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Small-angle light scattering study of quasi-two-dimensional structures of magnetic particles
Colloids and Surfaces A: Physicochemical and Engineering Aspects, 80 (1993) 19-27 0927-7757/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved.
19
Small-angle light scattering study of quasi-two-dimensional structures of magnetic particles Y.H. Hwang*, Department (Received
X-l. Wu
of Physics and Astronomy, 1 August
1992; accepted
University of Pittsburgh,
Pittsburgh,
PA 1.5260, USA
8 April 1993)
Abstract We studied quasi-two-dimensional magnetic domain structures using a small-angle light scattering technique. The structures are formed when magnetic particles in an aqueous suspension between two parallel glass plates are assembled and oriented in a magnetic field H perpendicular to the plates. Three different structure regions can be identified, and they are bounded by two critical field lines H,,(d) and HcL(4). Below HcI(4) the system consists of single particles; between H,,(4) and HC-($), the particles assemble into elongated disordered domains; and above Hc2(~), the domains develop a strong in-plane correlation. In the strong field limit H >>H,--, the structure factor of the magnetic fluid resembles that of a two-dimensional liquid. However, long lasting metastability and hysteresis observed in the experiment suggest that the system is essentially locked in a glassy state. Key words: Colloidal
physics; Domain
structure;
Instability;
Magnetic
Introduction A collection of magnetic particles interacting via a dipolar interaction is interesting from a statistical mechanics point of view. In three dimensions (3D) the dipolar interaction is truly long ranged in that the total pair interaction energy, or the second virial coefficient, diverges as the system size increases. As a result, in 3D the magnetic domain structure depends critically on the size and the shape of the sample container. In lower dimensions, however, the pair interaction potential can be integrated and vital coefficients of all orders can, in principle, be calculated [l]. A unique feature of dipolar interaction is its anisotropy. For two dipole moments fil and fi2 separated by a distance lil, the interaction potential is given by u(i) = _ fil. Fz - 3(F1r3 *Corresponding
author.
MF2
* 3
(1)
particle;
Phase transition
The interaction is attractive (ferromagnetic) when F /I fi2 and repulsive (antiferromagnetic) when F_Lji2. One consequence of the anisotropy of the dipolar interaction is a tendency for the particles to align into chains, with each dipole moment pointing towards a nearest neighbor along the chain. The degree of chain formation depends both on the number density of the particles, and on the strength of the dipolar interaction, In thermal equilibrium and in the absence of an external magnetic field H, the chain conformation is random and it resembles that of a flexible polymer. However, in the presence of H, the chains are stretched and aligned along the field [2]. There are two important dimensionless parameters A,, = pH/kB T and il,, = p2/D3 k, T, which characterize the particle-field and particle-particle interactions, where D is the diameter of the particle. The degree of dipolar alignment of individual particles is judged by the magnitude of A,,, and the ordering and correlation of the particles are determined by A,,,. To form a stable chain, there-
YH. Hwang and X-l. WujColloids Surfaces A: Physicochem. Eng. Aspects 80 (1993)
20
fore, requires ;IPP 3 1. A system of particular
interest
is a magnetic field consisting of superparamagnetic particles that interact via an induced dipolar inter-
tions on the spatial correlation of an array of dipolar chains has not previously been studied. The current experiment is the first attempt to
action. In this case p = xu,H, where x is the magnetic susceptibility and u0 (=no3/6) is the
investigate
volume of the particle. As can be seen, both iPf and 1,,, depend on HZ and the ratio of the two &r/&, (=6/rtx) depends only on the magnetic
ing measurements for suspension confined plates. The magnetic netic, i.e. they possess
property x of the particle, and is independent of H, T and D. For commercially available polystyrene encapsulated magnetic particles (available from Seradyn, Inc., Indianapolis, IN 46206) we have x w 0.07 and i,,/&,, >> 1, i.e. the induced dipole moments of individual particles are wellaligned with H before chaining occurs. Being essentially a one-dimensional object (imbedded in 3D), an aligned magnetic chain experiences strong transverse thermal fluctuations. We calculated the mean square amplitude fluctuations at a given wavevector q with the result [3] (pi)=
;
5
(2)
0
As can be seen, the transverse fluctuations increase as the length of the chain increases, or as the interparticle interaction decreases. It is interesting that even though the interaction between the particles is long ranged, the transverse fluctuations still behave as a random walk with the amplitude of the fluctuations given by fi zz m, where L/D is the number of particles in a chain. As recently pointed out by Halsey and Toor [4], thermal fluctuations not only alter the chain conformation but also induce an attractive force between the neighboring chains. The free energy F(I) due to two wavy chains separated by a distance 1was calculated by Halsey and Toor with the result
[51 F(I)=
-11.11
g%
(3)
This interaction has the same 1 dependence as the van der Waals interaction but the strength is orders of magnitude greater. The effect of chain fluctua-
19-27
such an effect.
In this article we report small-angle
light scatter-
an aqueous magnetic particle between two parallel glass particles are superparamagno magnetic dipole moment
at zero field. The particles are assembled and aligned by an external magnetic field H whose direction is perpendicular to the plates. In contrast to previous experiments for similar systems [6,7], we found that the magnetic fluid undergoes a series of structure “transitions” as H increases. Two critical field lines H,-, (4) and Hc2(~) were identified and they correspond to the onsets of particle ordering along the field direction and transverse correlation between the dipolar chains. The magnetic domain structures that we observed are not only sensitive to the magnitude of H but also show an interesting history-dependence on the field. By abruptly increasing H we can produce an array of chains which are more closely spaced but considerably more disordered compared to the equilibrium case, in which H is varied adiabatically, Long lasting metastability and history-dependent domain structures suggest the system is trapped in a glassy state. In certain respects our magnetic fluid is similar to an electrorheological (ER) fluid in that both systems are characterized by induced dipolar interactions. However, there are important differences between the two systems, and under certain circumstances they can behave very differently. The most notable difference is the boundary effect. In an ER fluid, multi-reflection of image charge on the walls makes a finite sized chain behave essentially like an infinite chain, and the dipolar interaction between the chains is short ranged due to mutual screening of adjacent dipole moments [4]. In our system, because the glass walls are not magnetically polarizable, the length of the chains is always finite and the dipolar interaction between the chains remains long-ranged and predominantly repulsive.
