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Langmuir 2004, 20, 8192-8199
Pressure versus Length Isotherms of Homogenous and Mixed One-Dimensional Dipolar Monolayers L. E. Helseth* and T. M. Fischer Department of Chemistry and Biochemistry, Florida State University, Tallahassee, Florida Received April 27, 2004. In Final Form: June 9, 2004 We demonstrate a novel method for compressing and expanding microscopic one-dimensional monolayers consisting of a finite number of aligned magnetic dipoles using a pair of microscopic magnetic barriers. By measuring the interaction between the beads and the barriers, we are able to determine the pressure of the dipolar monolayers. Our sensor can measure one-dimensional pressure in the femto and piconewton regime and is used to probe both homogeneous and mixed monolayers consisting of magnetic beads with diameters 1.0 µm and 2.8 µm. The larger beads appear to be well-described by a formalism taking into account magnetic dipolar interactions, whereas for smaller beads, such a simple picture does not hold. Upon compressing the monolayer above a certain density, it forms a bilayer. This process is governed by steric interactions or dipolar interactions, depending on the applied magnetic field. We also found oddeven effects, where the number of beads in the monolayer determines the initial structure of the bilayer.
1. Introduction The equation of state of an ideal gas was first presented by Robert Boyle in 1662 and in 1917 for two-dimensional systems by Irving Langmuir.1,2 It has proven a powerful approximation for the behavior of dilute particles. Attractive or repulsive interactions give rise to first- and second-order phase transitions, and the nature of these has been studied in detail for two and three dimensions. In this context, one-dimensional (1D) systems have, until recently, not attracted similar interest experimentally, which should be contrasted to the rich variety of theoretical studies of such systems.3 This recent increase in interest is due to their potential applications in microfluidics and nanofabrication.4,5 Moreover, such systems are also found in nature, in e.g., porous rocks or ion channels in the body, and for this reason alone it should be of importance to explore both model and natural systems experimentally.6,7 Suitable 1D potentials for trapping and manipulating dusty plasma crystals, as well as microscopic dielectric and magnetic beads, have been found and characterized in several different fields of physics.8-17 Due to the fact that they can be visualized directly using an optical (1) Boyle, R. A Defence of the Doctrine Touching the Spring and Weight of the Air; Printed by H. Hall for T. Robinson: Oxford, 1662. (2) Langmuir, I. J. Am. Chem. Soc. 1917, 39, 1848. (3) Mathematical Physics in One Dimension; Lieb, E. H., Mattis, D. C., Eds.; Academic Press: New York, 1966. (4) Karlsson, R.; Karlsson, M.; Karlsson, A.; Cans, A. S.; Bergenholtz, J.; A° kerman, B.; Ewing. A. G.; Voinova, M.; Orwar, O. Langmuir 2002, 18, 4186. (5) Gambardella, P.; Dallmeyer, A.; Maiti, K.; Malagoli, M. C.; Eberhardt, W.; Kern, K.; Carbone, C. Nature 2002, 416, 301. (6) Aidley, D. J.; Stansfield, P. R. Ion Channels: Molecules in Action; Cambridge University Press: New York, 1996. (7) Ka¨rger, J.; Ruthven, D. M. Diffusion in Zeolites and Microporous Solids; Wiley: New York, 1992. (8) Wei, Q. W.; Bechinger, C.; Leiderer, P. Science 2000, 287, 625. (9) Bechinger, C. Curr. Opin. Colloid Interface Sci. 2002, 7, 204. (10) Cui, B.; Diamant, H.; Lin, B. Phys. Rev. Lett. 2002, 89, 188302. (11) Cui, B.; Lin, B.; Sharma, S.; Rice, S. A. J. Chem. Phys. 2002, 116, 3119. (12) Furst, E. M.; Gast, A. P. Phys. Rev. E 2000, 61, 6732. (13) Helseth, L. E.; Wen, H. Z.; Fischer, T. M.; Johansen, T. H. Phys. Rev. E 2003, 68, 011402. (14) Liu, B.; Avinash, K.; Goree, J. Phys. Rev. Lett. 2003, 91, 25503. (15) Liu, B.; Avinash, K.; Goree, J. Phys. Rev. E 2004, 69, 036410. (16) Tartarkova, S. A.; Carruthers, A. E.; Dholakia, K. Phys. Rev. Lett. 2002, 89, 283901. (17) McGloin, D.; Carruthers, A. E.; Dholakia, K.; Wright, E. M. Phys. Rev. E 2004, 69, 021403.
