6018
Langmuir 1997, 13, 6018-6025
Articles Concentration-Dependent Sedimentation of Dilute Magnetic Fluids and Magnetic Silica Dispersions Liesbeth N. Donselaar, Albert P. Philipse,* and Jacques Suurmond Van ’t Hoff Laboratory for Physical and Colloid Chemistry, Debye Institute, Utrecht University, Padualaan 8, 3584 CH Utrecht, The Netherlands Received April 7, 1997. In Final Form: August 7, 1997X The concentration dependence of the sedimentation rate has been studied for a variety of colloids with a magnetic core and nonmagnetic shell. The nonmagnetic shell was either a silica layer or, in the case of magnetic fluids, a surfactant layer. It was found that the (linear) concentration dependence is very much affected by the thickness of the nonmagnetic shell and varied from negative for repulsive particles to positive for a sufficiently strong net attraction. This change in sign is in accordance with a recent theory for sedimentation of dipolar hard spheres. The sensitivity of concentration-dependent sedimentation to interaction details is also illustrated by the significant influence of any van der Waals attraction and surface charge, which competes with the effect of magnetic attraction. Our study shows that surface charge may be important in nonaqueous magnetic fluids.
1. Introduction The sedimentation velocity of colloidal particles in a viscous medium depends on particle concentration, an effect which has been studied over a long period of time.1-4 The most complete description of the sedimentation behavior seems to be that of Batchelor, who considered dilute homogeneous suspensions of rigid, neutral, impermeable spheres. Sedimentation experiments on uncharged spherical colloids seem to confirm Batchelor.5,6 It should be noted, however, that experimental results are generally not accurate enough to reproduce Batchelor’s theory quantitatively as is discussed by Kops-Werkhoven et al.5 In addition to hard sphere repulsions Batchelor also considered other types of interaction and calculated that particles with an attractive potential will settle faster than hard spheres at the same volume fraction. This effect has been studied by Jansen et al.7 for sterically stabilized silica spheres in toluene. With decrease of the temperature an attraction was induced between the particles, leading to a slight increase in the normalized sedimentation velocity as a function of the volume fraction as compared to hard spheres. The sticky attraction induced by varying the temperature is of short range and can be modeled by a narrow square well potential. However, other types of attraction may be present in colloidal dispersions, such as a dipolar attraction between magnetic particles. The aim of this report is to investigate the influence of such forces on the sedimentation rate. Several sedimentation experiments have been reported for magnetic fluids. These experiments were mainly * Corresponding author X Abstract published in Advance ACS Abstracts, October 1, 1997. (1) Kermack, W. O.; M’Kendrick, A. G.; Ponder, E. Proc. R. Soc. Edingburgh 1929, 49 (II), 170. (2) Burgers, J. M. Proc. Ned. Akad. Wet., Amsterdam 1942, 45, 9. (3) Batchelor, G. K. J. Fluid Mech. 1972, 52, 245. (4) Felderhof, B. U. J. Phys. A 1978, 11, 929 (5) Kops-Werkhoven, M. M.; Fijnaut, H. M. J. Chem. Phys. 1981, 74 (3), 1618. (6) Thies-Weesie, D. M. E.; Philipse, A. P.; Lekkerkerker, H. N. W. J. Colloid Interface Sci. 1996, 177, 427. (7) Jansen, J. W.; Kruif, C. G. de; Vrij, A. J. Colloid Interface Sci. 1986, 114 (2), 501.
