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Forced Rayleigh scattering experiments in concentrated magnetic fluids: effect of interparticle interactions on the diffusion coefficient
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 289 (2005) 39–42 www.elsevier.com/locate/jmmm
Forced Rayleigh scattering experiments in concentrated magnetic fluids: effect of interparticle interactions on the diffusion coefficient Guillaume Me´rigueta,, Emmanuelle Duboisa, Alain Bourdonb, Gilles Demouchyb, Vincent Dupuisb, Re´gine Perzynskib a
Laboratoire LI2C, Universite´ Pierre et Marie Curie, UMR CNRS 7612, case 51, 4 place Jussieu, 75252 Paris, Cedex 05, France b Laboratoire LMDH , Universite´ Pierre et Marie Curie, UMR CNRS 7603, 140 rue de Lourmel, 75015 Paris, France Available online 25 November 2004
Abstract The structure and dynamics of concentrated aqueous magnetic fluids with tunable interparticle interaction are investigated by small angle neutron scattering and forced Rayleigh scattering. The structure factor and the collective diffusion coefficient are determined. An effect of the interparticle interaction on the diffusion coefficient is observed and related to thermodynamic data with a good agreement. r 2004 Elsevier B.V. All rights reserved. PACS: 75.50.Mm; 82.70.Dd; 42.65.Es; 28.20.Cz Keywords: Magnetic liquids; Colloids; Forced Rayleigh scattering; Small angle neutron scattering
1. Introduction In the framework of a general study on concentrated magnetic colloids with well-defined isotropic repulsion and anisotropic dipolar interactions, we present here translational diffusion measurements. While the static structural properties of such concentrated dispersions, with and without an applied magnetic field, have been explored in detail in [1,2], their dynamical properties are less investigated. The magnetic fluids studied here are constituted of maghemite nanoparticles g-Fe2O3 coated with citrate molecules that ensure them a negative superficial charge. They are dispersed in water at pH 7 with a volume Corresponding author. Tel.: +33 144 27 3118; fax: +33 144 27 3834. E-mail address: [email protected] (G. Me´riguet).
fraction F ranging from 1% up to 25%. The particle diameters d follow a lognormal distribution characterized here by d 0 ¼ ðhln d iÞ ¼ 9:8 nm and s ¼ 0:25: The dipolar parameter g=F ¼ m0 m2s V =kT that quantifies the dipolar interactions between two particles compared to thermal energy, is 32 for this ferrofluid. In these systems, for a fixed g=F ratio, the interparticle interactions can be tuned by changing the osmotic pressure P of the suspensions. The latter is controlled by the ionic strength of the solution. Therefore, a phase diagram P–F [1] determines the structure and the properties of the dispersions. For high pressures, following a line at constant ionic strength, the dispersion is fluid, up to a limiting volume fraction above which the sample becomes solid. The structure factors S(q) plotted on Fig. 1 are obtained by small angle neutron scattering experiments (SANS) [1]. They are the counterparts of the radial distribution
0304-8853/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2004.11.012
ARTICLE IN PRESS 40
G. Me´riguet et al. / Journal of Magnetism and Magnetic Materials 289 (2005) 39–42 2
1% 4.90% 9.70% 15.80% 20.30% 25%
1.8 1.6
S (q)
1.4 1.2 1 0.8 0.6 0.4 0.2 0 0
0.02
0.04
0.06
q
0.08
0.1
° −1 (A )
Fig. 1. Structure factors S versus the scattering vector q for [citrate] ¼ 0.03 mol L1 and increasing volume fractions (SANS experiments, ILL, Grenoble, France).
