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Flexible magnetic filaments
Current Opinion in Colloid & Interface Science 10 (2005) 167 – 175 www.elsevier.com/locate/cocis
Flexible magnetic filaments A. Ce¯bers * Institute of Physics, Salaspils-1, LV-2169, Latvia Available online 18 August 2005
Abstract Flexible magnetic filaments, created recently in different labs, possess interesting physical properties such as novel buckling instabilities. These magnetic filaments may be used as sensors in different conditions or as mixers for microfluidics applications. This paper reviews recent achievements in this new field, based on the extension of the Kirchhoff model of elastic rods by including the orientating effect of the magnetic field upon the filaments. As an illustration of the various intriguing properties revealed by the model, self-propelling under the action of an AC magnetic field is discussed. D 2005 Elsevier Ltd. All rights reserved. Keywords: Magnetic filaments; Curvature elasticity; Buckling instability; Mixing; Self-propulsion
1. Introduction Filamentary objects are ubiquitous in nature carrying out several important functions inside the cell [1]. Magnetic filaments, composed of chains of ferromagnetic particles, are used by the magnetotactic bacteria for navigation purposes [2,3]. Chains of super-paramagnetic particles are mesoscopic elements, which determine the properties of the magnetorheological suspensions [4]. Magnetic filaments with interesting properties were created recently [5&&, 6&&,7,8&&] by linking micron size super-paramagnetic particles with some polymer. As a result, objects with a flexibility that is a billion times greater than that of different common materials are obtained.
2. Statics of magnetic filaments Colloidal chemistry aspects of the synthesis of magnetic filaments, which are super-paramagnetic particle chains irreversibly linked by some polymer (PAA [5&&,6&&,7] or covalent biotin – streptovidin binding [8&&]) are considered in
* Tel.: +371 9195961; fax: +371 7901214. E-mail address: [email protected]. 1359-0294/$ - see front matter D 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.cocis.2005.07.002
Refs. [6&&,7]. It is believed that the irreversible linking of the suspension of the super-paramagnetic particles is driven by depletion interactions in the polymer solution when the incubation time is short enough [7]. Rather exotic magnetic filaments are created by incorporating 50 nm size magnetosomes, produced by magnetotactic bacteria, in peptidenanotube template [9]. Magnetic filament can be considered as a Kirchhoff elastic rod taking into account the orientating magnetic field effect on the filament [10&&]. Therefore its energy reads Z Z 1 1 ð l 1Þ 2 2 2 Y Y 2 h t dl a H0 dl E¼ C 2 R2 8ð l þ 1Þ Z Kðl Þdl ð1Þ Here R is the radius of the curvature of the centre line of a filament, C is the curvature elasticity constant which can be expressed through the elastic properties of linkers [5&&], a is the radius of the circular cross section of the filament with total length 2L, H 0 is the applied magnetic field strength and l is the effective magnetic permeability of the filament. The Lagrange multiplier K, which has the meaning of compressional stress, accounts for local inextensibility of the filament. Different properties of the magnetic filaments were investigated utilizing this model. Its important feature lies in the appearance of volume torques due to the
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orienting effect of the applied field. The torque balance reads Y
h Yi Y dM Y þ t xF þ T ¼ 0 dl Y
2
ð2Þ Y
Þ 2 2 Here T ¼ 8ðl1 ðlþ1Þ a H0 sinð2#Þb Yis the torque per unit length due to the applied field, F is the stress in the Y filament, b ¼ ½Y t xnY is the binormal to the filament, and Y t ¼ ðcos#; sin#Þ is the tangent vector given by the angle #, which it makes with the field. The torque balance equation, accounting for the momentum stresses M = C / R [10&&] in the filament and Y the condition of its mechanical equilibrium dF =dl ¼ 0, && && predicts the buckling instability [5 ,10 ,11] for a filament orientated perpendicularly to the applied field. The critical ðl1Þ2 2 2 2 value of the magnetoelastic number Cm ¼ a L H0 8 ð lþ1 ÞC Y Y for the free F ¼ 0 and unclamped M ¼ 0 ends of the filament reads CmðnoÞ ¼ ð2n þ 1Þ2 k2 =8ð# ¼ k=2þ eðoÞ sin ðð2n þ 1Þkl=ð 2LÞÞ; odd modeÞ and CmðneÞ ¼ n2 k2 =2ð# ¼ k=2 þ eðeÞ cosðnkl=ð 2LÞÞ; even modeÞ. The lowest critical value, Cmn(e) = 0, corresponds to the rotation of the filament as whole in the direction of the magnetic field. The lowest critical value, k2 / 8, for no uniform deformation of the filament corresponds to the first odd mode (n = 0). The last condition corresponds to the appearance of the steady nontrivial solutions ofpthe Eq. ffiffiffiffiffiffiffiffiffi ffi (2) found in Ref. [12] 2L / l m = k, where lm ¼ L= 2Cm is the characteristic magnetic length of the filament. Among the other problems solved in Ref. [12], the bending of the filament containing the magnetic tip under the action of transversal force should be mentioned. The relation of these problems to the magneticsense organs of homing pigeons may be pointed out [12]. Development of the first odd mode leads to the formation of hairpins, which have been observed experimentally recently [5&&]. In the case of a filament with infinite length, the hairpin shape can be calculated theoretically [5&&,11]. This, for the maximal radius of the curvature of the filament, gives pffiffiffiffiffiffiffiffiffiffi L ¼ 2Cm : ð3Þ R max
Eq. (3), and the value of the curvature measured for the filaments that are linked with a polyacrylic acid [5&&] allows one to estimate the value of the magnetoelastic number, which is proportional to the square of the field strength. According to the experimental data [5&&], Cm = 12 at a quite moderate field 4k Oe is already well above the critical value for buckling instability. The family of hairpins of filaments with finite length calculated numerically [10&&,11] is shown in Fig. 1. The curvature at the tip of the hairpin obeys the theoretical relation given by Eq. (3) if the Cm number is large enough. Numerical studies reveal interesting properties of the filaments, namely, that the hairpin shape of the filament also exists if the lengths of their arms are
Fig. 1. Family of hairpins. Cm = 1.5(1); 3(2); 5(3); 12(4).