S0743-7463(97)00359-4 CCC: $14.00
performed to determine particle dimensions8 or to investigate the diffusion-sedimentation equilibrium in a gravitational field.9 The settling behavior of strongly flocculated magnetic iron oxide particles was qualitatively studied by Glasrud et al.10 Other sedimentation experiments were reported by Peterson and Krueger11 and by Lukashevich and Naletova.12,13 These workers investigated the sedimentation velocity of magnetic particles as a function of an applied magnetic field. Due to formation of aggregates the sedimentation velocity was increased. Hardly any research has been performed on the sedimentation behavior of stable magnetic dispersions in the absence of an external magnetic field. Therefore we performed a systematic study on the sedimentation behavior using various magnetic colloidal dispersions with a nonmagnetic outer shell. A schematic model of the particles is shown in Figure 1. The influence of the thickness of the nonmagnetic shell on the concentration dependence of the sedimentation behavior was investigated. The nonmagnetic layer is a steric barrier which reduces magnetic attractions between the particles. The dispersions are either magnetite particles encapsulated by a silica shell of variable thickness14 or magnetite particles stabilized with a thin surfactant layer (Figure 2). For studies on magnetic colloids also commercially available latex spheres may be used.15 For our purpose, however, these particles are too polydisperse in size and in particle mass density. Preliminary sedimentation experiments with a latex-dispersion yielded a very diffuse (8) Seifert, A.; Buske, N. J. Magn. Magn. Mater. 1993, 122, 115. (9) Raikher, Yu. L.; Shliomis, M. I. J. Magn. Magn. Mater. 1993, 122, 93. (10) Glasrud, G. G.; Navarrete, R. C.; Scriven, L. E.; Macosko C. W. AIChE J. 1993, 39 (4), 560. (11) Peterson, E. A.; Krueger, D. A. J. Colloid. Interface Sci. 1977, 62, 24. (12) Lukashevich, M. V.; Naletova, V. A.; Tyatyushkin, A. N. J. Magn. Magn. Mater. 1990, 85, 216. (13) Naletova, V. A.; Tyatyushkin, A. N. J. Magn. Magn. Mater. 1993, 122, 102. (14) Philipse, A. P.; Bruggen, M. P. B. van; Pathmamanoharan, C. Langmuir 1994, 10, 92. (15) Fermigier, M.; Gast, A. P. J. Colloid. Interface Sci. 1992, 154 (2), 522.
© 1997 American Chemical Society
Concentration-Dependent Sedimentation
Langmuir, Vol. 13, No. 23, 1997 6019
U/U0 ) (1 - Kφ)
Figure 1. Configuration used to calculate the pair interaction potential between colloidal particles with a magnetic core of radius a and a screening layer with thickness b - a. The screening layer could be a surfactant layer, an electrical double layer, or a grafted silica layer.
dispersion-solvent interface. Another disadvantage is that these particles do not possess a permanent moment. This paper is organized as follows. In section 2 Batchelor’s equation will be briefly described and an equation will be discussed for the sedimentation velocity as a function of the volume fraction for magnetic particles. The synthesis and characterization of the dispersions are reported in section 3 together with a description of the experimental setup. The results of the sedimentation experiments are discussed in section 4 and summarized in Section 5. 2. Theory 2.1. Sedimentation of Brownian Particles. The stationary sedimentation velocity U0 of a free particle of arbitrary shape, with volume Vp and friction factor f at infinite dilution is given by
U0 )
Vp(Fp - Fs)g f
where K is 6.55 for monodisperse hard spheres. Batchelor3 has modeled the influence of attractions by a square-well potential. This potential leaves the linearity in eq 5 unaffected. For spheres with long range double layer repulsion the concentration dependence of U may become nonlinear.17 Here we consider specifically magnetic attractions. In section 2.2. an equation for K will be derived,18 valid for colloidal spheres with weak magnetic interactions. 2.2. Sedimentation of Magnetic Particles with Weak Magnetic Interaction.18 The sedimentation velocity of interacting colloidal particles depends on the pair interaction potential, which determines the pair correlation function. The pair interaction potential for a certain µ2 is given by orientation of magnetic dipoles b µ1 and b
V(r b) )
(
)
µ 1b µ 2 3(µ b)(µ b2‚r b) µ0 b b1‚r 3 5 4π r r
(6)
where µ0 is the magnetic susceptibility of vacuum and r the distance between the dipoles. Equation 6 is the farfield approximation for two dipoles separated by a distance much larger than the dipole size. The model we use to calculate the normalized sedimentation velocity has magnetic particles with radius a and a nonmagnetic shell with thickness b - a (see Figure 1). For a configuration as in Figure 1 the pair interaction potential for two dipoles with identical magnetic moment µ is
µ2µ0
V(θ1,θ2,φ) )
(sin θ1 sin θ2 cos φ - 2 cos θ1 cos θ2) 4πr3 (7)
(1)
were Fp is the particle mass density, Fs the density of the solvent, and g the acceleration of gravity. For a sphere with hydrodynamic radius ah suspended in a liquid of viscosity ηs the friction factor is given by Stokes law
(5)
The pair-correlation function, which is the spherically averaged probability of the center of particles 1 and 2 being a distance r apart, is given by
g(r) ) ghs(r)
1 8π
∫ dθ1 ∫ dθ2 ∫ dφ exp(-βV(θ1,θ2,φ))
(2)
(8)
Assuming the hydrodynamic radius is equal to the radius a of a dry particle, eq 1 reduces to,
Here β is (kT)-1 where k is the Boltzmann constant and T the absolute temperature. The pair correlation function for hard spheres ghs is
f ) 6πηs ah
U0 )
2a2(Fp - Fs)g 9ηs
(3)
Equation 3 also applies to the sedimentation in a centrifuge, with g replaced by ω2l, with ω the angular velocity and l the distance between the center of rotation and the measuring cell. The sedimentation velocity can also be expressed in terms of the acceleration independent sedimentation coefficient,16 which is defined as
S0 )
2a2(Fp - Fs) 9ηs
(4)
Batchelor has shown that for a dilute dispersion of (nonmagnetic) colloids with low volume fraction (φ < 0.05) the mean settling velocity is3 (16) Rowe, A. J. Analytical ultracentrifugation in biochemistry and polymer science; Harding, S. E., Rowe, A. J., Horton, J. C., Eds.; RSC: Cambridge, 1992; Chapter 21.