YH. Hwang and X-l. Wu/Colloids Surfaces A: Physicochem. Eng. Aspects 80 (1993)
The strong experiments tive picture.
in-plane correlation observed in our lends support to the above qualita-
19-27
21
filter assembly. In normal operation, the input laser beam impinges normally on the sample cell and the scattered light intensity is projected onto a translucent
screen by a convex
lens L,. The use
Experimental
of an expanded laser beam and the lens L2 greatly reduces the speckle size of the scattered light,
The magnetic particles used in this experiment were purchased from Seradyn. The particles are spherical (D z 1 urn) and possess no permanent dipole moment. The stock solution contains 10%
making the intensity measurement more reliable. In this light scattering geometry, the measured scattering angle 8, is related to the focal length of L, by 8, = arctan(r/f), where r is the radial posi-
solid volume fraction. Other concentrations used in the experiment were diluted from the stock solution by distilled water. Because of a large specific gravity (1.3) the particles tend to settle in a matter of a few hours. To ascertain uniform solutions, all the samples were sonicated for several minutes before the measurement. The prepared sample was contained between two thick optical plates 5 cm in diameter. To prevent evaporation, the sample was sealed under compression by a stainless steel holder. Thin gaskets, with thickness L varying between 10 and
tion on the screen and f is the focal length of LZ. Owing to the index of refraction mismatch between the liquid/glass and glass/air interfaces the true scattering angle 0, and the measured angle 8, are related by the equation
50 urn, were used for sealing. Because of the large particle size used in the experiment, thin samples are crucial in order to reduce the multiple scattering contribution to our measurement. The assembled sample cell was placed in the center of a Helmholtz coil pair 20 cm in diameter with a maximum field of E 100 G. Small-angle light scattering measurements were carried out to study magnetic domain structures. The schematic diagram of the set-up is shown in Fig. 1. A 5 mW He-Ne laser (1. = 632.8 nm) is expanded to a beam diameter of 2 cm by a spatial
Helmholtz
8,=sin-’
sin 8,
The scattering wavevector q is given by q = (4xn/A) sin(Qs/2), with n being the refractive index of water. For f = 20 cm, a range of scattering angles 2” < 8 < 10” is accessible, which range of scattering wavevectors 20 000 cm- ‘. Since q << k,, with the input wavevector, q is almost the optical axis. As shown in Fig.
corresponds to a 4000 cm-’ < q < k, = 27m/i being perpendicular to 1, the sample cell
together with the Helmholtz coils can be rotated along the x axis, with the angle of rotation specified by Y. For Y = 0, the input laser beam impinged normally on the sample cell and 4 is in the x-y plane. In this geometry the in-plane density fluctuations
of magnetic
particles
for Y z 90”, ?j is almost
Coil
Assembly
(4)
( 1.33 >
Screen
Fig. I. Small-angle light scattering apparatus.
are probed.
normal
However,
to the sample cell
YH. Hwang and X-l. Wu/Colloids
22
so that it probes
anisotropic
scattering
Surfaces
A: Physicochem.
Eng. Aspects 80 (1993)
19-27
in the z (or
fi) direction. Observations
and discussion
For a given particle concentration 4 we found that the magnetic fluid undergoes a series of structure “transitions”
depending
As H increases, the system in the z direction, resulting
on the applied
H field.
first becomes ordered in elongated domains
whose in-plane positions are random. As H increases further, the domains become locally ordered in the x-y plane and the correlation between the domains increases monotonically. Associated with the onset of ordering in the longitudinal and the transverse directions we defined two critical field lines Hcl($) and H,--(4) which separate various structure regions. Our experiment is quite different from previous observations [6,7] in which only one critical field was identified. However, owing to the different criteria adopted in those experiments, there was no general agreement between the measurements. In the following we describe how the critical fields were determined in our experiment. In a weak magnetic field, particles are weakly interacting and their positions are random due to thermal fluctuations. In this weak field limit, there is no particle aggregation, and the scattering intensity distribution is isotropic. As H is increased above a certain threshold H,, , we observe an onset of optical anisotropy in the x-z plane. The anisotropic scattering is easily observable by rotating the sample assembly (sample plus Helmholtz coil) by an angle !P z 90”. As shown in Fig. 2, the scattering intensity is greatly enhanced along the x direction, whereas the scattering intensity along the z direction is nearly unaffected by H. The anisotropic scattering pattern in Fig. 2 is qualitatively the same as the diffraction pattern from a linear slit placed perpendicular to the optical axis. In our case, the anisotropic scattering is a clear indication of particle chaining along the field or z direction. By measuring the onset of the optical anisotropy as a function of 4, we found that H,-, (4)
2
L
1
Y
Fig. 2. Anisotropic light scattering in the x-z plane. The applied magnetic field H is in the z direction. The optical anisotropy is a result of particles chaining along the z direction.
is a monotonically decreasing function of 4. Figure 3 shows two measurements with different sample thicknesses L/D = 14 (triangles) and 34 (circles), where the thickness is normalized by the particle diameter. Experimentally, we found that H,-,(d) can be approximately fit by a power law H Cl = V’, with c(~ = a. Also shown in Fig. 3, H,-, (4) appears to depend sensitively on the sample thickness L/D: H,, decreases as L/D increases, indicating that the particles are more susceptible to chain formation as the system size along the 14
2 0
2
4
6
8
10
P (XI Fig. 3. The first critical field line H,, The triangles and circles are for measurements with L/D = 14 and 34 respectively. The inset is a double logarithmic plot for the same measurements.
YH. Hwang and X-l. Wu/Colloids Surfaces A: Physicochem. Eng. Aspects 80 (1993)
field direction
is increased,
a characteristic
19-27
23
of aniso-
tropic dipolar interaction. It is interesting to point out that despite ordering (aggregation) along the field direction, in the neighborhood of H,-,(4) the scattering intensity distribution is isotropic forward
direction
in the x-y plane and peaks in the (4 -0).