different. If the length of the shorter arm is small enough, the hairpin relaxes to a straight configuration. At the corresponding initial perturbation, shapes with several hairpins can be established as long living transient states, which relax to a final straight configuration along the direction of the field. Shapes with several buckles, observed in the experiments [5&&], may arise due to some stabilization of these transitory configurations due to the interaction with a solid wall. A similar behaviour was found recently for actin filaments in a nematic solvent (solution of fd viruses) [13] — namely, hairpins with unequal lengths of the arms relax to a straight configuration. The relations given above allow one to estimate the characteristic interaction strength k BTC of the filament with background with the director a nematic R 2 nYn : 12 kB T C tY Y nn dl. The critical length of the shortest arm required for the straightening of the hairpin can be estimated to be equal to the minimal curvature radius given by Eq. (3) (Fig. 1e [13]). Then the characteristic correlation pffiffiffiffiffiffiffiffiffi length of the filament in a nematic background, k ¼ lp =C, is about 5 Am. This value is in reasonable agreement with the picture shown in Fig. 1a [13]. Since the persistence length of the actin filament l p = C / k BT is 16 Am, the characteristic interaction strength of the actin filament with a nematic background can be estimated as 2.510 11 erg/cm, which corresponds to the weak anchoring of the nematic background to the actin filament characterized by the anchoring parameter x = k BTC / K ; 1.310 3. Its value allows one to presume that the anchoring of actin filaments to the nematic background is caused by steric interactions. The interaction energy of magnetic filaments with a nematic background is an important issue for the ferronematics — nematic liquid crystals with incorporated chains of ferromagnetic particles [14,15].
A. Ce¯bers / Current Opinion in Colloid & Interface Science 10 (2005) 167 – 175
169
Fig. 2. Bending of magnetic filament. Transversal force on the right end of filament is applied in y axis direction. Points — experimental data [8&&]. Dashed line — Cm = 0; F = 0.15.
Direct measurements of the elastic properties of magnetic filaments were carried out recently by studying their bending deformations due to the force applied by laser tweezers [8&&]. Filaments of super-paramagnetic particles linked by polyethylene glycol with different molecular weights were studied. The value of the elasticity constant C was found to depend on the type of experiment carried out — bending at applied transversal force, buckling at the compressional force or relaxation to straight configuration. In Ref. [16] by numerical calculation it was shown that the model of the magnetic filament (1) describes the set of experiments in Ref. [8&&] reasonably well. The shape of the filament is compared to the experimental data for Cm = 0.9 and F = 0.45 (the force applied transversally to the filament is scaled with C / L 2) in Fig. 2. An interesting property of the magnetic filaments is that they can be used as force sensors, since their stiffness depends on the magnetic field. The displacement of the filament y(x) can be found from the torque balance (Eq. (2)) Y at F ¼ Fa eYy , where F a is the force applied transversally to the field direction, and satisfying the boundary conditions y(0) = 0; y x (0) = 0; y xx (1) = 0 Fa y¼ 2Cm
pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ! sh 2Cm sh 2Cmð1 xÞ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi x : 2Cmch 2Cm
The stiffness of the filament k = F a / y(1) in dimensionless units is given by k¼
2Cm pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 1 tanh 2Cm = 2Cm
and can be changed due to the applied field by several orders of magnitude. The value of k for a 10 Am long filament with a curvature elasticity constant C = 10 12 dyn cm2 [8&&] at zero field is 3 pN/Am, which is of interest for different experimental techniques using force clamp and mapping. One such example is the measurement of the stiffness of the stereocilia [17], or the deformation of a substrate caused by the migration of cells [18]. Another important issue is the buckling instability of the filament caused by the applied compressional stress studied in Refs. [8&&,16]. The torques balance (Eq. (2)) accounting for the constraint of the filament end to the direction of the applied force gives the following equation for the critical force [16]: tg
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fc 2Cm ¼ Fc 2Cm :
ð4Þ
The following boundary conditions are applied: y( 1) = y x ( 1) = 0 and y(1) = y xx (1) = 0. The clamp of the fixed end of the filament in Ref. [8&&] is carried out by two polystyrene spheres held by the laser tweezers. According to Eq. (4) the critical force for the first mode of the buckling instability is F c = 2Cm + (2.247)2. Non-symmetrical buckled shapes calculated numerically [16], match those found in the experiment reasonably well. Since the buckling instability at compressional stress is studied in Ref. [8&&] only for a fixed value of the magnetic field, there is, at the present time, no experimental confirmation of the dependence of the critical force on the strength of the magnetic field. An interesting buckling instability for a magnetic filament of a different kind — a chain of ferromagnetic