functions in the reciprocal space. The value of S at q ¼ 0 is related to the compressibility of the suspension and the maximum gives the most probable distance dmp between particles (qmax ¼ 2p=dmp ). In this study, the compressibility is low and dmp F1=3 ; which means that the interparticle interaction is strongly repulsive. All the samples are fluids apart from the most concentrated one (F ¼ 25%) which is a solid. Yet, due to the remaining polydispersity, the solid is a glass, not a crystal, i.e. it presents an amorphous structure. While increasing F in the fluid phase, a drastic evolution of the rotational diffusion time trot can be measured by magneto-optical birefringence relaxation: this time is increased by 9 orders of magnitude while increasing the volume fraction toward the solid phase [3]! This behavior is also characteristic of a glass transition. For low pressures, following a line at constant ionic strength and increasing the volume fraction from low values, the dispersion—at first similar to a gas of nanoparticles—undergoes a liquid-gas transition (associated to a critical point). At higher F; the dispersion is then similar to a liquid (monophasic) and becomes a solid above a given threshold of volume fraction. In this area, the interactions are globally attractive, and the solid is there an attractive glass [1]. In the present paper, forced Rayleigh scattering is used to measure the collective translational diffusion coefficient Dtr of the nanoparticles along lines at constant ionic strength in the fluid area, i.e. for high osmotic pressures and globally repulsive interactions.
2. Experimental details The principle of the Rayleigh scattering is the following. Spatial modulations of concentration are induced by thermodiffusion (Ludwig–Soret effect) inside the colloid thanks to the standing interference fringes of two coherent laser beams at l ¼ 532 nm: Indeed, a
temperature grating arises due to the strong optical absorbance of the sample [4–7]. A He–Ne laser probe beam is diffracted by the spatial modulations of the optical index in the sample. The diffusion coefficient Dtr is deduced from the relaxation of the first-order intensity of the diffraction pattern when the two pump beams are switched off. In the citrated ferrofluids considered here, the Soret coefficient is negative, i.e. the nanoparticles move toward hot regions [5–7]. This technique is specific to ferrofluids: due to their strong Soret coefficient, it is possible to measure Dtr in concentrated samples, which is not possible, due to light absorption, with standard dynamic light scattering for example. In the present study, the sample is put in a 10 mm thick quartz cell. Neutron spin echo measurements can also be used, however it is a much more complicated and heavy experiment which probes more local scales [8]. Forced Rayleigh scattering allows measuring the relaxation time ttr of the concentration modulations for several scattering vectors q. The relaxation of the thermal modulations has also been measured [5–7] and is 103 faster. Consequently, the two phenomena can be decorrelated. Practically q is calculated from the spatial interfringe of the interference pattern. The quantity 1/ttr is found to be proportional to q2, which means that the motion is diffusive. The slope associated to this linear variation gives the diffusion coefficient Dtr.
3. Results and discussion To examine the effect of interparticle interaction on the collective diffusion coefficient, the interaction is tuned by adjusting the ionic strength and the volume fraction F of the nanoparticles. At low ionic strength ([citrate] ¼ 0.003 mol L1— strongly repulsive interparticle potential U) no spatial modulation of concentration is detected, in good agreement with recent theoretical works [9,10]. Indeed,
ARTICLE IN PRESS G. Me´riguet et al. / Journal of Magnetism and Magnetic Materials 289 (2005) 39–42
41
Table 1 Ionic strength and thermodynamic coefficients determined by SANS and FRS
Fig. 2. Translational diffusion coefficient obtained from forced Rayleigh scattering experiments versus F for two ionic strengths. The hard sphere limit is indicated as a reference as well as the value obtained from neutron spin echo experiments for this sample (’).
