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Studies of two-dimensional lattices using ferrofluid
Journal of Magnetism North-Holland
and Magnetic
Materials
37 (1983) 253-256
STUDIES OF TWO-DIMENSIONAL
LATTICES
253
USING FERROFLUID
*
A.T. SKJELTORP Institutefor Energy Technology, 2007 Kjeller, Noruq; Received
18 November
and Becton Center, Yale University, New Haven, CT 06520. USA
1982; in revised form 21 March
1983
By exerting a magnetic filed H normal to a ferrofluid layer, a triangular lattice of droplets is formed when H exceeds a by reducing H and the effects of dislocations and disclinations may be characteristic critical field. The lattice may be “heated’ studied. This offers a new experimental system for direct visual observations of phenomena related to two-dimensional melting.
1. Introduction Phase transitions in two-dimensional systems are sufficiently simple for theoretical models to produce quantitative predictions. In particular, recent models describing melting in two-dimensional lattices suggest that the movement of lattice defects is a controlling factor. There are relatively few experimental systems for direct visual inspections of such effects. These include polystyrene spheres trapped on the surface of water [I] and hard spheres forming a lattice on a vibrating solid surface [2]. Other systems for indirect observations include an electron layer floating on liquid helium [3], thin layers of liquid crystals [4] and gases adsorbed on graphite [5]. However, in all of these systems there are deviations from ideal models which may obscure theoretically important effects. The experiments have therefore produced conflicting results. This is also the case for computer simulations [6]. In the present work we introduce the use of ferrofluid as a new experimental system for direct visual inspection of two-dimensional phase transition studies. A ferrofluid has the remarkable property that a vertical magnetic field H produces a free-surface instability [7]. For H above a characteristic critical field, H,,, the surface becomes * Work supported
in part by NSF Grant no. DMR-798175 and in part by NATO Research Grant RG 133.8 1.
0304-8853/83/0000-0000/$03.00
0 1983 North-Holland
corrugated forming a regular hexagonal pattern of peaks and troughs. This instability is a special kind of first order two-dimensional phase transition reminiscent of a Benard instability [S]. The two phenomena are analogous in the sense that H corresponds to a temperature gradient AT induced in a fluid layer heated from below. When AT exceeds a characteristic critical temperature gradient AT,, convection starts. The order parameter here is the fluid velocity, whereas the height of the peaks above the initial surface is the order parameter for the ferrofluid. The most extensive theoretical work on the surface instability of a ferrofluid has been performed by Gailitis [9]. By using an energy minimizing principle including the magnetic energy, surface tension energy and hydrostatic energy, he was able to find the possible forms the surface may take for different field strengths as well as the perturbation amplitudes. In addition to the hexagonal lattice formed at H,-, the model also predicts a sudden transition form a hexagonal to a square lattice at a critical field H,,. The critical fields are found to be: H,?, = 87+~Apg)“~/.+
+ l)/(p
- I)‘,
(1)
H;,=H;,[l
l)‘/(p+
l)4]
(2)
+231(~-
Here, p is the magnetic permeability, CJ is the surface tension, Ap is the density difference across the interface and R is the gravitational accelera-
254
A. T. Skjeltorp / Studies of 20 lattices using ferrojluid
tion. The transitions given in eqs. (1) and (2) are for increasing fields. For decreasing fields, the transitions are shifted to lower critical field values, corresponding to a hysteresis effect. So far only one experiment has shown transition to a square lattice [lo], but no data or documentation has been given. Little attention has been devoted to thin layers of ferrofluid for which the troughs reach the substrate and the fluid breaks up into droplets. The critical fields for the onsets of the instabilities are also expected to be dependent on the layer thickness [lo]. The lattice formed represents a new experimental system with interesting properties. The present work is intended as an introduction to such studies.