ghs(r) ) 1
for r g 2b
)0
for r < 2b
(9)
When the magnetic interaction strength is small compared to kT, the pair-correlation function can be Taylor expanded
g(r) ) ghs(r)
1 8π
∫ dθ1 ∫ dθ2 ∫ dφ (1 - βV(θ1,θ2,φ) + 1 2 2 β V (θ1,θ2,φ) (10) 2
)
Because of symmetry arguments, orientation averaging over the term V(θ1,θ2,φ) yields zero; therefore the integration must be taken over β2 V2(θ1,θ2,φ). The final result is (17) Thies-Weesie, D. M. E.; Philipse, A. P.; Na¨gele, G.; Mandl B.; Klein, R. J. Colloid Interface Sci. 1995, 176, 43. (18) Dhont, J. K. G. An introduction to dynamics of colloids; Mo¨bius, D., Miller, R., Eds.; Elsevier: Amsterdam, 1996; Chapter 7.
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Figure 2. Transmission electron micrographs of (a) magnetic silica particles with a silica layer thickness of 50 nm and (b) magnetic MaS particles stabilized with a double surfactant layer in pentanol.20,21
( ( ) )
g(r) ) ghs(r) 1 +
µ2µ0β π
2
1 48r6
and a the radius of the magnetic nucleus, this becomes
(11)
To calculate the sedimentation velocity for this g(r) the following expression proposed by Batchelor3 can be used
U ) (1 - φ)U0 + V′ + V′′ + W + O(φ2)
(12)
with O(φ2) the higher order terms in volume fraction. V′, V′′, and W are defined as follows with b the total radius of the magnetic silica particle
V′ )
3 φU0 b2
∫b∞(g(r) - 1) r2 dr
V′′ ) W)
1 φU0 b3
1 φU0 2
(13) (14)
∫b∞ g(r)[∆As(r) + ∆Ac(r) + 2∆ Bs(r) + 2∆Bc(r)] r2 dr (15)
In eq 15 four hydrodynamic coefficients appear, defined by Batchelor.3 They refer to forces between two spheres moving parallel (∆As(r) and ∆Bs(r)) and perpendicular (∆Ac(r) and ∆Bc(r)) to their lines of centers. Substituting the pair correlation function of eq 11 in eq 12 finally yields for the sedimentation velocity
U )1U0
(
(
4 πa M ) µ ( 3 6.55 - 0.96 2
3
s
32πkTb3
))
0
2
φ + O(φ2) (17)
with b the radius of the total magnetic silica particle. Equation 17 was first found by Dhont.18 Note that the term multiplying φ is the parameter K in eq 5. According to eq 17 the sedimentation velocity increases due to magnetic attraction. This is expected for all attractive particles due to the excess of particles in the vicinity of a test sphere, which profit from the downward motion of this sphere. For strong magnetic interaction, i.e., µ2µ0/32πb3 > kT, the sedimentation velocity as a function of the volume fraction must be calculated numerically.18 In Figure 3 the calculated sedimentation coefficient K is plotted as a function of the magnetic moment µ and the silica layer thickness. Figure 3 predicts a strong effect due to the magnetic attraction in a very short range, 5 nm < b < 10 nm, of the screening layer thickness. 3. Experimental Section
U ) (1 - φ)U0 + φU0 -
3.1. Dispersions and Characterization. The sedimentation experiments were carried out with six magnetic dispersions. One system (coded FF7) contains bare magnetite grains stabilized with tetramethylammonium hydroxide (TMA) in water.19 The MaS particles are dispersed in pentanol and stabilized with a double surfactant layer,20,21 comprising chemisorbed oleic acid on the particle surface and a second layer of dodecylbenzenesulfonic acid (DBS). This second layer is weakly adsorbed, so there are also free DBS molecules present. On the magnetic
Using µ ) 4/3πa3Ms, with Ms the magnetization per volume
(19) Massart, R.; Cabuil, V J. Chim. Phys. 1987, 84 (7-8), 1247. (20) Bica, D.; Preda, T. Preparation Method for Alcohol-Based Fluids. Roman. Pat. No. 97224, 1989. (21) Bica, D.; Preda, T. Preparation Method for Alcohol-Based Fluids. Roman. Pat. No. 97559 1989.