This
suggests
that
the
elongated magnetic domains are uncorrelated in the x-y plane and the structure of the magnetic fluid resembles that of nematic ordering in liquid crystals. As H increases further, the forward scattering intensity gradually decreases and a second scattering peak emerges at a finite q (=qmax). Figure 4 shows a sequence of in-plane scattering intensity distributions at H = 0, 10, 20, and 40 G for a sample with 4 = 5% and L/D = 25. Correspondingly, in Fig. 5 we show the radial distribution of scattering intensities Z(q) obtained from Fig. 4. (The radial distribution of scattering intensity I(q) was obtained by a circular integration, or averaging, over a constant wavevector q in the x-y plane. The scheme is commonly used in small-angle neutron scattering measurements.) The decreasing forward scattering intensity indicates that the dominant interaction between the domains is repulsive, and the magnetic fluid becomes less
8.0
8.4
6.8
9.2
9.6
10.0
10.4
In(q (cm-'))
Fig. 5. Radial distribution of scattering intensity. The data is obtained from the scattering patterns in Fig. 4. The squares, circles, triangles, and crosses are for H = 0, 10, 20, and 40 G, respectively.
compressible as H increases. The repulsive interaction must also be responsible for strong in-plane domain correlation, which gives rise to the observed scattering peak. The appearance of the scattering peak at finite q allows us to define a second critical field line Hc2(c$). Figure 6 shows two measurements for samples with L/D = 14 (circles) and 34 (squares). again decays monotoniAs can be seen, H,, vs.C$I cally, and to a good approximation Hc2 z 4-"', with t12 ~4. It should be noted that, unlike the first critical field line, which shows strong sample thickness dependence, the second critical field line Hc-(@)appears to be almost independent of L/D. An interesting feature of our small-angle light
0
2
4
6
8
10
(P (Q
Fig. 4. Small-angle light scattering is applied adiabatically.
in the x-y
plane. The field
Fig. 6. The second critical field line H,, . The circles and squares are for samples with L/D = 14 and 34 respectively. The inset is a double logarithmic plot for the same measurements.
24
Y.H. Hwang and X-l. WufColioids
scattering measurements increases with H initially
is that above and eventually
Hcz, qmax saturates
at a constant value. This behavior has been observed for all the samples with different particle concentrations and sample thicknesses. Figure 7
Eng. Aspects
Surfaces A: Physicochem.
80 (1993) 19-27
It is interesting that Nch changes monotonically with $J and in the low concentration limit NC,,+ 1. The above observation allows us to construct a qualitative “phase” diagram for the magnetic fluid. As shown
in Fig. 8, for a fixed particle
concen-
shows typical measurements for two samples with the same L/D value (= 14) but different 4. The
tration the system undergoes several structure changes as H increases. In the weak field limit,
circles are for 4 = 5% and triangles are for 4 = 10%. If we assign a saturation field H, to the cross-over
thermal fluctuations dominate, and a magnetic fluid consists of single particles (SP) that interact weakly via a long-range dipole-dipole interaction.
regime where qmax begins to level off, we find that 3s is nearly a constant, independent of 4. proportional to the Since qmax is inversely separation between the domains average clearly indicate L, = 2~/4max, our measurements that L, decreases as H increases. The decrease in L, implies that the number of columns per unit area must be increased. This can happen if new columns are formed by nucleating monomers (or single particles) in the surrounding fluid. In the strong field limit H > H, all the monomers are depleted in the solution, qmax reaches a constant value and the scattering intensity distribution becomes independent of H. In this strong field limit we can estimate the average number of chains per column Nch by assuming zero monomer concentration and that chains are packed in the hexagonal form. This gives Nch = 6&-c~/D2qlfiax. Using the measured qmax, Nch is calculated for a number of concentrations, as shown in the inset of Fig. 7.
In the intermediate field, the particles aggregate to form elongated disordered domains (EDD). In this regime, even though particles are strongly interacting inside the domains, the interaction between the domains are considerably weaker. As H increases further, the interaction between the domains becomes progressively more repulsive, resulting in an array of correlated columns (CC) whose average spacing is determined by the crosssectional area of the sample, analogous to a colloidal crystal. What is different from many colloidal systems, however, is that the number density of the columns in the magnetic fluid is a non-conserved quantity. It increases as a result of secondary nucleation processes. In the strong field limit, all the monomers are consumed by the columns and the structure of the magnetic fluid becomes independent of H. In this regime the system on large
50
1
n
40
7 E
‘;; IJl 30 2 g
z I
.2 n
0
10
6
x 6 d
20
0
3 =
0
2
*
0
6
2
4
6
8
10
(0 (X)
0 0
10
20 H
30
40
50
60
(gauss)
Fig. 7. The plots of qmar vs. H. Triangles and circles are for samples with 4 = 10 and 5%, respectively. From qrnax we derive the number of chains per column, as shown in the inset.
Fig. 8. “Phase” diagram of magnetic fluid. SP, EDD, CC, and FDS stand for single particles, elongated disordered domains, correlated columns, and frozen domain structures, respectively. Different structure regions are separated by two critical field lines H,, (solid line) and Hc2 (dashed line), and the saturation field line Hs (dashhdot line).
YH. Hwang and X-l. Wu/Colloids
Surfaces
A: Physicochem.
Eng. Aspects 80 (1993)
19-27
25
length scales is still highly disordered and the dynamics is much slower than in the low field limit. It appears that the system is locked in a frozen domain state (FDS). All the measurements
above
were performed
at
the equilibrium condition: the external magnetic field H was varied adiabatically, allowing the system to relax after each small variation of the field. Non-equilibrium measurements were also carried out for the same magnetic fluid. In such a measurement His varied abruptly and the transient time for H is much less than 1 s. Figure 9 shows a series of measurements for a sample with 4 = 5% and L/D = 25. The magnetic field H is varied between 0 and 60 G. The measurements should be compared with the adiabatic process shown in Fig. 4. Several interesting features are worth mentioning. (1) We note that for the same concentration 4 and field strength H the non-equilibrium measurements result in qrnax values that are significantly larger than the equilibrium measurements, as shown in Fig. 10. This suggests that the average spacing between the columns and the width of the columns are smaller. A simple calculation shows that NC,, is of the order of unity as H jumps
Fig. 9. Small-angle is applied abruptly.