3. Results Fig. 1 shows a sequence of photographs of the lattices formed for sudden application of different fields. In fig. la, H = 165 Oe which is close to the lowest field for which a lattice is formed. It may be seen that there are domains of “perfect” lattices and decreasing lattice constants for increasing H. The lattice defects may have been due to field inhomogeneities, finite size effects or uneven thickness of the starting fluid layer, but this was not studied in detail. The droplet separation, a, was measured within the perfect domains for different H. Fig. 2 shows a dimensionless plot of ( u*)~ versus the hypercritical field h , . These quantities are defined as: a* = a,/a,
2. Experimental
h, = (H/H,)’
The present experiments were made using a commercial kerosene based ferrofluid with saturation magnetization 400 G [ll]. A thin ferrofluid layer (typically 1 mm) was prepared at the bottom of a rectangular quartz test chamber (4.5 x 4 x 1 cm3). This was placed inside a large solenoid producing a homogeneous magnetic field (O-750 Oe) normal to the fluid surface. By increasing the field above a certain critical field, the surface became unstable and fairly regular hexagonal patterns of droplets were formed. The final results were therefore quite different from the free surface instability described in the introduction in that there was practically no fluid left between the droplets. By increasing the thickness of the fluid layer, increasingly more fluid was left between the droplets and a second sublattice of smaller droplets could be formed. This was not studied in detail, and for the present experiments the layer thickness was always so thin that only one distinct pattern of droplets was formed. The observations were made with a beam of light through the cell projected onto a screen. Only static effects were recorded, whereas interesting dynamic effects observed will only be discussed qualitatively.
Here, a, = (4~&)(
(3) -
1.
~r/Apg)“~
(4)
(5)
is the spacing between the peaks expected for a hexagonal lattice on a free surface with no influence from a substrate [9]. For the present kerosene based ferrofluid, the surface tension was taken as u = 26 dyn/cm and the density differences across the interface ferrofluid and air Ap = 1.2 g/cm3. These values give a, = 1.08 cm. For the present film, the measured critical field used in eq. (4) was H, = 165 Oe. As expected, this is higher than the calculated value H,-, = 113 Oe for a free surface from eqn. (1) using the parameters given above and p = 3.1. The first point to note in fig. 2 is the first order character of the transition. To a good approximation (a*)* increases linearly with h, up to a hypercriticality of approximately 5. It is not clear to what extent a* may be considered as an order parameter here. However, using simple arguments it may be shown that to a first approximation the height of the conical droplets, 5, is proportional to l/a2 or (u*)~. For a free surface the height of the perturbation, 5, is the order parameter used in Gailitis treatment. For high fields, the lattice seems to saturate. A possible second transition to a quadratic lattice
A. T. Skjeltorp
/ Studies of 2D lattices using ferrofluid
0
5
255
10
h,
15
Fig. 2. Reduced reciprocal lattice constant cal field h, as defined in eqs. (3)-(4).
5mm
5rnm 1
Fig. 1. Ferrofluid lattices Oe and (c) 460 Oe.
formed
in fields: (a) 165 Oe, (b) 265
was not observed for the maximum field available (750 Oe). This is expected, as the critical field for this transition is calculated from eq. (2) to be H c2 = 890 Oe.
20 a* versus hypercriti-
4. Melting in two dimensions and use of ferrofluid The essential feature of the ferrofluid lattice once it is formed is that the individual droplet seems to be trapped in an energy well. The depth of the well determines the stability of the lattice as a two-dimensional system. This may be treated using an extension of the method employed by Gailitis [9]. The unique feature of the ferrofluid system is that the lattice constant may be chosen by varying the applied static field. Furthermore, once the lattice is formed, melting may be induced by reducing the field. As discussed earlier, the order parameter for the onset of the instability is the height of the corrugation above the flat surface. However, once separate droplets have been formed, and we want to use this system for studies of two-dimensional melting, the order parameter relates to the topology of the lattice, which will be discussed in the following. Recent models describing two-dimensional melting suggest that this is controlled by the behaviour of lattice defects [6]. Examples of these are shown schematically in fig. 3 for a hexagonal lattice. The dislocation in fig. 3a is formed by the addition of an extra row of lattice sites which does not change the long-range directional order. The disclinations shown in figs. 3b and 3c produce rotations of +~/3 and -a/3 around a central site with 5-fold and 7-fold symmetry, respectively. These defects are seen to change the long-range directional order. Finally, in fig. 3d, a pair of +~/3 and -v/3 disclinations is shown to form a translation dislocation.