the experiment is sensitive to mobility under temperature gradients. It depends on the interaction potential U, both through its derivative with respect to temperature (thermodiffusion term) and to volume fraction (mass diffusion term). For strongly repulsive potentials U, the variation of U with volume fraction is strong and masks the effects due to temperature: no modulation of concentration is induced. For higher ionic strengths (here [citrate]X0.03 mol L1), the measurements of the diffusion coefficient Dtr is possible and Dtr increases with F (see Fig. 2). At low volume fractions, the collective translational diffusion coefficient can be approximated to Dtr ðFÞ ¼ D0 ð1 þ ðkT -kF ÞFÞ;
(1)
where
Citrate concentration (mol L1)
kT (SANS)
kT (FRS)
0.003 0.03 0.08
96710 3073 1672
— 3278 1773
value of kT the more repulsive the interaction potential U. For hard spheres, kT equals 8 leading to Dtr ðFÞ ¼ D0 ð1 þ 1:45FÞ [12]. This ideal behavior is plotted on Fig. 2 as a solid line. The noticeable difference between hard spheres and the two experimental data series with different ionic strengths is due to the increase of the collective diffusion coefficient with the repulsion range and intensity. For the experimental series, assuming the Dtr is linear with F; the slopes, i.e. kT–kF, are close to the values expected using SANS data for the second virial coefficient and assuming a hard sphere behavior for the friction term despite a poor experimental accuracy on the diffusion coefficient values. Besides, the samples studied here are concentrated, thus they may lay out of the linear regime range which would explain the deviation between the coefficients. In conclusion, an effect of the interaction potential variation through the ionic strength on the collective diffusion coefficient is evidenced by thermal diffusion forced Rayleigh scattering on ferrofluids. The magnitude of this effect is in good agreement with the second virial coefficient determined by SANS experiments for high osmotic pressure suspensions. A complementary study on low osmotic pressure suspensions is currently carried out. In addition, the understanding of the interparticle interaction influence on the Soret coefficient would be of interest.
D0 is the diffusion coefficient at infinite dilution which
can be obtained from spin echo experiments (D0 ¼ 2.0
1011 m2 s1) at a much more local scale [11]. kT and kF account, respectively, for thermodynamic and hydrodynamic interactions. kF is a friction coefficient evaluated to 6.55 for hard spheres [12]. kT is the dimensionless second virial coefficient that can be derived by small angle neutron scattering for small volume fractions through the structure factor value in the thermodynamic limit (q-0) according to 1=Sðq ! 0; FÞ ¼ 1 þ kT F:
(2)
The second virial coefficients determined by SANS at ILL are listed in Table 1 below. The greater the
References [1] F. Cousin, E. Dubois, V. Cabuil, Phys. Rev. E 68 (2003) 021405. [2] F. Gazeau, E. Dubois, F. Boue´, R. Perzynski, Phys. Rev. E 65 (2002) 031043. [3] G. Me´riguet, E. Dubois, V. Dupuis, R. Perzynski, Third International Symposium on Slow Dynamics in Complex Systems, AIP Conf. Proc. 708 (2003) 124. [4] J.C. Bacri, A. Cebers, A. Bourdon, G. Demouchy, B.M. Heegaard, B. Kashevsky, R. Perzynski, Phys. Rev. E 52 (4) (1995) 3936. [5] J. Lenglet, A. Bourdon, J.-C. Bacri, G. Demouchy, Phys. Rev. E 65 (2002) 031408.
ARTICLE IN PRESS 42
G. Me´riguet et al. / Journal of Magnetism and Magnetic Materials 289 (2005) 39–42
[6] S. Alves, G. Demouchy, A. Be´e, D. Talbot, A. Bourdon, A.M. Figueiredo Neto, Philos. Mag. 83 (17–18) (2003) 2059. [7] G. Demouchy, A. Mezulis, A. Be´e, D. Talbot, J.-C. Bacri, A. Bourdon, J. Phys. D: Appl. Phys. 37 (2004) 1417. [8] F. Gazeau, F. Boue´, E. Dubois, R. Perzynski, J. Phys. Condens. Matter. 15 (2003) S1305.
[9] E. Bringuier, A. Bourdon, Phys. Rev. E 67 (2003) 0114041. [10] J.K.G. Dhont, J. Chem. Phys. 120 (3) (2004) 1632. [11] G. Me´riguet, E. Dubois, V. Dupuis, B. Farago, R. Perzynski, to be published. [12] G.K. Batchelor, J. Fluid Mech. 52 (1972) 245; G.K. Batchelor, J. Fluid Mech. 74 (1976) 1.