[{ ( ) } { ( ) }]
βµ2µ0 2 9 1 + + + 2 2 32πb3 βµ2µ0 2 -1.55 - 0.035 + O(φ2) (16) 32πb3
Concentration-Dependent Sedimentation
Langmuir, Vol. 13, No. 23, 1997 6021 gradient field an alternating force on the sample is produced proportional to the magnitude of the gradient field and the magnetic moment of the sample.23 Dried samples of known mass were wrapped in Scotch tape and stuck to the sample probe. Liquid samples were put in glass cups of 2 mm diameter and 5 mm length, closed with a glass plate of 2.5 mm diameter fixed with UV glue (Tectonite350). The glue hardens in 10 min when exposed to UV light. The electrophoretic mobility of particles was determined by means of a DELSA-440SX instrument (Coulter Electronics, Inc.). The electrophoretic mobility is determined by measuring the Doppler shift of scattered light caused by particles moving in an electric field.24 For a Debye length κ-1, which is large in comparison to the radius R of the particle, the mobility Ue.m. is related to the zeta potential ξ via the Hu¨ckel equation
Ue.m. ) 20ξ/3η Figure 3. Sedimentation constant K, in eq 5, plotted as a function of the magnetic moment µ for various values of the screening layer thickness. b is the total radius of the magnetic sphere. The sedimentation coefficient are obtained by numerical calculations.18 When K < 0, the formula obtained for weak magnetic interaction eq 17 is valid. particles in the aqueous dispersion FFWa a very thin silica layer (indistinguishable with transmission electron microscopy) was precipitated.14 The other three dispersions (coded FWS40St, FWS130St and FWS200St) contain magnetic colloids coated with a silica layer. These particles were synthesised by precipitation of silica, using the Sto¨ber method,22 on magnetic grains obtained by the Massart method.19 A detailed description of the synthesis can be found elsewhere.14 These magnetic silica spheres are sterically stabilized by grafting with octadecyl alcohol and dispersed in cyclohexane. System SD250S contains nonmagnetic silica spheres sterically stabilized with octadecyl alcohol.6 Silica particles grafted with octadecyl alcohol and dispersed in cyclohexane are uncharged. Characterization results (see Table 1) were obtained as follows. Transmission electron microscopy (TEM) was performed with a Philips CM10H electron microscope. Samples were prepared by dipping 200 mesh Cu-grids, coated with a Formvar/carbon film, in very dilute dispersions. The obtained micrographs were analyzed with an interactive image analysis system. The area of about 100 particles was measured to calculate the number average radius of the spheres and the standard deviation. Static light scattering measurements were done on a Fica-50 light scattering photometer using vertically polarized incident and detected light (λo ) 546 nm). The scattered average light intensity was measured as a function of the scattering angle. The Guinier approximation was used to calculate the optical radius. The samples were made dust free by centrifuging them for 5 min at 1000 rpm. Dynamic light scattering (DLS) was done on very dilute dispersions with an argon laser (Spectra Physics Series 2000) operating at 514.5 nm. The dispersions were made dust free by filtering over a Fluoropore filter (pore diameter 1.0 µm). Autocorrelation functions were measured with a Malvern Multibit K7025 128 points correlator at scattering angles between 35° and 140°. From a second-order cumulant fit of the autocorrelation functions diffusion coefficients were obtained from which the hydrodynamic radius aDLS was calculated using the Stokes-Einstein equation
aDLS )
kT 6πηsD0
(18)
where k is the Boltzmann constant, T the absolute temperature, ηs the solvent viscosity, and D0 the free-particle diffusion coefficient. Magnetization measurements were done with an alternating gradient magnetometer (AGM), MicroMag 2900, from Princeton Measurements Corp. The magnetic moment of a sample was measured as a function of the magnetic field which ranges from 0 to 160 × 104 A m-1. With the application of an alternating (22) Sto¨ber, W.; Fink, A.; Bohn, E. J. Colloid Interface Sci. 1968, 26, 62.