A. T. Skjeltorp
256
/ Studies of 2D lattices using ferrofluid
dislocation
+w3
g-w3
Fig. 3. Defects in hexagonal two-dimensional lattices. (a) A dislocation with Burgers vector 6. (b), (c): 71/3 and - ?r/3 disclinations with 5-fold and 7-fold symmetry around the central site. (d) Combination of the n/3 and - n/3 disclination forming a dislocation.
The Kosterlitz-Thouless model for melting [ 121 predicts that unbinding of pairs of dislocations leads to melting. Halperin and Nelson [13] introduced also the possibility of unpairing of disclinations from a new so-called “hexatic” intermediate phase before melting. To identify the transitions, a positionaland a directional order parameter is introduced with correlation functions G(r) and O(r), respectively [4]. The term quasi-long-range order has been used for an algebraic decay with distance r for these correlations. This is in contrast to three-dimensional crystals for which the correlations remain finite. For the two defect-mediated melting processes described above, unpairing of dislocations and disclinations, the decay of the correlation functions with distance, changes from algebraic to exponential for G(r) and O(r), respectively. A glance at fig. 1 shows that the ferrofluid lattices contain both dislocations and disclinations of the types shown in fig. 3. By “heating” (reducing H) a particular lattice, “melting” was seen to
take place involving the movement of many lattice defects at the same time. The domains with perfect lattices changed both in position, orientation and process. The direct vissize during this “melting” ual observations of these effects may be useful in relation to the models of lattice-defect mediated melting described above. However, for this it is necessary to do more quantitative measurements of the correlation functions, by using digital imaging techniques. Work on this is in progress. Another interesting possibility is to study commensurate-incommensurate transitions [6]. This may be realized by bonding thin permalloy discs to a substrate forming a regular lattice. In a field, this will produce energy wells for which the separation may be chosen at will in relation to the free ferrofluid lattice. Hence the use of ferrofluid opens many possibilities for model studies of twodimensional systems.
References [I] P. Pieranski,
Phys. Rev. Lett. 45 (1980) 569. [2] P. Pieranski et al., Phil. Mag. A37 (1978) 107. [3] C.C. Grimes and G. Adams, Phys. Rev. Lett. 42 (1979) 795. [4] R. Pindak and D. Moncton, Phys. Today (May 1982) 57. [5] For a recent review, see SK. Sinha, in: Nonlinear Phenomena at Phase Transitions and Instabilities, ed. T. Riste (Plenum, New York, 1982) p. 433. [6] For a recent review, see J.M. Kosterlitz. in: Nonlinear Phenomena at Phase Transitions and Instabilities, ed. T. Riste (Plenum, New York, 1982) p. 397. [7] A review of the static and dynamic properties of ferrofluids is given by M.I. Shliomis and Yu. L. Kaikher. IEEE Trans. Magn. MAG-16 (1980) 237. [S] M.G. Velarde and C. Normand, Sci. Am. 243 (1980) 79. (91 A. Gailitis, J. Fluid Mech. 82 (1977) 401. [lo] M.D. Cowley and R.E. Rosensweig, J. Fluid Mech. 30 (1967) 671. [ 1 l] Obtained from Ferrofluidics Corp., 40 Simon St., Nashua, NH 03061, USA. [ 121 J.M. Kosterlitz and D.J. Thouless. J. Phys. C 6 (1973) 118. [13] B.I. Halperin and D.R. Nelson, Phys. Rev. Lett. 41 (1978) 121.