Journal of Magnetism and Magnetic Materials 289 (2005) 39–42 www.elsevier.com/locate/jmmm
Forced Rayleigh scattering experiments in concentrated magnetic fluids: effect of interparticle interactions on the diffusion coefficient Guillaume Me´rigueta,, Emmanuelle Duboisa, Alain Bourdonb, Gilles Demouchyb, Vincent Dupuisb, Re´gine Perzynskib a
Laboratoire LI2C, Universite´ Pierre et Marie Curie, UMR CNRS 7612, case 51, 4 place Jussieu, 75252 Paris, Cedex 05, France b Laboratoire LMDH , Universite´ Pierre et Marie Curie, UMR CNRS 7603, 140 rue de Lourmel, 75015 Paris, France Available online 25 November 2004
Abstract The structure and dynamics of concentrated aqueous magnetic fluids with tunable interparticle interaction are investigated by small angle neutron scattering and forced Rayleigh scattering. The structure factor and the collective diffusion coefficient are determined. An effect of the interparticle interaction on the diffusion coefficient is observed and related to thermodynamic data with a good agreement. r 2004 Elsevier B.V. All rights reserved. PACS: 75.50.Mm; 82.70.Dd; 42.65.Es; 28.20.Cz Keywords: Magnetic liquids; Colloids; Forced Rayleigh scattering; Small angle neutron scattering
1. Introduction In the framework of a general study on concentrated magnetic colloids with well-defined isotropic repulsion and anisotropic dipolar interactions, we present here translational diffusion measurements. While the static structural properties of such concentrated dispersions, with and without an applied magnetic field, have been explored in detail in [1,2], their dynamical properties are less investigated. The magnetic fluids studied here are constituted of maghemite nanoparticles g-Fe2O3 coated with citrate molecules that ensure them a negative superficial charge. They are dispersed in water at pH 7 with a volume Corresponding author. Tel.: +33 144 27 3118; fax: +33 144 27 3834. E-mail address: [email protected] (G. Me´riguet).
fraction F ranging from 1% up to 25%. The particle diameters d follow a lognormal distribution characterized here by d 0 ¼ ðhln d iÞ ¼ 9:8 nm and s ¼ 0:25: The dipolar parameter g=F ¼ m0 m2s V =kT that quantifies the dipolar interactions between two particles compared to thermal energy, is 32 for this ferrofluid. In these systems, for a fixed g=F ratio, the interparticle interactions can be tuned by changing the osmotic pressure P of the suspensions. The latter is controlled by the ionic strength of the solution. Therefore, a phase diagram P–F [1] determines the structure and the properties of the dispersions. For high pressures, following a line at constant ionic strength, the dispersion is fluid, up to a limiting volume fraction above which the sample becomes solid. The structure factors S(q) plotted on Fig. 1 are obtained by small angle neutron scattering experiments (SANS) [1]. They are the counterparts of the radial distribution
0304-8853/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2004.11.012
ARTICLE IN PRESS 40
G. Me´riguet et al. / Journal of Magnetism and Magnetic Materials 289 (2005) 39–42 2
1% 4.90% 9.70% 15.80% 20.30% 25%
1.8 1.6
S (q)
1.4 1.2 1 0.8 0.6 0.4 0.2 0 0
0.02
0.04
0.06
q
0.08
0.1
° −1 (A )
Fig. 1. Structure factors S versus the scattering vector q for [citrate] ¼ 0.03 mol L1 and increasing volume fractions (SANS experiments, ILL, Grenoble, France).