microscope, colloidal spheres and dusty plasma crystals have been shown to be excellent for probing, e.g., diffusion, hydrodynamic interactions, electrostatic interactions, and light-induced interactions in one dimension.8-17 However, there exist to our knowledge no measurements of pressure versus length isotherms for 1D systems equivalent to isotherms in two and three dimensions, i.e., subjected to a compression or expansion by an external barrier or piston. The aim of the current study is to present a method for compressing and expanding microscopic one-dimensional colloidal monolayers consisting of a finite number of aligned magnetic dipoles and to use a calibrated force transducer to evaluate the measured one-dimensional pressure versus length isotherms. 2. Experimental Methods The magnetic potential well was created using bismuthsubstituted ferrite garnet films (Lu2.5Bi0.5Ga0.1Fe5O12) of thickness 4 µm and magnetization Ms ) 105 A/m grown on top of a 0.5 mm thick (100) gadolinium gallium garnet (GGG) substrate. In such garnet films, the magnetization vector lies in the plane of the garnet film along one of the magnetic easy axes, where in-plane magnetized domains with magnetization vector in the positive or negative y-direction are separated by, e.g., Bloch walls with a magnetic moment perpendicular to the film. The garnet films possess inverse magneto-elasticity, which means that they react to stress by a change in the direction of the magnetization. In the films used here, cracks arise due to the mismatch in lattice constant between the GGG substrate and the film. These cracks are straight cuts, giving rise to a stress distribution in the garnet film (stress lines), which maps directly into a corresponding distribution of the z-component of the magnetization vector. The stress lines are typically a few nanometers wide, 10-1000 µm long, and are aligned along the crystallographic easy axis.18 We found experimentally that the magnetization vector rotates about the x-axis (i.e. stress line), from lying in the y-direction far away, to the z-direction close to the stress line. To minimize the energy associated with the crack, the magnetization vector point in the opposite (z) direction on either side of the crack, thereby creating a dipolar structure capable of attracting paramagnetic beads (see Figure 1). We also observed that the stress lines generate domain walls that are oriented perpendicular to the line. Far (18) It should be emphasized that there are many ways to create stress lines in garnet films. For example, it is possible to induce stress patterns by controlled impurity implantation or by generating patterns with a focused ion beam.
10.1021/la048949c CCC: $27.50 © 2004 American Chemical Society Published on Web 07/30/2004
Isotherms of One-Dimensional Dipolar Monolayers
Figure 1. Drawing of the experimental setup with a single magnetic barrier. The magnetization vector in the film far away from the stress line is pointing in the y-direction when x > 0, but is rotating toward the positive (y > 0) or negative (y < 0) z-direction when we approach the stress line. Moreover, the polarity of this structure changes sign at the barrier x ) 0 (both the y- and z-component of the magnetization vector). The corresponding magnetic field lines far away from x ) 0 are also shown in the figure, together with an aligned magnetic bead. A polarization microscope is used to visualize the stress line, the magnetic barriers, and the beads. Only the objective is shown in the figure, and the liquid covering the magnetic film is omitted for clarity.
Langmuir, Vol. 20, No. 19, 2004 8193 water at a density of 107 beads/mL were confined within the walls of this ring. The two kinds of (super)paramagnetic beads used here were manufactured by Dynal, coated with a carboxylic acid (COOH-) group. They have radius ab ) 1.4 µm (Dynabeads M270 with magnetic susceptibility χb ) 0.17) and as ) 0.5 µm (Dynabeads MyOne with magnetic susceptibility χs ) 0.3), respectively. The beads and magnetic barriers were visualized by a Leica DMPL polarization microscope used in transmission (the magnetic film is transparent to visible light), and the images were captured by a Hamamatsu CCD camera. Our CCD camera has a frame rate of 1/30 s and can measure bead velocities up to about 50 µm/s. Beyond this velocity, the images of the beads are no longer clear, and accurate data cannot be extracted. After letting the system equilibrate for 20 min, one observes that the beads reside directly on top of the stress lines. The magnetic barriers in the magnetic film were visualized by their polar Faraday rotation, and in this way, we could determine the location of the barriers, as well as their relative position to the magnetic beads, with an accuracy of about 0.5 µm.