light scattering
in the x-y
plane. The field
0
10
20 30 40 H (gouss)
50
60
Fig. 10. Equilibrium and non-equilibrium qmnx measurements. Triangles are for the equilibrium measurements and circles are for the non-equilibrium measurements. The inset is the number of chains per column for the same measurements.
from 0 to 60 G, and NC,, is a factor of two smaller than the corresponding equilibrium measurements, as shown in the inset of Fig. 10. This implies that if the field is increased abruptly, single chains are formed and they do not coalesce into columns as suggested by a recent theory for ER fluids [4]. (2) Unlike the equilibrium measurements, the non-equilibrium data shown in Fig. 10 indicate that the saturation field Hs shifts to a much higher value if it still exists. (3) We also note that the scattering peaks in the non-equilibrium measurements are weaker and broader, suggesting that the in-plane structures of magnetic columns are more disordered compared to the equilibrium case. The enhanced disorder may be expected because the system does not have time to anneal itself. The non-equilibrium structures that we observed seem to have a very long lifetime. We were not able to observe any structure relaxation over a period of 12 h once the structure had formed. The long-lasting metastability suggests the energy barrier for the structure relaxation is much greater than the thermal energy k, T, and the system is essentially trapped in a local minimum of free energy. Dynamic light scattering was also carried out to measure the in-plane density fluctuations of the magnetic domains as a function of the H field. The measured intensity-intensity autocorrelation functions are strongly non-exponential, with char-
YH. Hwang and X-l. Wu/Colloids Surfaces A: Physicochem. Eng. Aspects 80 (1993)
26
acteristic
relaxation
H increases.
Both
times that increase the static
and
scattering experiments, therefore, the notion that the metastability
sharply
dynamic
as
light
seem to support and hysteresis
in a field perpendicular two-dimensional solid [S]. Although
19-27
to the layer. In this case a phase has been observed
it is difficult to single out the reason
for the lack of long-range dimensional
for a similar system [7]. Using an optical transmis-
bled into columnar
sion measurement Lemaire et al. [7] found that the turbidity increased rapidly upon reaching a
ticles per column N, is not a constant. The particle number fluctuations are given approximately by
certain field strength and they interpreted the sharp increase as the onset of aggregation. Their measurements showed that Hc, is a linear increasing function of C$ instead of a decreasing function of 4, in contradiction to what we observed. Since the turbidity measurement gives the total scattering cross-section, it is not very sensitive to the domain structures in a magnetic fluid. Therefore, the criterion adopted in the experiment of Lemaire et al.
l/A. Therefore, for short columns, as in our experiment, the number fluctuations can be quite large. As a result, the coupling constant between the magnetic domains is effectively a random variable. Secondly, the surface roughness on the glass substrates may provide anchoring for the magnetic columns. Such an anchoring effect is likely to be incommensurate with the bulk ordering, giving rise to an overall disordered structure. Thirdly, the size and magnetic susceptibility of the particles are not uniform in our sample, and this presents another source of disorder for the interaction. Finally, conformational fluctuations of dipolar chains could cause significant entanglement between the chains. If the energy barrier for disentanglement is large enough the disorder may persist almost indefinitely.
may be somewhat ambiguous. However, in our experiment the anisotropic scattering probes directly the uniaxial growth of the magnetic domains, and the measurement is likely to be the most accurate determination of Hc, . Qualitatively, our observation of H,-,is also consistent with the second virial coefficient calculation [4], which predicts
system,
order for our quasi-two-
that we observed are a result of disorder. The concentration dependence of H,, that we observed is different from a previous measurement
we offer several
possibilities
that may contribute to the disorder that we observed. First, when magnetic particles are assemstructures,
the number
of par-
Hci = 12kaT 73x*4(1
- 4)‘03[7
+
112
2(L/D) + 12(L/D)2]
(5) This expression shows that H,-,is a decreasing function of both 4 and L/D,consistent with our observation. Quantitatively, however, our measurement of H,, does not agree with the above calculation. Most notably the exponent ~1, is a factor of two smaller than the predicted value. The physical reasons for this discrepancy are not entirely clear. An interesting finding of our experiment is that the FDS pre-empts long-range order (crystalline order) in the system. The behavior is significantly different from a single layer of magnetic particles
Conclusion We performed small-angle light scattering measurements to probe the domain structures of aqueous magnetic particles in parallel plates. In contrast to previous measurements we found that the magnetic particles undergo a series of structure changes: single particles -+ elongated disordered domains -+ transversely correlated columns, as H increases. The various structure regions are separated by two critical field lines H,, and HC2. It appears that even above Hc2 there are still significant numbers of monomers present in the solution. Therefore, as H increases new columns are nucleated and they enhance the transverse correlation between the columns. Above the saturation field
Y.H. Hwang and X-l. WulColloids
Surfaces
A: Physicochem.
Eng. Aspects 80 (1993)
Ifs the monomer concentration approaches zero and magnetic domains become frozen. In this frozen state, an array of magnetic columns are
locally ordered
but globally
disordered.
27
19-27
stimulating discussions and research is partially supported
comments.
The
by a grant
from
Exxon Co.
Associated
with the disorder the system exhibits interesting Many features and hysteresis. metastability
References 1
observed in our experiment indicate the system falls out of thermal equilibrium quickly as H increases and behaves somewhat like a two-dimen-
2
sional glass.
3 4 5 6
Acknowledgments
I
We thank Professor W.I. Goldburg, Professor M. Widom, Dr. C. Yeung, and Mr. M.X. Liu for
8
M. Widom and H. Zhang, Mater. Res. Proc., 248 (1992) 235. G. Helgesen, A.T. Skjeltorp, P.M. Mom, R. Botet and R. Jullien, Phys. Rev. Lett., 61 (1988) 1736. Y.H. Hwang and X-l. Wu, in preparation. T.C. Halsey and W. Toor, Phys. Rev. Lett., 65 (1990) 2820. T.C. Halsey and W. Toor, J. Stat. Phys., 61 (1990) 1257. R. Tao, J.T. Woestman and N.K. Jaggi, Appl. Phys. Lett., 55 (1989) 1844. E. Lemaire, Y. Grasselli and G. Bossis, J. Phys. II France, 2 (1992) 359. G. Helgesen and A.T. Skjeltorp, Spontaneous Formation of Space-Time Structure and Criticality, Kluwer, Dordrecht, 1991. p. 391.
