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Investigation of Brownian and Néel relaxation in magnetic fluids
Journal of Magnetism and Magnetic Materials 201 (1999) 102}104
Investigation of Brownian and NeH el relaxation in magnetic #uids R. KoK titz *, W. Weitschies , L. Trahms, W. Semmler Institut fu( r Diagnostikforschung GmbH an der Freien Universita( t Berlin, Spandauer Damm 130, D-14050 Berlin, Germany Physikalisch-Technische Bundesanstalt, Abbestra}e 2 - 12, D-10587 Berlin, Germany Received 20 May 1998; received in revised form 4 September 1998
Abstract Relaxation of magnetic #uids was studied at room temperature with a DC-SQUID-magnetometer in a time window ranging from a few milliseconds to seconds. Freeze-dried samples and unstable liquid samples with sediments showed NeH el relaxation signals which are in excellent accordance with the time dependence expected from theory. The Brownian relaxation of a stable ferro#uid was shorter than the observed time window. The relaxation of a stable magnetic #uid with increased viscosity is well described by a superposition of two exponential decays. The di!erent time dependencies of the NeH el and Brownian relaxation processes can be utilised to determine the type of relaxation process and hence allow conclusions on the mobility of the particles. 1999 Elsevier Science B.V. All rights reserved. Keywords: Brownian relaxation; NeH el relaxation
1. Introduction In the last decades relaxation of magnetic #uids was investigated in the frequency [1] and the time [2,3] domain. Recently, relaxation processes in magnetic #uids have been utilised for the sensitive binding speci"c quanti"cation of biological binding processes [4]. In this novel application the relaxation signal of magnetic nanoparticles, attached as labels to biological substances, is measured in the time domain using superconducting quantum interference devices (SQUIDs). For the interpretation
* Corresponding author. Fax: #49-30-30390499. E-mail address: [email protected] (R. KoK titz)
of complex relaxation signals obtained from such measurements an adequate understanding of the basic relaxation processes in magnetic #uids is needed. Here we investigate the relaxation of magnetic nanoparticles in the solid phase and in solution under various experimental conditions.
2. Theoretical background The magnetisation of ferro#uids can relax by two di!erent mechanisms, the so-called NeH el relaxation and the Brownian relaxation. NeH el relaxation is caused by the reorientation of the magnetisation vector inside the magnetic core against an energy barrier [5]. Brownian relaxation is due to
0304-8853/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 0 6 5 - 7
R. Ko( titz et al. / Journal of Magnetism and Magnetic Materials 201 (1999) 102}104
rotational di!usion of the whole particle in the carrier liquid [6]. The time constant of NeH el relaxation depends much stronger on the particle diameter than the Brownian relaxation time. The e!ective relaxation time is dominated by the faster relaxation process. While both, NeH el and Brownian relaxation of an ensemble of identical particles are described simply by an exponential decay with time, relaxation in real systems is usually nonexponential, due to the distribution of sizes and shapes of the particles. Using the so-called critical volume approximation, Chantrell et al. [7] derived a formula for the time-dependent magnetisation of a system of magnetic nanoparticles after magnetisation in a "eld H for a time t :
t (1) M(t)"C ln 1# , t
where the characteristic time t depends on t
and H . For the case of weak magnetising "elds
[8] Berkov et al. obtained in a generalised theoretical approach (1) with t "t which was con
"rmed experimentally [8]. A compact formula for Brownian relaxation of a system with a distribution of hydrodynamic particle sizes is not known.