3. Magnetic Barriers In this section, we will attempt to calibrate the force from the magnetic barriers, and also try to find a model that can describe them reasonable accurately. We model the field from a single magnetic barrier as consisting of four semi-infinite magnetic surface charges displayed in Figures 1 and 2. The surface charge is defined as the scalar product of the magnetization vector and a unit vector perpendicular to the film, i.e., σ ) M B ‚e bz ) Mz(x,y,z ) 0). In general, the magnetic field from a distribution of magnetic surface charges is given by
H B D(x,y,z) )
∫A ∫σrbr3dx1dy1
1 4π
b r ) (x - x1)e bx + (y - y1)e by + ze bz (1)
Figure 2. Schematic diagram of the two barriers confining a dipolar monolayer consisting of three beads (side view). In the top view we have drawn the magnetic surface charges (the arrows show the magnetic field), but not the beads. Note that at the position of the two barriers, the magnetic surface charges change sign, thereby creating a localized transition that repels aligned magnetic beads. from the line, the domain wall is of Bloch type, separating regions where the magnetization vector is pointing in the positive (x > 0) or negative (x < 0) y-direction, see Figure 1. The plus and minus signs in Figure 1 and 2 show the magnetic surface charges generated by the z-component of the magnetization vector. The magnetic beads always rest directly on top of the stress line, here defined to be located at y ) z ) 0, with a net magnetic moment in the negative y-direction. Aligned beads residing on top of the stress lines are always repelled from the domain wall oriented perpendicular to the stress line since it requires a change in the magnetic moment of the beads, thus increasing their energy. Therefore, two such domain walls separated by a distance L may act as magnetic barriers confining beads (see Figure 2). The typical domain wall coercivity is about 100 A/m, and the domain wall is displaced from its original position by a small external field in the y-direction at a typical rate 1 µm per A/m. A weak magnetic field in the positive y-direction will displace the barrier in Figure 1 in the positive x-direction, while a field in the opposite direction will also change the direction of the displacement. Since a system consisting of two barriers separate domains of different magnetization vector, the two barriers always move in opposite directions and can therefore be used to squeeze the magnetic beads. To probe dipolar monolayers with the magnetic structure, a small ring of diameter about 1 cm was put on top of the magnetic film, and (super)paramagnetic beads immersed in deionized, pure
where the integration is taken over the surface of the magnetic film. The simplest possible model for the magnetic field generated by the magnetization distribution near a stress line (located at y ) 0) and a magnetic barrier (located at x ) 0) consistent with our observations of the polar Faraday rotation is found by assuming two opposing magnetic monopole lines displaced by a distance b from the x-axis and changing sign at the position x1 ) 0, i.e., σ ) bMssign(x1)[δ(y1 - b) - δ(y1 + b)], where Ms is the magnitude of the magnetization of the magnetic film. Here we will treat b as an adjustable parameter that must be consistent with independent measurements of the polar Faraday rotation. A straightforward calculation gives the following y and z components of the magnetic field:
HD y (x,y,z) )
Ms(y - b)b
x (y - b)2 + z2 2πxx2 + (y - b)2 + z2 Ms(y + b)b
x (2) (y + b)2 + z2 2πxx2 + (y + b)2 + z2 HD z (x,y,z) )
Mszb
x (y - b)2 + z2 2πxx2 + (y - b)2 + z2 Mszb
x (3) (y + b)2 + z2 2πxx2 + (y + b)2 + z2 The x-component of the field is negligible in all the cases studied here. The force on a single magnetic bead is then found by noting that its dipole is associated with an energyE ) b ‚H B , where m b ) (4π/3)a3χH B is the magnetic moment, - µ0m
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and the total field is a sum of the field from the magnetic film and the external field, Hex, H B )H BD + H B ex. The force from the magnetic barriers on the beads can be evaluated asF B ) -∇E. Let us first consider the case when H B ex ) He bz and the bead is located far from the barrier (x . 0), which results in the following fields when b is small
HD y (x,y,z) ) HD z (x,y,z) )
Msb2 z2 - y2 π (y2 + z2)2
Msb2 2yz π (y2 + z2)2
(4)
(5)