19
Small-angle light scattering study of quasi-two-dimensional structures of magnetic particles Y.H. Hwang*, Department (Received
X-l. Wu
of Physics and Astronomy, 1 August
1992; accepted
University of Pittsburgh,
Pittsburgh,
PA 1.5260, USA
8 April 1993)
Abstract We studied quasi-two-dimensional magnetic domain structures using a small-angle light scattering technique. The structures are formed when magnetic particles in an aqueous suspension between two parallel glass plates are assembled and oriented in a magnetic field H perpendicular to the plates. Three different structure regions can be identified, and they are bounded by two critical field lines H,,(d) and HcL(4). Below HcI(4) the system consists of single particles; between H,,(4) and HC-($), the particles assemble into elongated disordered domains; and above Hc2(~), the domains develop a strong in-plane correlation. In the strong field limit H >>H,--, the structure factor of the magnetic fluid resembles that of a two-dimensional liquid. However, long lasting metastability and hysteresis observed in the experiment suggest that the system is essentially locked in a glassy state. Key words: Colloidal
physics; Domain
structure;
Instability;
Magnetic
Introduction A collection of magnetic particles interacting via a dipolar interaction is interesting from a statistical mechanics point of view. In three dimensions (3D) the dipolar interaction is truly long ranged in that the total pair interaction energy, or the second virial coefficient, diverges as the system size increases. As a result, in 3D the magnetic domain structure depends critically on the size and the shape of the sample container. In lower dimensions, however, the pair interaction potential can be integrated and vital coefficients of all orders can, in principle, be calculated [l]. A unique feature of dipolar interaction is its anisotropy. For two dipole moments fil and fi2 separated by a distance lil, the interaction potential is given by u(i) = _ fil. Fz - 3(F1r3 *Corresponding
author.
MF2
* 3
(1)
particle;
Phase transition
The interaction is attractive (ferromagnetic) when F /I fi2 and repulsive (antiferromagnetic) when F_Lji2. One consequence of the anisotropy of the dipolar interaction is a tendency for the particles to align into chains, with each dipole moment pointing towards a nearest neighbor along the chain. The degree of chain formation depends both on the number density of the particles, and on the strength of the dipolar interaction, In thermal equilibrium and in the absence of an external magnetic field H, the chain conformation is random and it resembles that of a flexible polymer. However, in the presence of H, the chains are stretched and aligned along the field [2]. There are two important dimensionless parameters A,, = pH/kB T and il,, = p2/D3 k, T, which characterize the particle-field and particle-particle interactions, where D is the diameter of the particle. The degree of dipolar alignment of individual particles is judged by the magnitude of A,,, and the ordering and correlation of the particles are determined by A,,,. To form a stable chain, there-
YH. Hwang and X-l. WujColloids Surfaces A: Physicochem. Eng. Aspects 80 (1993)
20
fore, requires ;IPP 3 1. A system of particular
interest
is a magnetic field consisting of superparamagnetic particles that interact via an induced dipolar inter-
tions on the spatial correlation of an array of dipolar chains has not previously been studied. The current experiment is the first attempt to
action. In this case p = xu,H, where x is the magnetic susceptibility and u0 (=no3/6) is the
investigate
volume of the particle. As can be seen, both iPf and 1,,, depend on HZ and the ratio of the two &r/&, (=6/rtx) depends only on the magnetic
ing measurements for suspension confined plates. The magnetic netic, i.e. they possess
property x of the particle, and is independent of H, T and D. For commercially available polystyrene encapsulated magnetic particles (available from Seradyn, Inc., Indianapolis, IN 46206) we have x w 0.07 and i,,/&,, >> 1, i.e. the induced dipole moments of individual particles are wellaligned with H before chaining occurs. Being essentially a one-dimensional object (imbedded in 3D), an aligned magnetic chain experiences strong transverse thermal fluctuations. We calculated the mean square amplitude fluctuations at a given wavevector q with the result [3] (pi)=
;
5
(2)
0
As can be seen, the transverse fluctuations increase as the length of the chain increases, or as the interparticle interaction decreases. It is interesting that even though the interaction between the particles is long ranged, the transverse fluctuations still behave as a random walk with the amplitude of the fluctuations given by fi zz m, where L/D is the number of particles in a chain. As recently pointed out by Halsey and Toor [4], thermal fluctuations not only alter the chain conformation but also induce an attractive force between the neighboring chains. The free energy F(I) due to two wavy chains separated by a distance 1was calculated by Halsey and Toor with the result
[51 F(I)=
-11.11
g%
(3)
This interaction has the same 1 dependence as the van der Waals interaction but the strength is orders of magnitude greater. The effect of chain fluctua-
19-27
such an effect.
In this article we report small-angle
light scatter-
an aqueous magnetic particle between two parallel glass particles are superparamagno magnetic dipole moment
at zero field. The particles are assembled and aligned by an external magnetic field H whose direction is perpendicular to the plates. In contrast to previous experiments for similar systems [6,7], we found that the magnetic fluid undergoes a series of structure “transitions” as H increases. Two critical field lines H,-, (4) and Hc2(~) were identified and they correspond to the onsets of particle ordering along the field direction and transverse correlation between the dipolar chains. The magnetic domain structures that we observed are not only sensitive to the magnitude of H but also show an interesting history-dependence on the field. By abruptly increasing H we can produce an array of chains which are more closely spaced but considerably more disordered compared to the equilibrium case, in which H is varied adiabatically, Long lasting metastability and history-dependent domain structures suggest the system is trapped in a glassy state. In certain respects our magnetic fluid is similar to an electrorheological (ER) fluid in that both systems are characterized by induced dipolar interactions. However, there are important differences between the two systems, and under certain circumstances they can behave very differently. The most notable difference is the boundary effect. In an ER fluid, multi-reflection of image charge on the walls makes a finite sized chain behave essentially like an infinite chain, and the dipolar interaction between the chains is short ranged due to mutual screening of adjacent dipole moments [4]. In our system, because the glass walls are not magnetically polarizable, the length of the chains is always finite and the dipolar interaction between the chains remains long-ranged and predominantly repulsive.