functions in the reciprocal space. The value of S at q ¼ 0 is related to the compressibility of the suspension and the maximum gives the most probable distance dmp between particles (qmax ¼ 2p=dmp ). In this study, the compressibility is low and dmp F1=3 ; which means that the interparticle interaction is strongly repulsive. All the samples are fluids apart from the most concentrated one (F ¼ 25%) which is a solid. Yet, due to the remaining polydispersity, the solid is a glass, not a crystal, i.e. it presents an amorphous structure. While increasing F in the fluid phase, a drastic evolution of the rotational diffusion time trot can be measured by magneto-optical birefringence relaxation: this time is increased by 9 orders of magnitude while increasing the volume fraction toward the solid phase [3]! This behavior is also characteristic of a glass transition. For low pressures, following a line at constant ionic strength and increasing the volume fraction from low values, the dispersion—at first similar to a gas of nanoparticles—undergoes a liquid-gas transition (associated to a critical point). At higher F; the dispersion is then similar to a liquid (monophasic) and becomes a solid above a given threshold of volume fraction. In this area, the interactions are globally attractive, and the solid is there an attractive glass [1]. In the present paper, forced Rayleigh scattering is used to measure the collective translational diffusion coefficient Dtr of the nanoparticles along lines at constant ionic strength in the fluid area, i.e. for high osmotic pressures and globally repulsive interactions.
2. Experimental details The principle of the Rayleigh scattering is the following. Spatial modulations of concentration are induced by thermodiffusion (Ludwig–Soret effect) inside the colloid thanks to the standing interference fringes of two coherent laser beams at l ¼ 532 nm: Indeed, a
temperature grating arises due to the strong optical absorbance of the sample [4–7]. A He–Ne laser probe beam is diffracted by the spatial modulations of the optical index in the sample. The diffusion coefficient Dtr is deduced from the relaxation of the first-order intensity of the diffraction pattern when the two pump beams are switched off. In the citrated ferrofluids considered here, the Soret coefficient is negative, i.e. the nanoparticles move toward hot regions [5–7]. This technique is specific to ferrofluids: due to their strong Soret coefficient, it is possible to measure Dtr in concentrated samples, which is not possible, due to light absorption, with standard dynamic light scattering for example. In the present study, the sample is put in a 10 mm thick quartz cell. Neutron spin echo measurements can also be used, however it is a much more complicated and heavy experiment which probes more local scales [8]. Forced Rayleigh scattering allows measuring the relaxation time ttr of the concentration modulations for several scattering vectors q. The relaxation of the thermal modulations has also been measured [5–7] and is 103 faster. Consequently, the two phenomena can be decorrelated. Practically q is calculated from the spatial interfringe of the interference pattern. The quantity 1/ttr is found to be proportional to q2, which means that the motion is diffusive. The slope associated to this linear variation gives the diffusion coefficient Dtr.
3. Results and discussion To examine the effect of interparticle interaction on the collective diffusion coefficient, the interaction is tuned by adjusting the ionic strength and the volume fraction F of the nanoparticles. At low ionic strength ([citrate] ¼ 0.003 mol L1— strongly repulsive interparticle potential U) no spatial modulation of concentration is detected, in good agreement with recent theoretical works [9,10]. Indeed,
ARTICLE IN PRESS G. Me´riguet et al. / Journal of Magnetism and Magnetic Materials 289 (2005) 39–42
41
Table 1 Ionic strength and thermodynamic coefficients determined by SANS and FRS
Fig. 2. Translational diffusion coefficient obtained from forced Rayleigh scattering experiments versus F for two ionic strengths. The hard sphere limit is indicated as a reference as well as the value obtained from neutron spin echo experiments for this sample (’).