This is the field of a dipolar line structure, and it is responsible for attracting the beads toward the stress line. Experimentally, we observed that by applying a magnetic field in the z-direction, the beads will reach a new equilibrium position yeq. For example, between the barriers of Figure 2, the beads are displaced in positive y-direction when we apply a magnetic field in positive z-direction, whereas outside the barriers, the beads are displaced in the negative y-direction in order to align with the magnetic field. This behavior can be explained by considering the force on a single magnetic bead in the in the y-direction,
Fy ≈ -
[
16µ0 3 2 4 πzH 2 y a χMs b 2 (z - 3y2) 12 3 3π (y + z ) M b2y s
]
(6)
On the basis of this formula, we expect that the bead comes to rest an equilibrium distance yeq ) sign(x)z/x3 when the external magnetic field is sufficiently strong. Experimentally, we observe that the equilibrium position in an external field H > 2 kA/m is 0.5a > |yeq| > a (varying a little bit in different experiments), which suggests that the center of the bead is resting at a distance of about z ) a above the magnetic film. All our experiments have been done either directly on top of the stress line (y ) 0) in a negligible external magnetic field, or at yeq in a strong external field (H > 2 kA/m). To find the force from the barrier in the x-direction on a single bead when y ) 0 and H ) 0, we note that only the y-component of the field is important
HD y (x,y ) 0,z) ) -
Msb2
x
b + z πxx2 + b2 + z2 2
2
(7)
which gives the following x-component of the force on a single magnetic bead:
Fax (x,z) )
8µ0 3 (Msb2)2 x aχ 2 3π b + z2 (x2 + b2 + z2)2
(8)
We note that the interaction between the beads and the barrier is always repulsive and falls off as 1/x3 at large distances. By pushing the beads in front of a single barrier at different velocities, we measured the velocity, v, and the distance, x, simultaneously. Since inertia is negligible for our beads, the magnetic force must be equal to the hydrodynamic drag ηfav, where f is the hydrodynamic drag coefficient. To obtain an estimate for f, we measured the diffusion coefficient, D, by recording the position of single particles performing Brownian motion on top of the garnet film. From this, we could obtain the mean
Figure 3. Hydrodynamic measurements of the force from a single magnetic barrier on a paramagnetic bead. The circles represent 1.0 µm beads, and the squares represent 2.8 µm beads. The lines correspond to theoretical fits according to eq 8.
square deviation as a function of time, which subsequently allowed us to find the diffusion coefficient. This procedure has been discussed in detail in our previous paper, ref 13. Then we related D to the drag coefficient through StokesEinstein’s relationship, D ) kT/fηa, where k is Boltzmann’s constant, T ) 300 K is the temperature, and η ) 10-3 Ns/m2 is the viscosity of water. We found that f is about 30 for the large beads, which will be used here also for the small ones. The drag force is shown in Figure 3, where the circles show the experimental data points for beads of diameter 1.0 µm, whereas the squares correspond to beads of diameter 2.8 µm. For the large beads, we measured forces up to about 1.2 pN at a distance x ) 5 µm, whereas for the small beads, Fx ) 0.8 pN is reached when x ) 1.5 µm. Due to the limited temporal resolution, the experimental accuracy using this method is estimated to be 10% of the measured force. The solid line shows the theoretical fit using eq 8, as ) 0.5 µm and bs ) 0.14 µm, whereas the dashed line corresponds to ab ) 0.5 µm and bb ) 0.30 µm. The factor of 2 in difference between the obtained values for b could be due to the fact that we have assumed that both beads have the same hydrodynamic drag coefficient and also experience a homogeneous magnetic field. Resolving these issues would require a detailed understanding of the coupling (electrostatic and hydrodynamic) of the beads to the magnetic film, as well as a detailed understanding of the magnetic behavior of the beads in an inhomogeneous field. However, this is outside the scope of the current study and will not be pursued here. The other geometry of interest is that of a strong external magnetic field in the z-direction, H B ex ) He bz. As discussed above, this leads to an equilibrium position yeq ≈ a (at z ≈ a), and the beads are no longer located directly above the stress line. Moreover, mainly the z-component of the field couples to the magnetic moment, which is a reasonable approximation when H > 2 kA/m. Then, the force is found to be
2 Fbx (x,yeq,z) ) µ0χa3HMszb 3
{
}
1 1 - 2 2 2 3/2 [x + (yeq - b) + z ] [x + (yeq + b)2 + z2]3/2 2