and Magnetic
Materials
37 (1983) 253-256
STUDIES OF TWO-DIMENSIONAL
LATTICES
253
USING FERROFLUID
*
A.T. SKJELTORP Institutefor Energy Technology, 2007 Kjeller, Noruq; Received
18 November
and Becton Center, Yale University, New Haven, CT 06520. USA
1982; in revised form 21 March
1983
By exerting a magnetic filed H normal to a ferrofluid layer, a triangular lattice of droplets is formed when H exceeds a by reducing H and the effects of dislocations and disclinations may be characteristic critical field. The lattice may be “heated’ studied. This offers a new experimental system for direct visual observations of phenomena related to two-dimensional melting.
1. Introduction Phase transitions in two-dimensional systems are sufficiently simple for theoretical models to produce quantitative predictions. In particular, recent models describing melting in two-dimensional lattices suggest that the movement of lattice defects is a controlling factor. There are relatively few experimental systems for direct visual inspections of such effects. These include polystyrene spheres trapped on the surface of water [I] and hard spheres forming a lattice on a vibrating solid surface [2]. Other systems for indirect observations include an electron layer floating on liquid helium [3], thin layers of liquid crystals [4] and gases adsorbed on graphite [5]. However, in all of these systems there are deviations from ideal models which may obscure theoretically important effects. The experiments have therefore produced conflicting results. This is also the case for computer simulations [6]. In the present work we introduce the use of ferrofluid as a new experimental system for direct visual inspection of two-dimensional phase transition studies. A ferrofluid has the remarkable property that a vertical magnetic field H produces a free-surface instability [7]. For H above a characteristic critical field, H,,, the surface becomes * Work supported
in part by NSF Grant no. DMR-798175 and in part by NATO Research Grant RG 133.8 1.
0304-8853/83/0000-0000/$03.00
0 1983 North-Holland
corrugated forming a regular hexagonal pattern of peaks and troughs. This instability is a special kind of first order two-dimensional phase transition reminiscent of a Benard instability [S]. The two phenomena are analogous in the sense that H corresponds to a temperature gradient AT induced in a fluid layer heated from below. When AT exceeds a characteristic critical temperature gradient AT,, convection starts. The order parameter here is the fluid velocity, whereas the height of the peaks above the initial surface is the order parameter for the ferrofluid. The most extensive theoretical work on the surface instability of a ferrofluid has been performed by Gailitis [9]. By using an energy minimizing principle including the magnetic energy, surface tension energy and hydrostatic energy, he was able to find the possible forms the surface may take for different field strengths as well as the perturbation amplitudes. In addition to the hexagonal lattice formed at H,-, the model also predicts a sudden transition form a hexagonal to a square lattice at a critical field H,,. The critical fields are found to be: H,?, = 87+~Apg)“~/.+
+ l)/(p
- I)‘,
(1)
H;,=H;,[l
l)‘/(p+
l)4]
(2)
+231(~-
Here, p is the magnetic permeability, CJ is the surface tension, Ap is the density difference across the interface and R is the gravitational accelera-
254
A. T. Skjeltorp / Studies of 20 lattices using ferrojluid
tion. The transitions given in eqs. (1) and (2) are for increasing fields. For decreasing fields, the transitions are shifted to lower critical field values, corresponding to a hysteresis effect. So far only one experiment has shown transition to a square lattice [lo], but no data or documentation has been given. Little attention has been devoted to thin layers of ferrofluid for which the troughs reach the substrate and the fluid breaks up into droplets. The critical fields for the onsets of the instabilities are also expected to be dependent on the layer thickness [lo]. The lattice formed represents a new experimental system with interesting properties. The present work is intended as an introduction to such studies.