YH. Hwang and X-l. Wu/Colloids Surfaces A: Physicochem. Eng. Aspects 80 (1993)
The strong experiments tive picture.
in-plane correlation observed in our lends support to the above qualita-
19-27
21
filter assembly. In normal operation, the input laser beam impinges normally on the sample cell and the scattered light intensity is projected onto a translucent
screen by a convex
lens L,. The use
Experimental
of an expanded laser beam and the lens L2 greatly reduces the speckle size of the scattered light,
The magnetic particles used in this experiment were purchased from Seradyn. The particles are spherical (D z 1 urn) and possess no permanent dipole moment. The stock solution contains 10%
making the intensity measurement more reliable. In this light scattering geometry, the measured scattering angle 8, is related to the focal length of L, by 8, = arctan(r/f), where r is the radial posi-
solid volume fraction. Other concentrations used in the experiment were diluted from the stock solution by distilled water. Because of a large specific gravity (1.3) the particles tend to settle in a matter of a few hours. To ascertain uniform solutions, all the samples were sonicated for several minutes before the measurement. The prepared sample was contained between two thick optical plates 5 cm in diameter. To prevent evaporation, the sample was sealed under compression by a stainless steel holder. Thin gaskets, with thickness L varying between 10 and
tion on the screen and f is the focal length of LZ. Owing to the index of refraction mismatch between the liquid/glass and glass/air interfaces the true scattering angle 0, and the measured angle 8, are related by the equation
50 urn, were used for sealing. Because of the large particle size used in the experiment, thin samples are crucial in order to reduce the multiple scattering contribution to our measurement. The assembled sample cell was placed in the center of a Helmholtz coil pair 20 cm in diameter with a maximum field of E 100 G. Small-angle light scattering measurements were carried out to study magnetic domain structures. The schematic diagram of the set-up is shown in Fig. 1. A 5 mW He-Ne laser (1. = 632.8 nm) is expanded to a beam diameter of 2 cm by a spatial
Helmholtz
8,=sin-’
sin 8,
The scattering wavevector q is given by q = (4xn/A) sin(Qs/2), with n being the refractive index of water. For f = 20 cm, a range of scattering angles 2” < 8 < 10” is accessible, which range of scattering wavevectors 20 000 cm- ‘. Since q << k,, with the input wavevector, q is almost the optical axis. As shown in Fig.
corresponds to a 4000 cm-’ < q < k, = 27m/i being perpendicular to 1, the sample cell
together with the Helmholtz coils can be rotated along the x axis, with the angle of rotation specified by Y. For Y = 0, the input laser beam impinged normally on the sample cell and 4 is in the x-y plane. In this geometry the in-plane density fluctuations
of magnetic
particles
for Y z 90”, ?j is almost
Coil
Assembly
(4)
( 1.33 >
Screen
Fig. I. Small-angle light scattering apparatus.
are probed.
normal
However,
to the sample cell
YH. Hwang and X-l. Wu/Colloids
22
so that it probes
anisotropic
scattering
Surfaces
A: Physicochem.
Eng. Aspects 80 (1993)
19-27
in the z (or
fi) direction. Observations
and discussion
For a given particle concentration 4 we found that the magnetic fluid undergoes a series of structure “transitions”
depending
As H increases, the system in the z direction, resulting
on the applied
H field.
first becomes ordered in elongated domains
whose in-plane positions are random. As H increases further, the domains become locally ordered in the x-y plane and the correlation between the domains increases monotonically. Associated with the onset of ordering in the longitudinal and the transverse directions we defined two critical field lines Hcl($) and H,--(4) which separate various structure regions. Our experiment is quite different from previous observations [6,7] in which only one critical field was identified. However, owing to the different criteria adopted in those experiments, there was no general agreement between the measurements. In the following we describe how the critical fields were determined in our experiment. In a weak magnetic field, particles are weakly interacting and their positions are random due to thermal fluctuations. In this weak field limit, there is no particle aggregation, and the scattering intensity distribution is isotropic. As H is increased above a certain threshold H,, , we observe an onset of optical anisotropy in the x-z plane. The anisotropic scattering is easily observable by rotating the sample assembly (sample plus Helmholtz coil) by an angle !P z 90”. As shown in Fig. 2, the scattering intensity is greatly enhanced along the x direction, whereas the scattering intensity along the z direction is nearly unaffected by H. The anisotropic scattering pattern in Fig. 2 is qualitatively the same as the diffraction pattern from a linear slit placed perpendicular to the optical axis. In our case, the anisotropic scattering is a clear indication of particle chaining along the field or z direction. By measuring the onset of the optical anisotropy as a function of 4, we found that H,-, (4)
2
L
1
Y
Fig. 2. Anisotropic light scattering in the x-z plane. The applied magnetic field H is in the z direction. The optical anisotropy is a result of particles chaining along the z direction.
is a monotonically decreasing function of 4. Figure 3 shows two measurements with different sample thicknesses L/D = 14 (triangles) and 34 (circles), where the thickness is normalized by the particle diameter. Experimentally, we found that H,-,(d) can be approximately fit by a power law H Cl = V’, with c(~ = a. Also shown in Fig. 3, H,-, (4) appears to depend sensitively on the sample thickness L/D: H,, decreases as L/D increases, indicating that the particles are more susceptible to chain formation as the system size along the 14
2 0
2
4
6
8
10
P (XI Fig. 3. The first critical field line H,, The triangles and circles are for measurements with L/D = 14 and 34 respectively. The inset is a double logarithmic plot for the same measurements.
YH. Hwang and X-l. Wu/Colloids Surfaces A: Physicochem. Eng. Aspects 80 (1993)
field direction
is increased,
a characteristic
19-27
23
of aniso-
tropic dipolar interaction. It is interesting to point out that despite ordering (aggregation) along the field direction, in the neighborhood of H,-,(4) the scattering intensity distribution is isotropic forward
direction
in the x-y plane and peaks in the (4 -0).