103
4. Results and discussion Fig. 1 shows excellent agreement between the relaxation signal of the freeze dried sample 3 containing 100 lmol Fe after magnetisation in a weak "eld of 8 A/m for 10 s and the theoretical time dependence "tted to (1) with t "t . The
relaxation measured for sample 4 (iron content: 100 nmol) magnetised with a stronger "eld of 1.6 kA/m is depicted in the inset of Fig. 1. Obviously, this relaxation signal is not well described by a "t using (1) with t "t . This suggests, that in
this case the assumption of a weak magnetising "eld is not valid. Considering t as a free parameter a good "t to the experimental data is obtained with t "7t thus con"rming [7]. This indicates, that
the relaxation after magnetisation with a strong "eld resembles the relaxation of a sample magnetised in a decreased "eld for a longer time. The relaxation signal of the unstable liquid sample 1 magnetised with 400 A/m is shown in Fig. 2. A curve "t according to Eq. (1) yields t " 1.5t . The intermediate value for t corresponds
3. Method and materials Relaxation measurements at room temperature starting a few milliseconds after switching o! a magnetising "eld were performed with a DCSQUID-magnetometer in a magnetically shielded environment using a specially designed twin coil for the magnetisation of the sample [8]. We investigated a water based magnetic #uid of magnetic iron oxide nanoparticles with a mean core diameter of approximately 10 nm (standard deviation of lognormal distribution p+0.5) and a mean hydrodynamic diameter of approximately 60 nm. This #uid (sample 0) was used as the basis for a number of modi"ed samples: an unstable sample 1 accomplished by chemically induced aggregation and subsequent sedimentation of the particles; a high viscosity sample 2 with a solution of 87% glycerol in water as carrier liquid and freeze dried samples 3 and 4 of di!erent concentrations for relaxation measurements of nanoparticles in a solid phase.
Fig. 1. Relaxation signal of sample 3 after magnetisation with 8 A/m for 10 s and curve "t according to Eq. (1) with t " t . *B(100 s) was set to zero. Inset: Relaxation signal of
sample 4 after magnetisation with 1.6 kA/m for 10 s and curve "ts according to Eq. (1) with t "t and t "7t .
104
R. Ko( titz et al. / Journal of Magnetism and Magnetic Materials 201 (1999) 102}104
of two exponential decays (q "0.25 s, q "1.12 s) "ts the measurement well. The relaxation of the freeze dried sample 4 is not well described by a double exponential decay.
5. Conclusions and summary
Fig. 2. Relaxation signal of the unstable sample 1 with sediment after magnetisation with 400 A/m for 10 s and curve "t according to Eq. (1). Inset: Relaxation signal and various "ts for the glycerol based sample 2 magnetised using the same parameters.
to the intermediate magnetising "eld. The good agreement between "t and measured data indicates that the signal is due to NeH el relaxation, caused by aggregated particles forming the sediment. For the stable #uid of sample 0 no relaxation signal could be detected. This is due to the hydrodynamic particle size of the suspended particles, which yields an estimated Brownian relaxation time less than 100 ls and is much shorter than the starting time of the measurement (200 ms). In the inset of Fig. 2 the relaxation signal of sample 2 is plotted against a logarithmic time scale. Evidently, Eq. (1) does not describe the relaxation of sample 2, also a curve "t using a single exponential decay is unsatisfactory, but yields a rough estimate of the mean Brownian relaxation time of 0.52 s. However, a superposition
The measurements indicate that the time dependencies of Brownian and NeH el relaxation are clearly di!erent. This is due to the narrow distribution of Brownian relaxation times compared to the extremely wide distribution of NeH el-relaxation times. The presence of a NeH el relaxation signal gives evidence that at least for a fraction of particles the Brownian mechanism is inhibited. This is the case e.g. for magnetic nanoparticles "xed to a solid phase or unstable aggregated magnetic #uids containing sediments. Since the relaxation signal yields information about the underlying relaxation process it can consequently be utilised to distinguish between particles bound to a solid phase and free particles. This is an important feature for the analysis of biological binding processes [4].
References [1] P.C. Fannin, S.W. Charles, J. Phys. D 22 (1989) 187. [2] H. Bogardus, D.A. Krueger, D. Thompson, J. Appl. Phys. 49 (6) (1978) 3422. [3] R. KoK titz, P.C. Fannin, L. Trahms, J. Magn. Magn. Mater. 149 (1995) 42. [4] W. Weitschies, R. KoK titz, T. Bunte, L. Trahms, Pharm. Pharmacol. Lett. 7 (1) (1997) 5. [5] L. NeH el, Ann. Geophys. 5 (1949) 99. [6] P. Debye, Polar Molecules, Chemical Catalog Company, New York, 1929. [7] R.W. Chantrell, S.R. Hoon, B.K. Tanner, J. Magn. Magn. Mater. 38 (1983) 133. [8] D.V. Berkov, R. KoK titz, J. Phys.: Condens. Matter 8 (1996) 1257.