3. Results Fig. 1 shows a sequence of photographs of the lattices formed for sudden application of different fields. In fig. la, H = 165 Oe which is close to the lowest field for which a lattice is formed. It may be seen that there are domains of “perfect” lattices and decreasing lattice constants for increasing H. The lattice defects may have been due to field inhomogeneities, finite size effects or uneven thickness of the starting fluid layer, but this was not studied in detail. The droplet separation, a, was measured within the perfect domains for different H. Fig. 2 shows a dimensionless plot of ( u*)~ versus the hypercritical field h , . These quantities are defined as: a* = a,/a,
2. Experimental
h, = (H/H,)’
The present experiments were made using a commercial kerosene based ferrofluid with saturation magnetization 400 G [ll]. A thin ferrofluid layer (typically 1 mm) was prepared at the bottom of a rectangular quartz test chamber (4.5 x 4 x 1 cm3). This was placed inside a large solenoid producing a homogeneous magnetic field (O-750 Oe) normal to the fluid surface. By increasing the field above a certain critical field, the surface became unstable and fairly regular hexagonal patterns of droplets were formed. The final results were therefore quite different from the free surface instability described in the introduction in that there was practically no fluid left between the droplets. By increasing the thickness of the fluid layer, increasingly more fluid was left between the droplets and a second sublattice of smaller droplets could be formed. This was not studied in detail, and for the present experiments the layer thickness was always so thin that only one distinct pattern of droplets was formed. The observations were made with a beam of light through the cell projected onto a screen. Only static effects were recorded, whereas interesting dynamic effects observed will only be discussed qualitatively.
Here, a, = (4~&)(
(3) -
1.
~r/Apg)“~
(4)
(5)
is the spacing between the peaks expected for a hexagonal lattice on a free surface with no influence from a substrate [9]. For the present kerosene based ferrofluid, the surface tension was taken as u = 26 dyn/cm and the density differences across the interface ferrofluid and air Ap = 1.2 g/cm3. These values give a, = 1.08 cm. For the present film, the measured critical field used in eq. (4) was H, = 165 Oe. As expected, this is higher than the calculated value H,-, = 113 Oe for a free surface from eqn. (1) using the parameters given above and p = 3.1. The first point to note in fig. 2 is the first order character of the transition. To a good approximation (a*)* increases linearly with h, up to a hypercriticality of approximately 5. It is not clear to what extent a* may be considered as an order parameter here. However, using simple arguments it may be shown that to a first approximation the height of the conical droplets, 5, is proportional to l/a2 or (u*)~. For a free surface the height of the perturbation, 5, is the order parameter used in Gailitis treatment. For high fields, the lattice seems to saturate. A possible second transition to a quadratic lattice
A. T. Skjeltorp
/ Studies of 2D lattices using ferrofluid
0
5
255
10
h,
15
Fig. 2. Reduced reciprocal lattice constant cal field h, as defined in eqs. (3)-(4).
5mm
5rnm 1
Fig. 1. Ferrofluid lattices Oe and (c) 460 Oe.
formed
in fields: (a) 165 Oe, (b) 265
was not observed for the maximum field available (750 Oe). This is expected, as the critical field for this transition is calculated from eq. (2) to be H c2 = 890 Oe.
20 a* versus hypercriti-
4. Melting in two dimensions and use of ferrofluid The essential feature of the ferrofluid lattice once it is formed is that the individual droplet seems to be trapped in an energy well. The depth of the well determines the stability of the lattice as a two-dimensional system. This may be treated using an extension of the method employed by Gailitis [9]. The unique feature of the ferrofluid system is that the lattice constant may be chosen by varying the applied static field. Furthermore, once the lattice is formed, melting may be induced by reducing the field. As discussed earlier, the order parameter for the onset of the instability is the height of the corrugation above the flat surface. However, once separate droplets have been formed, and we want to use this system for studies of two-dimensional melting, the order parameter relates to the topology of the lattice, which will be discussed in the following. Recent models describing two-dimensional melting suggest that this is controlled by the behaviour of lattice defects [6]. Examples of these are shown schematically in fig. 3 for a hexagonal lattice. The dislocation in fig. 3a is formed by the addition of an extra row of lattice sites which does not change the long-range directional order. The disclinations shown in figs. 3b and 3c produce rotations of +~/3 and -a/3 around a central site with 5-fold and 7-fold symmetry, respectively. These defects are seen to change the long-range directional order. Finally, in fig. 3d, a pair of +~/3 and -v/3 disclinations is shown to form a translation dislocation.