(19)
with the dielectric constant, 0 the permittivity constant, and η the viscosity. The total charge on the sphere Q may be estimated from
Q ≈ 4 π0ξR
(20)
The measurements were done using a He-Ne laser, λ0 is 633 nm, and the Doppler spectra were measured at four fixed nominal angles: 7.5°, 15°, 22.5°, and 30°. 3.2. Determination of the Volume Fraction. The volume fraction φ of (magnetic) silica dispersions is obtained from the weight concentration of the colloids and the particle mass density. The particle mass density Fp of the FWS dispersions was determined by weighting 25.0 mL of the dispersions with known mass fraction and 25.0 mL of the solvent in the same flask. For magnetic fluids we need to follow a different procedure, because the solid concentrations are difficult to determine from dried residues which also contain an unknown amount of TMA or surfactant. The magnetic particles were reduced to iron salts by mixing a known volume with concentrated hydrochloric acid. The iron concentration was determined using atomic absorption spectroscopy (AAS). The weight concentration of the iron oxide, cn, can easily be calculated on the basis of the iron concentration. Due to partial oxidation the real weight concentration is in the range cmin and cmax, respectively, the concentration for pure magnetite and maghemite. The ratio cmax and cmin is 0.97 indicating that the oxidation to maghemite hardly affects the estimate of φ. The concentration of the ferrofluid can be calculated via
c ) cn +
(
)
cnFl (a + δ)3 -1 Fn a3
(21)
here cn is the weight concentration of the nucleus, Fl the density of the stabilising layer, Fn the density of the nucleus, a the radius of the nucleus, and δ the thickness of the stabilizing layer. The particle density Fp is given by
Fp ) (Fn - Fl)
a3 + Fl (a + δ)3
(22)
The radius a of the nucleus can be determined by electron microscopy. The thickness δ is estimated for FF7 and FFWa to be δ ≈ 1 nm and for MaS δ ≈ 5 nm. The density of the stabilizing layer is assumed to be equal to the density of the solvent. The calculated densities can be found in Table 1. In the calculation of the density the assumption is made that maghemite is not hydrated (γ-Fe2O3‚xH2O); the density of the hydrated iron oxide is much lower (2.44-3.60 g/mL25 ). 3.3. Sedimentation. The sedimentation measurements under gravity were carried out as described by Thies-Weesie et al.6 Weighted amounts of the dispersions SD250S, FWS130St, FWS200St, and cyclohexane were poured into cylindrical glass (23) Instruction manuel: MicroMag 2900, Princeton Measurements Corporation. (24) Xu, R. Langmuir 1993, 9, 2955. (25) Weast, R. C. Handbook of Chemistry and Physics, 66th ed.; CRC Press: Cleveland, OH, 1975.
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Table 1. Particle Radii and Mass Densities system
Fp/g mL-1
TEM/nm
SD250S MaS FF7 FFWa FWS40St FWS130St FWS200St
1.78 ( 0.01 1.6 ( 0.4a 3.3 ( 0.2a 3.3 ( 0.2a 1.99 ( 0.05 1.90 ( 0.05 1.87 ( 0.01
229.9 ( 9.7 7(2 5 ( 0.5 5 ( 0.5 39.3 ( 4.1 103.0 ( 9.6 149.2 ( 5.7
SLS/nm b b b 74 145.0 211.7
DLS/nm 265.8 15-45c 70 119.5 183.5
a Calculation described in section 3.2 b Particles too small to be characterized with this technique. c Varies with concentration.
tubes of 1 cm diameter and 15 cm length and carefully closed. The tubes were immersed in a water bath, placed on a heavy marble table to minimize vibrations. The equipment was placed in a thermostatic room to keep the temperature of the sedimenting dispersions constant at 22.0 ( 0.1 °C. Before the measurements were started, it was checked to ensure that the tubes were hanging perfectly vertical. The sedimentation experiments with FWS40St were carried out in tubes with diameter of 1 and 10 cm length in a thermostated room. The dispersion-solvent interface was monitored using a Zeiss Ni-40 leveling instrument. To increase the visibility of the interfaces, the tubes were illuminated with the lamp of a slide projector. After the measurement the dispersions were homogenized again by shaking and ultrasonification. After shaking, the samples showed no flocs or other inhomogeneities which could indicate coagulation of particles. For most concentrations the sedimentation velocity was measured three times and averaged. The procedure to measure the sedimentation velocity as a function of the volume fraction was checked by reproducing the experiments of Thies-Weesie et al.26 for the nonmagnetic colloidal dispersion SD250S, for which the measured Stokes velocity was in good agreement with the calculated velocity.26 Because small particles of about 5-10 nm do not sediment significantly in the gravitational field, an analytical ultracentrifuge (Beckman, model E) was used for the systems FF7, FFWa, and MaS. With a photoelectric scan system the sedimentation velocity of the dispersions as a function of the volume fraction was measured. This was done by recording the maximal change in absorption (the dispersion-solvent interface) in the centrifuge cell with time. For the scan system a wavelength of 550 nm was used. Rotorspeeds between 6000 and 8000 rpm were used and the measuring time was in the order of 10 min. The temperature was kept at 25.0 °C. The samples were prepared by mixing a weighted amount of a concentrated dispersion with a weighted amount of the solvent. The particles of FFWa were dispersed in distillated water and the colloids of MaS were dispersed in 1-pentanol. The composition of the dispersion of FF7 was not well-known, because an unknown amount of free TMA molecules was still present in the ferrofluid. Therefore two series of the FF7 dispersion were measured. In one series the dispersions were made by diluting the stock solution with double distilled water and the other by diluting it with an aqueous solution (pH ) 10) of TMA in double distilled water.