In the limit b f 0, this expression becomes
(9)
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yeqz Fbx (x,yeq,z) ) 4µ0χa3HMsb2 2 (10) [x + yeq2 + z2]5/2 At the equilibrium position, the force is seen to be always repulsive and falls off as 1/x5/2 at large distances. Equations 8 and 9 provide us with the possibility to find the force on a single bead in the vicinity of the barrier by measuring the distance x. This will be useful for the measurement of the pressure of an ensemble of beads placed above the stress line and confined between two barriers. 5. Homogeneous Monolayer Consider a general system composed of N dipoles, each of radius ai and separated by a distance dij. The magnetic moments are aligned along an external magnetic field, H, directed perpendicular to the system, and therefore the system has an energy given by N N-1
Ed )
∑ ∑ Eij j>i i)1
Eij ) - µ0m b i(ri)‚H B j(ri)
(11)
Here, µ0 is the permeability of water and the magnetic moment is mi(ri) ) (4π/3)a3χi[H + HD(ri)]. The magnetic field at position ri from dipole j (located at rj) is given by b j(rj)/(4πd3ij). Here, we neglect the influence of H B j(ri) ) -m the field generated by the other magnetic particles on the magnetic moment of particle i, which in the current situation is a good approximation to within an accuracy of 5%. If the particles are identical and separated by a distance d, the expression for the energy becomes
Ed )
4π 9d3
N-1 N
µ0Neffa6χ2H2
Neff )
∑ ∑ i)1 j>i
1
(i - j)3
(12)
where Neff > N when N g 8 and Neff f Nζ(3) (ζ(3) ≈ 1.2) for N f ∞. Let the dipolar system now be confined between two barriers separated by a distance L. The distance from the barrier to the nearest colloidal particle is t. Assume that the distance between the barriers changes by a small distance dL, which means that a work dW ) PddL is done by the barrier on the dipole system. Here, the change in distance dt between the barriers and the two nearest dipoles is negligible, at least an order smaller than dL. Since dW ) -dEd, the one-dimensional pressure of the dipole system is found to be
Pd ) -
dEd 3 ) E dL L - 2t d
L > 2Na
(13)
The force from the magnetic barrier is used as the pressure sensor since its magnitude equals the pressure (P ) Fx) of the dipolar monolayer. The word “pressure” used here is not a quantity related to the size or the statistical behavior of the system. Instead, it reflects the magnitude of the forces acting on the system. Moreover, we will measure the pressure isotherms. The word “isotherm” is also justified since the temperature is constant in our experiments. If we increase the temperature, the colloids will perform more vigorous Brownian motion (similar to the behavior of molecules in a gas or a liquid subjected to increasing temperature). As discussed previously, the force from the barriers has been calibrated by hydrodynamic drag measurements, and we can therefore measure it by measuring the distance,
t, between the beads and the barriers. In some cases, it is not sufficient to only measure the distance between the barriers and the nearest particle due to the long-range nature of the repulsive force. However, we found that it was always sufficient to measure the distance between the three nearest beads and then sum up their contributions in order to find the force, Fx. Note that we measure instantaneous pressure directly from microscopic images and that thermal excitations therefore do not influence the accuracy as long as the beads can be clearly resolved. On the other hand, it should be mentioned that it appeared to be simpler to determine the distance between small beads and the barriers than between large beads and the barriers. This may be related to the change in stray light due to light scattering. In general, the accuracy changes a little bit from experiment to experiment, depending on which barrier system is used and the exact illumination conditions. We typically found an accuracy of the order 50 fN for large beads and about 3 fN for small beads. We would like to emphasize that a system of large beads does not show random behavior due to lack of Brownian motion, and the reproducibility is therefore good, as we have checked through many measurements. For the systems composed of either small beads or a mixture of small and large beads, Brownian motion does play a role. To ensure that our results are correct, we have measured each pressure isotherm two or more times, and the error estimates are therefore the result of numerous measurements. Instead of using the magnetic force