A. T. Skjeltorp
256
/ Studies of 2D lattices using ferrofluid
dislocation
+w3
g-w3
Fig. 3. Defects in hexagonal two-dimensional lattices. (a) A dislocation with Burgers vector 6. (b), (c): 71/3 and - ?r/3 disclinations with 5-fold and 7-fold symmetry around the central site. (d) Combination of the n/3 and - n/3 disclination forming a dislocation.
The Kosterlitz-Thouless model for melting [ 121 predicts that unbinding of pairs of dislocations leads to melting. Halperin and Nelson [13] introduced also the possibility of unpairing of disclinations from a new so-called “hexatic” intermediate phase before melting. To identify the transitions, a positionaland a directional order parameter is introduced with correlation functions G(r) and O(r), respectively [4]. The term quasi-long-range order has been used for an algebraic decay with distance r for these correlations. This is in contrast to three-dimensional crystals for which the correlations remain finite. For the two defect-mediated melting processes described above, unpairing of dislocations and disclinations, the decay of the correlation functions with distance, changes from algebraic to exponential for G(r) and O(r), respectively. A glance at fig. 1 shows that the ferrofluid lattices contain both dislocations and disclinations of the types shown in fig. 3. By “heating” (reducing H) a particular lattice, “melting” was seen to
take place involving the movement of many lattice defects at the same time. The domains with perfect lattices changed both in position, orientation and process. The direct vissize during this “melting” ual observations of these effects may be useful in relation to the models of lattice-defect mediated melting described above. However, for this it is necessary to do more quantitative measurements of the correlation functions, by using digital imaging techniques. Work on this is in progress. Another interesting possibility is to study commensurate-incommensurate transitions [6]. This may be realized by bonding thin permalloy discs to a substrate forming a regular lattice. In a field, this will produce energy wells for which the separation may be chosen at will in relation to the free ferrofluid lattice. Hence the use of ferrofluid opens many possibilities for model studies of twodimensional systems.
References [I] P. Pieranski,
Phys. Rev. Lett. 45 (1980) 569. [2] P. Pieranski et al., Phil. Mag. A37 (1978) 107. [3] C.C. Grimes and G. Adams, Phys. Rev. Lett. 42 (1979) 795. [4] R. Pindak and D. Moncton, Phys. Today (May 1982) 57. [5] For a recent review, see SK. Sinha, in: Nonlinear Phenomena at Phase Transitions and Instabilities, ed. T. Riste (Plenum, New York, 1982) p. 433. [6] For a recent review, see J.M. Kosterlitz. in: Nonlinear Phenomena at Phase Transitions and Instabilities, ed. T. Riste (Plenum, New York, 1982) p. 397. [7] A review of the static and dynamic properties of ferrofluids is given by M.I. Shliomis and Yu. L. Kaikher. IEEE Trans. Magn. MAG-16 (1980) 237. [S] M.G. Velarde and C. Normand, Sci. Am. 243 (1980) 79. (91 A. Gailitis, J. Fluid Mech. 82 (1977) 401. [lo] M.D. Cowley and R.E. Rosensweig, J. Fluid Mech. 30 (1967) 671. [ 1 l] Obtained from Ferrofluidics Corp., 40 Simon St., Nashua, NH 03061, USA. [ 121 J.M. Kosterlitz and D.J. Thouless. J. Phys. C 6 (1973) 118. [13] B.I. Halperin and D.R. Nelson, Phys. Rev. Lett. 41 (1978) 121.