4. Results and Discussion 4.1. Gravitational Sedimentation. During the sedimentation of the nonmagnetic silica spheres SD250S a sharp interface was observed between dispersion and the clear supernatant. The height of the interface descends linearly with time (Figure 4). The normalized sedimentation velocity U/U0 of the silica dispersion as a function of the volume fraction φ is plotted in Figure 5. The value of U0 determined from extrapolation of U to infinite dilution (Table 2), is in good agreement with the calculated value (Table 2) and with the value U0 ) (15.76 ( 0.07) × 10-8 m s-1 obtained by Thies-Weesie6 for the same system. (For calculation of U0 the hydrodynamic radius determined with DLS is used.) The correspondence between theory and experiment and with the already (26) Thies-Weesie, D. M. E. Sedimentation and liquid permeation of inorganic colloids; Thesis, Utrecht University, 1995.
Figure 4. Height of the interface between the dispersion SD250S, containing nonmagnetic particles, and the solvent, cyclohexane, against time. The results of two volume fractions, 0.11 and 0.04, are shown.
Figure 5. Normalized sedimentation velocity U/U0 versus the volume fraction of the system SD250S, containing nonmagnetic silica particles dispersed in cyclohexane. The solid line is a linear fit.
reported value6 demonstrates that the procedure used to measure the sedimentation velocity as described in section 3.3 is reliable. The results of the measured sedimentation velocities for the systems FWS40St, FWS130St, and FWS200St are shown in Figure 6. Although the interface between dispersion and particle-free solvent was some what diffuse, it was still possible to measure reproducibly the height of the interface with time. The sediments of FWS40St and FWS200St were amorphous, whereas the sediment of FWS130St showed a beautiful crystalline iridescence. This indicates that at least the FWS200St particles are slightly attractive which counteracts colloidal crystallization. (The FWS40St particles are too small to see Bragg reflections anyhow.) Apparently the FWS130St spheres exhibit a hard-sphere crystallization. So in this case the silica shell masks the magnetic attractions. The values of the slopes of U/U0 versus the volume fraction (K in eq 15) are listed in Table 2 together with the measured and calculated Stokes velocities. The calculated U0 values for FWS130St and FWS200St are in good agreement with the experimental U0 values. According to the theory (eq 17) in section 2.2, the coefficient K is a function of the dipole moment of the particles. We define an effective µsed as the dipole moment which according to Figure 3 would correspond with the measured K, for a given radius b. These magnetic moments can be found in Table 3, together with calculated values (µtheor) and values which are measured with the
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Langmuir, Vol. 13, No. 23, 1997 6023
Table 2. Calculated and Measured Values for U0, k in Eq 23, and K system SD250S MaS
U0 (10-9 m s-1)
158 ( 6 0.59 ( 0.008
165 ( 3 0.056 ( 0.002
0.23 ( 0.03 0.23 ( 0.03 0.23 ( 0.03 14 ( 2 36 ( 5 80 ( 5
1.08 ( 0.04 1.04 ( 0.06 1.09 ( 0.04 28 ( 3 33 ( 1 69 ( 1
FF7-H2O FF7-TMA FFWa FWS40St FWS130St FWS200St a
measurement
theoretical U0 (10-9 m s-1)
k (g L-1)
K + 3.5 ( 0.2 + 146 ( 20a + 77 ( 10b - (14 ( 2) × 102 a - (16 ( 3) × 102 a - (14 ( 1) × 102 a + 4.1 ( 1.7 + 5.9 ( 1.1 + 3.0 ( 0.5
+0.13 ( 0.02 - 0.58 ( 0.09 - 0.65 ( 0.11 - 0.59 ( 0.06
Calculated by using a theoretical radius. b Calculated by using the radius as determined with DLS.