from the barrier, we may determine the instantaneous dipolar pressure by measuring the magnetic field, H, as well as the distances, dij, between the beads by inspecting the microscopic images and then inserting it into eqs 11 and 13. If the pressure in the system is purely governed by dipolar interactions, we therefore expect the values found from eq 13 to be equal to the force, Fx. 5.1. Large Beads. In this section, we report findings on a homogeneous monolayer consisting of beads with a diameter of 2.8 µm. In the measurements of these large beads, the z-component of the magnetic field was used to compress the barriers (i.e., alter L). These barriers function more or less independently of the particular number of particles inside them (i.e., they are not influenced by the magnetic beads). Thus, the particular magnetic field needed to reach the close-packed density was determined by the garnet film, not the number of beads. This is just like a barrier system in a Langmuir through, where the barriers operate independently of the 2D monolayer inside them. Parts a and c of Figure 4 show a monolayer consisting of 12 beads during compression, whereas Figure 4 d-f shows a similar process for a monolayer consisting of 15 beads. In both cases, we apply a strong external magnetic field in the z-direction so that the beads are located at an equilibrium position yeq ) a, and eq 9 should be used to calculate the force on single beads. It should also be emphasized that the magnetic field, H, has been varied at each stage of the compression (from H ) 3840 A/m to H ) 8400 A/m), see Figure 4. Figure 5 shows the pressure versus length isotherms for the N ) 12 system, where the circles represent the force from the magnetic barriers on the monolayer, whereas the dashed line is the corresponding dipolar pressure found using eqs 11 and 13. In Figure 5, the pressure values labeled a-c correspond to the images a-c in Figure 4. One observes a single phase of relatively homogeneous density as the monolayer is compressed (Figure 4a). However, when one reaches the closest packed density 1/2a (L ) 33 µm), steric interactions
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Figure 4. Homogeneous monolayers consisting of N ) 12 (a-c) and N ) 15 (d-f) beads of diameter 2.8 µm. In a-c, a monolayer consisting of 12 beads is compressed, and picture a shows the most expanded state in an external field H ) 3840 A/m. Image b shows the monolayer around the close-packed density (H ) 7920 A/m), whereas c visualizes the monolayerbilayer coexistence region (H ) 8400 A/m). Image d shows a relatively compressed state (H ) 3840 A/m) when N ) 15, whereas in e, we have compressed the beads beyond the closepacked configuration (H ) 4560 A/m), and a monolayer-bilayer coexistence region shows up. Finally, in f, a bilayer has formed (H ) 4800 A/m).
Figure 5. Pressure versus length isotherms for a homogeneous monolayer consisting of 2.8 µm beads. The circles represent the force from the magnetic barriers on a monolayer consisting of 12 beads obtained using eq 9, whereas the dashed line is the corresponding dipolar pressure using eq 13. Here a, b, and c correspond to images a-c in Figure 4.
become important and a transition from a monolayer to a bilayer with a phase coexistence region is observed (Figure 4b and c) at a pressure of about 2 pN. Repeating this experiment for N ) 15, gives the pressure versus length isotherm displayed in Figure 6. The circles represent the force from the magnetic barriers on the monolayer, whereas the dashed line is the corresponding dipolar pressure found using eqs 11 and 13. The points d-f correspond to the images d-f in Figure 4. Note that a coexistence pressure of about 0.8 pN (L ) 45 µm) is observed in Figure 6, which is lower than that found in Figure 5. This is explained by the fact that the magnetic field in the case where N ) 15 was H ) 4500 A/m at the close-packed density, whereas for N ) 12 it was H ) 7900 A/m. Thus, the pressure needed to squeeze the monolayer into a bilayer can be controlled by using an external magnetic field. By comparing parts c and e of Figure 4,
Helseth and Fischer
Figure 6. Pressure isotherms for two homogeneous monolayers consisting of 2.8 µm beads. The circles represent the force from the magnetic barriers on a monolayer consisting of 12 beads obtained using eq 9, whereas the dashed line is the corresponding dipolar pressure using eq 13. Here d, e, and f correspond to the images d-f in Figure 4. The squares and the solid line show similar data for N ) 5.