This
suggests
that
the
elongated magnetic domains are uncorrelated in the x-y plane and the structure of the magnetic fluid resembles that of nematic ordering in liquid crystals. As H increases further, the forward scattering intensity gradually decreases and a second scattering peak emerges at a finite q (=qmax). Figure 4 shows a sequence of in-plane scattering intensity distributions at H = 0, 10, 20, and 40 G for a sample with 4 = 5% and L/D = 25. Correspondingly, in Fig. 5 we show the radial distribution of scattering intensities Z(q) obtained from Fig. 4. (The radial distribution of scattering intensity I(q) was obtained by a circular integration, or averaging, over a constant wavevector q in the x-y plane. The scheme is commonly used in small-angle neutron scattering measurements.) The decreasing forward scattering intensity indicates that the dominant interaction between the domains is repulsive, and the magnetic fluid becomes less
8.0
8.4
6.8
9.2
9.6
10.0
10.4
In(q (cm-'))
Fig. 5. Radial distribution of scattering intensity. The data is obtained from the scattering patterns in Fig. 4. The squares, circles, triangles, and crosses are for H = 0, 10, 20, and 40 G, respectively.
compressible as H increases. The repulsive interaction must also be responsible for strong in-plane domain correlation, which gives rise to the observed scattering peak. The appearance of the scattering peak at finite q allows us to define a second critical field line Hc2(c$). Figure 6 shows two measurements for samples with L/D = 14 (circles) and 34 (squares). again decays monotoniAs can be seen, H,, vs.C$I cally, and to a good approximation Hc2 z 4-"', with t12 ~4. It should be noted that, unlike the first critical field line, which shows strong sample thickness dependence, the second critical field line Hc-(@)appears to be almost independent of L/D. An interesting feature of our small-angle light
0
2
4
6
8
10
(P (Q
Fig. 4. Small-angle light scattering is applied adiabatically.
in the x-y
plane. The field
Fig. 6. The second critical field line H,, . The circles and squares are for samples with L/D = 14 and 34 respectively. The inset is a double logarithmic plot for the same measurements.
24
Y.H. Hwang and X-l. WufColioids
scattering measurements increases with H initially
is that above and eventually
Hcz, qmax saturates
at a constant value. This behavior has been observed for all the samples with different particle concentrations and sample thicknesses. Figure 7
Eng. Aspects
Surfaces A: Physicochem.
80 (1993) 19-27
It is interesting that Nch changes monotonically with $J and in the low concentration limit NC,,+ 1. The above observation allows us to construct a qualitative “phase” diagram for the magnetic fluid. As shown
in Fig. 8, for a fixed particle
concen-
shows typical measurements for two samples with the same L/D value (= 14) but different 4. The
tration the system undergoes several structure changes as H increases. In the weak field limit,
circles are for 4 = 5% and triangles are for 4 = 10%. If we assign a saturation field H, to the cross-over
thermal fluctuations dominate, and a magnetic fluid consists of single particles (SP) that interact weakly via a long-range dipole-dipole interaction.
regime where qmax begins to level off, we find that 3s is nearly a constant, independent of 4. proportional to the Since qmax is inversely separation between the domains average clearly indicate L, = 2~/4max, our measurements that L, decreases as H increases. The decrease in L, implies that the number of columns per unit area must be increased. This can happen if new columns are formed by nucleating monomers (or single particles) in the surrounding fluid. In the strong field limit H > H, all the monomers are depleted in the solution, qmax reaches a constant value and the scattering intensity distribution becomes independent of H. In this strong field limit we can estimate the average number of chains per column Nch by assuming zero monomer concentration and that chains are packed in the hexagonal form. This gives Nch = 6&-c~/D2qlfiax. Using the measured qmax, Nch is calculated for a number of concentrations, as shown in the inset of Fig. 7.
In the intermediate field, the particles aggregate to form elongated disordered domains (EDD). In this regime, even though particles are strongly interacting inside the domains, the interaction between the domains are considerably weaker. As H increases further, the interaction between the domains becomes progressively more repulsive, resulting in an array of correlated columns (CC) whose average spacing is determined by the crosssectional area of the sample, analogous to a colloidal crystal. What is different from many colloidal systems, however, is that the number density of the columns in the magnetic fluid is a non-conserved quantity. It increases as a result of secondary nucleation processes. In the strong field limit, all the monomers are consumed by the columns and the structure of the magnetic fluid becomes independent of H. In this regime the system on large
50
1
n
40
7 E
‘;; IJl 30 2 g
z I
.2 n
0
10
6
x 6 d
20
0
3 =
0
2
*
0
6
2
4
6
8
10
(0 (X)
0 0
10
20 H
30
40
50
60
(gauss)
Fig. 7. The plots of qmar vs. H. Triangles and circles are for samples with 4 = 10 and 5%, respectively. From qrnax we derive the number of chains per column, as shown in the inset.
Fig. 8. “Phase” diagram of magnetic fluid. SP, EDD, CC, and FDS stand for single particles, elongated disordered domains, correlated columns, and frozen domain structures, respectively. Different structure regions are separated by two critical field lines H,, (solid line) and Hc2 (dashed line), and the saturation field line Hs (dashhdot line).
YH. Hwang and X-l. Wu/Colloids
Surfaces
A: Physicochem.
Eng. Aspects 80 (1993)
19-27
25
length scales is still highly disordered and the dynamics is much slower than in the low field limit. It appears that the system is locked in a frozen domain state (FDS). All the measurements
above
were performed
at
the equilibrium condition: the external magnetic field H was varied adiabatically, allowing the system to relax after each small variation of the field. Non-equilibrium measurements were also carried out for the same magnetic fluid. In such a measurement His varied abruptly and the transient time for H is much less than 1 s. Figure 9 shows a series of measurements for a sample with 4 = 5% and L/D = 25. The magnetic field H is varied between 0 and 60 G. The measurements should be compared with the adiabatic process shown in Fig. 4. Several interesting features are worth mentioning. (1) We note that for the same concentration 4 and field strength H the non-equilibrium measurements result in qrnax values that are significantly larger than the equilibrium measurements, as shown in Fig. 10. This suggests that the average spacing between the columns and the width of the columns are smaller. A simple calculation shows that NC,, is of the order of unity as H jumps
Fig. 9. Small-angle is applied abruptly.
light scattering
in the x-y
plane. The field
0
10
20 30 40 H (gouss)
50
60
Fig. 10. Equilibrium and non-equilibrium qmnx measurements. Triangles are for the equilibrium measurements and circles are for the non-equilibrium measurements. The inset is the number of chains per column for the same measurements.