Figure 6. Sedimentation velocity U versus the volume fraction φ of the magnetic silica dispersions FWS200St, FWS130St, and FWS40St dispersed in cyclohexane. The corresponding U0 and K values are given in Table 2. The solid lines are linear fits. Table 3. Magnetic Moments of the Colloidal Particles system
µtheora (10-19 A m2)
µmagnb (10-19 A m2)
µsedc (10-19 A m2)
MaS FF7 FFWa FWS40St FWS130St FWS200St
5.2 2.0 2.0 14 14 14
1.0 ( 0.2 2.8 ( 0.3 2.6 ( 0.3 2.3 ( 0.4 1.9 ( 0.4 1.7 ( 0.4
50 nm
behave as hard spheres. This is in good agreement with K ) 5.9 ( 1.1 as found for FWS130St. However for the system FWS200St a K value of 3.0 ( 0.5 was obtained. The significant difference with 6.55 cannot be explained by magnetic attractions and is most probably caused by van der Waals attraction.6 This also explains the K value of 3.5 ( 0.5 found for the silica dispersion SD250S, which contains relatively large particles with radius of 229.9 ( 9.7 nm. The dispersion FWS40St shows a slightly lower K value than that expected for hard spheres. This result is in disagreement with Figure 3 if it is assumed that the particles have a magnetic moment of 1.5 × 10-18 A m2 as calculated (Table 3). Electron micrographs show that the amount of magnetite per silica sphere is polydisperse. Possibly the sedimentation behavior is most strongly affected by the particles with a high magnetic moment, resulting in a lower K value than that predicted. So the results in section 4.1 suggest the interesting possibility that K has a maximum for a certain silica layer thickness. Below this thickness magnetic interaction decreases K, and for a larger shell thickness the van der Waals attraction becomes noticeable. This possibility is also consistent with the (reproducible) observation of colloidal crystal formation for FWS130St particles. For much smaller or much larger particles, slight attractions disturb the crystallization. 4.2. Sedimentation in a Centrifugal Field. To measure sedimentation velocities of small particles, an analytical ultracentrifuge was used. The sedimentation results for the magnetic fluids FF7-H2O, FF7-TMA, and FFWa are given in Figure 7 and Figure 8. When the normalized sedimentation velocity is plotted as a function of the iron concentration, the slope k (Table 2) is related to the coefficient K (eq 5) by
k)
KMMac 3MFeFp
(23)
with MMa the molar mass of magnetite and MFe the molar mass of iron. (The calculation of the weight concentration c and the particle density Fp is described in section 3.2.) In Figure 8 the normalized sedimentation velocity U/U0 is plotted against the volume fraction. The data points are fitted with linear fits and the corresponding slopes K are listed in Table 2. According to Table 2, U0 and K for the two FF7 series are almost the same. So the difference in TMA concentration did not affect the sedimentation behavior. Although the particles FFWa possess different surface properties, because a very thin silica layer is precipitated on the magnetic grains, the measured U0 and K are comparable with the values obtained for FF7. The sedimentation coefficient as a function of the iron concentration of the ferrofluid MaS, containing magnetite particles in 1-pentanol, can be found in Figure 9. The calculated value of k (eq 23) is given in Table 2. In this
6024 Langmuir, Vol. 13, No. 23, 1997
Donselaar et al.
Figure 7. Sedimentation coefficient S versus the iron concentration for FF7-H2O, FF7 TMA, and FFWa. The dispersions FF7-H2O were made by diluting a concentrated stock solution with double distilled water. The dispersions FF7-TMA were made by diluting a concentrated stock solution with a mixture of double distilled water and TMA with pH ) 10. FFWa contains magnetite particles with a very thin silica layer. The corresponding U0 and K values are given in Table 2. The solid line is the result of a linear fit.
table two K values can be found for MaS one which is calculated by using the estimated radius (section 3.2) and the other with the measured DLS radius. But although two different K values are obtained for MaS, they are still both strongly positive in contrast to FF7 where a strongly negative K was found. In Figure 10 the normalized sedimentation velocity U/U0 is given as a function of the volume fraction. It is striking that K is positive, in contrast with the strongly negative value found for the aqueous ferrofluid FF7. Several reasons could explain the result. Firstly the magnetic moment per particle is much smaller than that found for FF7 (Table 3). Secondly, the screening layer thickness of MaS is larger because it consists of a double surfactant layer where both surfactants are large in comparison to the TMA ion. However, the repulsive barrier of the surfactant layer is not sufficient to explain the measured K ≈ 100, because even if the particle (plus surfactant layer) would act as hard spheres the coefficient is only K ≈ 6. The large K value very likely is due to charge effects. Experimental17 and theoretical work18,27 showed that the sedimentation velocity of charged particles at low ionic strength is given by
U/U0 ) 1 - pφa
(24)
with p a positive constant and 0 < R < 1. In deionized suspensions the sedimentation velocity shows a φ1/3 dependence. Electrophoreses demonstrates that the MaS particles carry charges. The electrophoretic mobility is (27) Na¨gele, G. Phys. Rep. 1996, 272, 216-372.