one clearly observes that the structures in the coexistence region are different, which gives rise to different steric interactions. Unfortunately, at higher pressures, it becomes increasingly difficult to measure the distance, t, as the beads overlap with the position of the magnetic barrier. Consequently, our sensor becomes more inaccurate, which is the reason for the larger error bars (0.2 pN) in the coexistence region in Figures 5 and 6. Thus, we cannot draw any certain conclusions about the magnitude of the steric interactions. A third example of a pressure versus length isotherm for N ) 5 is shown in Figure 6, where the squares represent the force from the magnetic barriers on a monolayer consisting of N ) 5 beads, and the solid line is the dipolar pressure. At the close-packed density (L ) 18 µm), a pressure of about 0.7 pN is observed at a magnetic field H ) 5760 A/m. In all the examples above, we observed good agreement between the force from the barriers and the dipolar pressure. This suggests that a monolayer consisting of large paramagnetic beads is well described by eqs 11 and 13 and that thermal fluctuations play a minor role. 5.2. Small Beads. When the bead size decreases, the gravity also decreases and the influence of the thermal excitations increase. To this end, we studied homogeneous monolayers consisting of beads with a diameter of 1.0 µm. Two examples of such monolayers are shown in Figure 7, where a shows an expanded and b a compressed state for N ) 16 both in a constant external magnetic field in the z-direction, H ) 3840 A/m. Figure 7c and d show expanded and compressed states when N ) 17 both in a constant external field in the z-direction, H ) 5280 A/m. Here (and in the rest of this study), the compression and expansion is achieved with a small in-plane magnetic field ( 30 µm, the inaccuracy of the measured pressure is too large and cannot be determined for this system with our method. For comparison, we see that even Boyle’s law (dotted line) gives too-high pressure, and we therefore believe that our method cannot monitor pressures below 10 fN accurately. One may also naively think that the monolayer possess no significant dipolar interactions. However, we think that such a conclusion is not entirely justified, since we observe a well-ordered monolayer exhibiting the monolayerbilayer transition seen in Figure 8b at L ) 25 µm. Here, two beads pop out at Nd ) 20 µm, long before the closepacked condition has been reached (2Na ) 16 µm). Upon repeating this experiment in zero external field (y ) 0), we never observed this phenomena. Instead the beads are pressed out at the close-packed density by steric repulsion similar to that in Figure 4. This suggests that dipolar interactions are at least partially responsible for the squeeze-out when the external magnetic field is sufficiently strong.
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Figure 8. Theoretical (a) and experimental (b) pressure versus length isotherms. See the text for details.
By compressing the monolayer further, we see that a pressure larger than the uncertainty can be definitely detected when L < 20 µm. At L ) 13 µm, a dense bilayer with a structure similar to that in Figure 4f has developed. Upon repeating the whole experiment at H ) 5280 A/m with N ) 17 beads, we observed that only one bead pops out initially at L ) 30 µm, see Figure 8d. This should be compared to Figure 8b, where two beads are squeezed out. These observations demonstrate the presence of oddeven effects upon squeezing the beads out of the monolayer. An odd N (N ) 2n - 1 where n ) 0,1,2,...) results in one single bead being squeezed out during compression, whereas an even N (N ) 2n) results in two beads, as dictated by symmetry. We found that the excited beads jump on top of the monolayer, trying to align with field from the resting beads as well as with the magnetic field above the stress line, see Figure 7b and d. If this process is purely governed by gravity and dipolar forces, the gain in dipolar energy should compensate for the increase in gravitational potential energy (4π∆Fg2a4/3) obtained by lifting the bead a distance of the order 2a, where ∆F ) 600 kg/m3 is the difference in density of the beads and the water, and g ) 10 N/kg. Assuming the interaction between the squeezed-out bead(s) and the resting monolayer to be negligible, the critical barrier length Lc ≈ Ndc (neglecting the small change in L during squeeze-out) is given by
Lc ≈ N
[
]
µ0a2 2 2 χ H ζ(3) 6g∆F
1/3
(14)
We find that Lc ≈ 36 µm when N ) 16 and Lc ≈ 50 µm when N ) 17. These values are higher than the experimental values 25 µm (N ) 16) and 30 µm (N ) 17), and
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Figure 9. Mixed monolayers consisting of a combination of 2.8 µm and 1.0 µm beads in zero external magnetic field (H ) 0) (see the text for details).
Figure 11. Pressure versus length isotherms for a monolayer consisting of six small and four large beads (see the text for details).
Figure 10. Pressure versus length isotherms for a monolayer consisting of four small and five large beads (see the text for details).
it is therefore reasonable to believe that thermal excitations also play a role in stalling the formation of a bilayer. 6. Mixed Monolayer In the case of a mixture of two different magnetic particles with different susceptibility, the situation is more complicated, and one must, in general, evaluate eq 11 explicitly. We will assume here that the pressure is still given by eq 13. Figure 9 shows two examples of mixed monolayers consisting of a finite number of beads. Here, only a very weak magnetic field in the y-direction is applied (