from 0 to 60 G, and NC,, is a factor of two smaller than the corresponding equilibrium measurements, as shown in the inset of Fig. 10. This implies that if the field is increased abruptly, single chains are formed and they do not coalesce into columns as suggested by a recent theory for ER fluids [4]. (2) Unlike the equilibrium measurements, the non-equilibrium data shown in Fig. 10 indicate that the saturation field Hs shifts to a much higher value if it still exists. (3) We also note that the scattering peaks in the non-equilibrium measurements are weaker and broader, suggesting that the in-plane structures of magnetic columns are more disordered compared to the equilibrium case. The enhanced disorder may be expected because the system does not have time to anneal itself. The non-equilibrium structures that we observed seem to have a very long lifetime. We were not able to observe any structure relaxation over a period of 12 h once the structure had formed. The long-lasting metastability suggests the energy barrier for the structure relaxation is much greater than the thermal energy k, T, and the system is essentially trapped in a local minimum of free energy. Dynamic light scattering was also carried out to measure the in-plane density fluctuations of the magnetic domains as a function of the H field. The measured intensity-intensity autocorrelation functions are strongly non-exponential, with char-
YH. Hwang and X-l. Wu/Colloids Surfaces A: Physicochem. Eng. Aspects 80 (1993)
26
acteristic
relaxation
H increases.
Both
times that increase the static
and
scattering experiments, therefore, the notion that the metastability
sharply
dynamic
as
light
seem to support and hysteresis
in a field perpendicular two-dimensional solid [S]. Although
19-27
to the layer. In this case a phase has been observed
it is difficult to single out the reason
for the lack of long-range dimensional
for a similar system [7]. Using an optical transmis-
bled into columnar
sion measurement Lemaire et al. [7] found that the turbidity increased rapidly upon reaching a
ticles per column N, is not a constant. The particle number fluctuations are given approximately by
certain field strength and they interpreted the sharp increase as the onset of aggregation. Their measurements showed that Hc, is a linear increasing function of C$ instead of a decreasing function of 4, in contradiction to what we observed. Since the turbidity measurement gives the total scattering cross-section, it is not very sensitive to the domain structures in a magnetic fluid. Therefore, the criterion adopted in the experiment of Lemaire et al.
l/A. Therefore, for short columns, as in our experiment, the number fluctuations can be quite large. As a result, the coupling constant between the magnetic domains is effectively a random variable. Secondly, the surface roughness on the glass substrates may provide anchoring for the magnetic columns. Such an anchoring effect is likely to be incommensurate with the bulk ordering, giving rise to an overall disordered structure. Thirdly, the size and magnetic susceptibility of the particles are not uniform in our sample, and this presents another source of disorder for the interaction. Finally, conformational fluctuations of dipolar chains could cause significant entanglement between the chains. If the energy barrier for disentanglement is large enough the disorder may persist almost indefinitely.
may be somewhat ambiguous. However, in our experiment the anisotropic scattering probes directly the uniaxial growth of the magnetic domains, and the measurement is likely to be the most accurate determination of Hc, . Qualitatively, our observation of H,-,is also consistent with the second virial coefficient calculation [4], which predicts
system,
order for our quasi-two-
that we observed are a result of disorder. The concentration dependence of H,, that we observed is different from a previous measurement
we offer several
possibilities
that may contribute to the disorder that we observed. First, when magnetic particles are assemstructures,
the number
of par-
Hci = 12kaT 73x*4(1
- 4)‘03[7
+
112
2(L/D) + 12(L/D)2]
(5) This expression shows that H,-,is a decreasing function of both 4 and L/D,consistent with our observation. Quantitatively, however, our measurement of H,, does not agree with the above calculation. Most notably the exponent ~1, is a factor of two smaller than the predicted value. The physical reasons for this discrepancy are not entirely clear. An interesting finding of our experiment is that the FDS pre-empts long-range order (crystalline order) in the system. The behavior is significantly different from a single layer of magnetic particles
Conclusion We performed small-angle light scattering measurements to probe the domain structures of aqueous magnetic particles in parallel plates. In contrast to previous measurements we found that the magnetic particles undergo a series of structure changes: single particles -+ elongated disordered domains -+ transversely correlated columns, as H increases. The various structure regions are separated by two critical field lines H,, and HC2. It appears that even above Hc2 there are still significant numbers of monomers present in the solution. Therefore, as H increases new columns are nucleated and they enhance the transverse correlation between the columns. Above the saturation field
Y.H. Hwang and X-l. WulColloids
Surfaces
A: Physicochem.
Eng. Aspects 80 (1993)
Ifs the monomer concentration approaches zero and magnetic domains become frozen. In this frozen state, an array of magnetic columns are
locally ordered
but globally
disordered.
27
19-27
stimulating discussions and research is partially supported
comments.
The
by a grant
from
Exxon Co.
Associated
with the disorder the system exhibits interesting Many features and hysteresis. metastability
References 1
observed in our experiment indicate the system falls out of thermal equilibrium quickly as H increases and behaves somewhat like a two-dimen-
2
sional glass.
3 4 5 6
Acknowledgments
I
We thank Professor W.I. Goldburg, Professor M. Widom, Dr. C. Yeung, and Mr. M.X. Liu for
8
M. Widom and H. Zhang, Mater. Res. Proc., 248 (1992) 235. G. Helgesen, A.T. Skjeltorp, P.M. Mom, R. Botet and R. Jullien, Phys. Rev. Lett., 61 (1988) 1736. Y.H. Hwang and X-l. Wu, in preparation. T.C. Halsey and W. Toor, Phys. Rev. Lett., 65 (1990) 2820. T.C. Halsey and W. Toor, J. Stat. Phys., 61 (1990) 1257. R. Tao, J.T. Woestman and N.K. Jaggi, Appl. Phys. Lett., 55 (1989) 1844. E. Lemaire, Y. Grasselli and G. Bossis, J. Phys. II France, 2 (1992) 359. G. Helgesen and A.T. Skjeltorp, Spontaneous Formation of Space-Time Structure and Criticality, Kluwer, Dordrecht, 1991. p. 391.