Figure 8. Normalized sedimentation velocity U/U0 as a function of the volume fraction φ of the aqueous magnetic dispersions FF7-H2O, FF7-TMA, and FFWa (see caption of Figure 7). The solid line is the result of a linear fit.
Figure 9. Sedimentation coefficient S as a function of the iron concentration of the ferrofluid MaS in 1-pentanol. The solid line is the result of a linear fit.
17 × 10-10 m2/(V s) and the zeta potential (eq 19) is 66 mV, corresponding to a number of elementary charges per sphere of order 1. The data do not allow a conclusion whether eq 24 applies. Note that a fit of eq 24 with a linear relation produces a high apparent K value. Also the extrapolation to infinite dilution is uncertain, because dU/dφ in eq 24 diverges at φ f 0. (This probably accounts for the difference in calculated and measured U0 for MaS in Table 2.) Fitting the results with a linear fit when R ) 1/3 K values larger than 100 can be obtained at volume fractions smaller than 1 × 10-3. The dispersions FF7 and FFWa have a high ionic strength; hence the thickness of the electric double layer is small and R ) 1, so the results can be fitted with a linear fit.17 In Table 3 three magnetic moments, obtained by
Concentration-Dependent Sedimentation
Figure 10. Normalized sedimentation velocity (U/U0) against the volume fraction of the ferrofluid MaS in 1-pentanol. The solid line is the result of a linear fit.
calculation (µtheor), by measuring with magnetometer (µmag), and by using K and the total radius b of the particles in Figure 3 (µsed) are given. The radius b is the radius as estimated in section 3.2. For the calculation of µtheor for the systems MaS, FF7, and FFWa the bulk magnetization for magnetite and the radius as measured with TEM are used. For FF7 and FFWa the magnetic moments µmagn and µsed are in good agreement. As can be seen in Figure 3, for particles with a radius of about 5 nm, a very small difference in magnetic moment gives rise to a strong increase or decrease of the sedimentation coefficient K. 5. Summary and Conclusions The thickness of the nonmagnetic shell is very important for the sedimentation behavior of the investigated coreshell colloids. Magnetic silica particles with a radius of about 100 nm show hard sphere behavior. Particles with radii larger than 150 nm show a weaker concentration dependence of the sedimentation velocity probably as a result of van der Waals forces. For magnetic silica colloids with a radius smaller than 40 nm, weakening of the concentration dependence of the sedimentation velocity is caused by magnetic attraction. The sedimentation velocity of the aqueous ferrofluid (FF7) increases as a function of the concentration. Such
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an effect has not been observed before for inorganic colloidal particles. It reminds one of the sedimentation velocity with concentration for self-associating species like β-lactoglobulin A,28 though these are polymers with a high flexibility. A negative value of K does not imply that the dispersion is unstable. The ferrofluid FF7 is stable for years, and no settling of flocs or aggregation is observed, although after ultracentrifugation it was not possible to redisperse the sediment again, indicating that as a result of the high centrifugal force the particles are irreversible aggregated. This suggests that the FF7 particles form reversible aggregates in a weak secondary minimum and centrifugation causes irreversible flocculation in a primary minimum. There is no typical sedimentation behavior for magnetic fluids as is shown by the ferrofluid MaS, which contains magnetite particles stabilized with a double surfactant layer in pentanol. Instead of a negative sedimentation coefficient K, as found for the aqueous ferrofluid FF7, a high positive value is obtained which very likely is the result of some surface charge. This seems to contradict the common belief that nonaqueous magnetic fluids are only sterically stabilized; a few charges may generate longrange double layer forces because the dielectric constant is low and κ-1 is large. The observed sedimentation behavior of the magnetic particles can be described qualitatively by the equations of Batchelor3 and Dhont.18 Sedimentation is a very sensitive to additional (van der Waals) attractions or charge on the particles, which complicates the sedimentation analysis. Acknowledgment. W. Wijting is thanked for synthesizing system FWS40St and for performing sedimentation measurements. The ferrofluid MaS is a gift from Dr. D. Bica and Professor L. Vekas (University of Timisoara, Romania). Dr. J. Dhont and Dr. P. Scholten are thanked for helpful discussions. LA970359+ (28) Adams, E. T., Jr. In Analytical ultracentrifugation in biochemistry and polymer science; Harding, S. E., Rowe, A. J., Horton, J. C., Eds.; RSC: Cambridge, 1992; Chapter 22.