Investigation of Brownian and NeH el relaxation in magnetic #uids R. KoK titz *, W. Weitschies , L. Trahms, W. Semmler Institut fu( r Diagnostikforschung GmbH an der Freien Universita( t Berlin, Spandauer Damm 130, D-14050 Berlin, Germany Physikalisch-Technische Bundesanstalt, Abbestra}e 2 - 12, D-10587 Berlin, Germany Received 20 May 1998; received in revised form 4 September 1998
Abstract Relaxation of magnetic #uids was studied at room temperature with a DC-SQUID-magnetometer in a time window ranging from a few milliseconds to seconds. Freeze-dried samples and unstable liquid samples with sediments showed NeH el relaxation signals which are in excellent accordance with the time dependence expected from theory. The Brownian relaxation of a stable ferro#uid was shorter than the observed time window. The relaxation of a stable magnetic #uid with increased viscosity is well described by a superposition of two exponential decays. The di!erent time dependencies of the NeH el and Brownian relaxation processes can be utilised to determine the type of relaxation process and hence allow conclusions on the mobility of the particles. 1999 Elsevier Science B.V. All rights reserved. Keywords: Brownian relaxation; NeH el relaxation
1. Introduction In the last decades relaxation of magnetic #uids was investigated in the frequency [1] and the time [2,3] domain. Recently, relaxation processes in magnetic #uids have been utilised for the sensitive binding speci"c quanti"cation of biological binding processes [4]. In this novel application the relaxation signal of magnetic nanoparticles, attached as labels to biological substances, is measured in the time domain using superconducting quantum interference devices (SQUIDs). For the interpretation
* Corresponding author. Fax: #49-30-30390499. E-mail address: [email protected] (R. KoK titz)
of complex relaxation signals obtained from such measurements an adequate understanding of the basic relaxation processes in magnetic #uids is needed. Here we investigate the relaxation of magnetic nanoparticles in the solid phase and in solution under various experimental conditions.
2. Theoretical background The magnetisation of ferro#uids can relax by two di!erent mechanisms, the so-called NeH el relaxation and the Brownian relaxation. NeH el relaxation is caused by the reorientation of the magnetisation vector inside the magnetic core against an energy barrier [5]. Brownian relaxation is due to
0304-8853/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 0 6 5 - 7
R. Ko( titz et al. / Journal of Magnetism and Magnetic Materials 201 (1999) 102}104
rotational di!usion of the whole particle in the carrier liquid [6]. The time constant of NeH el relaxation depends much stronger on the particle diameter than the Brownian relaxation time. The e!ective relaxation time is dominated by the faster relaxation process. While both, NeH el and Brownian relaxation of an ensemble of identical particles are described simply by an exponential decay with time, relaxation in real systems is usually nonexponential, due to the distribution of sizes and shapes of the particles. Using the so-called critical volume approximation, Chantrell et al. [7] derived a formula for the time-dependent magnetisation of a system of magnetic nanoparticles after magnetisation in a "eld H for a time t :
t (1) M(t)"C ln 1# , t
where the characteristic time t depends on t
and H . For the case of weak magnetising "elds
[8] Berkov et al. obtained in a generalised theoretical approach (1) with t "t which was con
"rmed experimentally [8]. A compact formula for Brownian relaxation of a system with a distribution of hydrodynamic particle sizes is not known.