the experiment is sensitive to mobility under temperature gradients. It depends on the interaction potential U, both through its derivative with respect to temperature (thermodiffusion term) and to volume fraction (mass diffusion term). For strongly repulsive potentials U, the variation of U with volume fraction is strong and masks the effects due to temperature: no modulation of concentration is induced. For higher ionic strengths (here [citrate]X0.03 mol L1), the measurements of the diffusion coefficient Dtr is possible and Dtr increases with F (see Fig. 2). At low volume fractions, the collective translational diffusion coefficient can be approximated to Dtr ðFÞ ¼ D0 ð1 þ ðkT -kF ÞFÞ;
(1)
where
Citrate concentration (mol L1)
kT (SANS)
kT (FRS)
0.003 0.03 0.08
96710 3073 1672
— 3278 1773
value of kT the more repulsive the interaction potential U. For hard spheres, kT equals 8 leading to Dtr ðFÞ ¼ D0 ð1 þ 1:45FÞ [12]. This ideal behavior is plotted on Fig. 2 as a solid line. The noticeable difference between hard spheres and the two experimental data series with different ionic strengths is due to the increase of the collective diffusion coefficient with the repulsion range and intensity. For the experimental series, assuming the Dtr is linear with F; the slopes, i.e. kT–kF, are close to the values expected using SANS data for the second virial coefficient and assuming a hard sphere behavior for the friction term despite a poor experimental accuracy on the diffusion coefficient values. Besides, the samples studied here are concentrated, thus they may lay out of the linear regime range which would explain the deviation between the coefficients. In conclusion, an effect of the interaction potential variation through the ionic strength on the collective diffusion coefficient is evidenced by thermal diffusion forced Rayleigh scattering on ferrofluids. The magnitude of this effect is in good agreement with the second virial coefficient determined by SANS experiments for high osmotic pressure suspensions. A complementary study on low osmotic pressure suspensions is currently carried out. In addition, the understanding of the interparticle interaction influence on the Soret coefficient would be of interest.
D0 is the diffusion coefficient at infinite dilution which
can be obtained from spin echo experiments (D0 ¼ 2.0
1011 m2 s1) at a much more local scale [11]. kT and kF account, respectively, for thermodynamic and hydrodynamic interactions. kF is a friction coefficient evaluated to 6.55 for hard spheres [12]. kT is the dimensionless second virial coefficient that can be derived by small angle neutron scattering for small volume fractions through the structure factor value in the thermodynamic limit (q-0) according to 1=Sðq ! 0; FÞ ¼ 1 þ kT F:
(2)
The second virial coefficients determined by SANS at ILL are listed in Table 1 below. The greater the
References [1] F. Cousin, E. Dubois, V. Cabuil, Phys. Rev. E 68 (2003) 021405. [2] F. Gazeau, E. Dubois, F. Boue´, R. Perzynski, Phys. Rev. E 65 (2002) 031043. [3] G. Me´riguet, E. Dubois, V. Dupuis, R. Perzynski, Third International Symposium on Slow Dynamics in Complex Systems, AIP Conf. Proc. 708 (2003) 124. [4] J.C. Bacri, A. Cebers, A. Bourdon, G. Demouchy, B.M. Heegaard, B. Kashevsky, R. Perzynski, Phys. Rev. E 52 (4) (1995) 3936. [5] J. Lenglet, A. Bourdon, J.-C. Bacri, G. Demouchy, Phys. Rev. E 65 (2002) 031408.
ARTICLE IN PRESS 42
G. Me´riguet et al. / Journal of Magnetism and Magnetic Materials 289 (2005) 39–42
[6] S. Alves, G. Demouchy, A. Be´e, D. Talbot, A. Bourdon, A.M. Figueiredo Neto, Philos. Mag. 83 (17–18) (2003) 2059. [7] G. Demouchy, A. Mezulis, A. Be´e, D. Talbot, J.-C. Bacri, A. Bourdon, J. Phys. D: Appl. Phys. 37 (2004) 1417. [8] F. Gazeau, F. Boue´, E. Dubois, R. Perzynski, J. Phys. Condens. Matter. 15 (2003) S1305.
[9] E. Bringuier, A. Bourdon, Phys. Rev. E 67 (2003) 0114041. [10] J.K.G. Dhont, J. Chem. Phys. 120 (3) (2004) 1632. [11] G. Me´riguet, E. Dubois, V. Dupuis, B. Farago, R. Perzynski, to be published. [12] G.K. Batchelor, J. Fluid Mech. 52 (1972) 245; G.K. Batchelor, J. Fluid Mech. 74 (1976) 1.