO, the superior repulsive effect arises from the normal stress of PEO solution.

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