103
4. Results and discussion Fig. 1 shows excellent agreement between the relaxation signal of the freeze dried sample 3 containing 100 lmol Fe after magnetisation in a weak "eld of 8 A/m for 10 s and the theoretical time dependence "tted to (1) with t "t . The
relaxation measured for sample 4 (iron content: 100 nmol) magnetised with a stronger "eld of 1.6 kA/m is depicted in the inset of Fig. 1. Obviously, this relaxation signal is not well described by a "t using (1) with t "t . This suggests, that in
this case the assumption of a weak magnetising "eld is not valid. Considering t as a free parameter a good "t to the experimental data is obtained with t "7t thus con"rming [7]. This indicates, that
the relaxation after magnetisation with a strong "eld resembles the relaxation of a sample magnetised in a decreased "eld for a longer time. The relaxation signal of the unstable liquid sample 1 magnetised with 400 A/m is shown in Fig. 2. A curve "t according to Eq. (1) yields t " 1.5t . The intermediate value for t corresponds
3. Method and materials Relaxation measurements at room temperature starting a few milliseconds after switching o! a magnetising "eld were performed with a DCSQUID-magnetometer in a magnetically shielded environment using a specially designed twin coil for the magnetisation of the sample [8]. We investigated a water based magnetic #uid of magnetic iron oxide nanoparticles with a mean core diameter of approximately 10 nm (standard deviation of lognormal distribution p+0.5) and a mean hydrodynamic diameter of approximately 60 nm. This #uid (sample 0) was used as the basis for a number of modi"ed samples: an unstable sample 1 accomplished by chemically induced aggregation and subsequent sedimentation of the particles; a high viscosity sample 2 with a solution of 87% glycerol in water as carrier liquid and freeze dried samples 3 and 4 of di!erent concentrations for relaxation measurements of nanoparticles in a solid phase.
Fig. 1. Relaxation signal of sample 3 after magnetisation with 8 A/m for 10 s and curve "t according to Eq. (1) with t " t . *B(100 s) was set to zero. Inset: Relaxation signal of
sample 4 after magnetisation with 1.6 kA/m for 10 s and curve "ts according to Eq. (1) with t "t and t "7t .
104
R. Ko( titz et al. / Journal of Magnetism and Magnetic Materials 201 (1999) 102}104
of two exponential decays (q "0.25 s, q "1.12 s) "ts the measurement well. The relaxation of the freeze dried sample 4 is not well described by a double exponential decay.
5. Conclusions and summary
Fig. 2. Relaxation signal of the unstable sample 1 with sediment after magnetisation with 400 A/m for 10 s and curve "t according to Eq. (1). Inset: Relaxation signal and various "ts for the glycerol based sample 2 magnetised using the same parameters.
to the intermediate magnetising "eld. The good agreement between "t and measured data indicates that the signal is due to NeH el relaxation, caused by aggregated particles forming the sediment. For the stable #uid of sample 0 no relaxation signal could be detected. This is due to the hydrodynamic particle size of the suspended particles, which yields an estimated Brownian relaxation time less than 100 ls and is much shorter than the starting time of the measurement (200 ms). In the inset of Fig. 2 the relaxation signal of sample 2 is plotted against a logarithmic time scale. Evidently, Eq. (1) does not describe the relaxation of sample 2, also a curve "t using a single exponential decay is unsatisfactory, but yields a rough estimate of the mean Brownian relaxation time of 0.52 s. However, a superposition
The measurements indicate that the time dependencies of Brownian and NeH el relaxation are clearly di!erent. This is due to the narrow distribution of Brownian relaxation times compared to the extremely wide distribution of NeH el-relaxation times. The presence of a NeH el relaxation signal gives evidence that at least for a fraction of particles the Brownian mechanism is inhibited. This is the case e.g. for magnetic nanoparticles "xed to a solid phase or unstable aggregated magnetic #uids containing sediments. Since the relaxation signal yields information about the underlying relaxation process it can consequently be utilised to distinguish between particles bound to a solid phase and free particles. This is an important feature for the analysis of biological binding processes [4].
References [1] P.C. Fannin, S.W. Charles, J. Phys. D 22 (1989) 187. [2] H. Bogardus, D.A. Krueger, D. Thompson, J. Appl. Phys. 49 (6) (1978) 3422. [3] R. KoK titz, P.C. Fannin, L. Trahms, J. Magn. Magn. Mater. 149 (1995) 42. [4] W. Weitschies, R. KoK titz, T. Bunte, L. Trahms, Pharm. Pharmacol. Lett. 7 (1) (1997) 5. [5] L. NeH el, Ann. Geophys. 5 (1949) 99. [6] P. Debye, Polar Molecules, Chemical Catalog Company, New York, 1929. [7] R.W. Chantrell, S.R. Hoon, B.K. Tanner, J. Magn. Magn. Mater. 38 (1983) 133. [8] D.V. Berkov, R. KoK titz, J. Phys.: Condens. Matter 8 (1996) 1257.