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particles, is reported in Ref. [3]. The curvature elasticity of these objects can be determined by magnetodipolar interactions alone [11]. In considering the interaction between the nearest neighbours in the chain of the ferromagnetic particles [11], for the curvature elasticity constant, the value m 2 / 2d 2 is obtained. Here m is the magnetic moment of the particle, but d is its diameter. This relation predicts the number of the particles in the chain N at which the formation of a closed ring with the radius R is energetically advantageous:
2m2 1 m2 2k <0: þ 3 4 d2 R d
ð5Þ
In this case, the gain in the energy of magnetodipolar interactions due to the approaching ends of the filament is larger than the loss due to the bending of the chain. Eq. (5) shows that this will take place when N > k2 / 2, close to the exact result N = 4 when the energy of the closed ring structure becomes less than that of the straight chain [19]. In the general case, the effective curvature elasticity constant of the chain of ferromagnetic particles depends on the orientation of the dipolar moment with respect to its axis, due to the anisotropy of the magnetodipolar interactions. This anisotropy of the curvature elasticity is observed by breaking the ring of ferromagnetic particles in the magnetic field perpendicular to its plane when the field strength increases above a critical value [20]. Investigation of the interesting issue of the mechanics of the flexible magnetic filaments with curvature elasticity depending on the orientation of filament and anisotropic hydrodynamic friction is under way at present [21]. The value of the curvature elasticity constant allows one to predict the critical compressional stress at which the chain buckles: K = k2C / (2L)2. A numerical estimate for the magnetosome of ten magnetite particles with a size 0.1 Am, which corresponds to the real case of magnetotactic bacteria [3], yields a critical compressional stress value of about 3.4 pN. How the properties of the magnetic filaments are used for magnetic field sensing by the magnetotactic organisms is an active field of research [3,12]. It should be noted that the closed structures of ferromagnetic particle chains are not necessarily ideal rings. Closed structures with several kinks are observed for the chains of the ferromagnetic particles extracted from magnetotactic bacteria [22]. Sometimes, short chains are attached to the kinks forming branched filaments [22]. This correlates with the fact, arising from the elasticity theory of the Kirchhoff rod, that the elastic energy of the ring E r = 19.7392C / 2L is larger than the energy of a loop with a fixed point and kink angle 81- — E l = 14.05555C / 2L [23]. The phase separation in systems of filaments with branching points is considered from a general point of view in Ref. [24]. The curvature elasticity constant due to magnetodipolar interactions allows one to estimate the persistence length of the chain of magnetic particles l p = kd / 2, where k = m 2 /
d 3k BT is the magnetodipolar interaction parameter. Correction to this first order term for the persistence length of the chain of dipolar particles by slowly converging asymptotic series with respect to the magnetodipolar interaction parameter k is given in Ref. [25]. k, for d = 21 nm Fe3O4 particles, equals to 16. This correlates with the experimental observation by cryogen transmission electron microscopy of rather long chains containing about 8 colloidal Fe3O4 particles [26]. The values of the magnetodipolar interaction parameter for colloids of Fe0.75C0.25 particles with sizes 6.9 and 8.2 nm and thickness of surfactant layer 7 nm are 4.9 and 10.54, respectively. This corresponds to the persistence length of chains containing 2 –3 and 5 – 6 particles, which is in good agreement with the cryogenic electron micrographs in Ref. [27&]. An interesting example of magnetic particle chain formation is given in Ref. [28], where the chaining of 90 Am Ni particles in the horizontal plane is observed under the action of a vertical AC magnetic field. This somewhat unexpected result can be explained by the orientation of the easy magnetization axes of the ferromagnetic particle in the direction perpendicular to the field if the characteristic particle reorientation time is greater than the period of AC magnetic field. An important feature of the flexible magnetic filament model (1) lies in the appearance of normal stress due to the action of the magnetic torques. This was confirmed recently [29] by the observation of the dependence of the Plateau angle in 2D magnetic foam on the strength of the magnetic field. It may be described within the limit of a small deformation of the foam by taking into account in the condition of the force balance acting on the vertex, in addition to the tangential forces due to the surface tension also the normal stresses caused by the action of the magnetic torques on the Plateau borders of the foam.
3. Dynamics of magnetic filaments Magnetic filaments have interesting dynamic properties, which were studied by applying AC magnetic fields [30,31]. In the simplest case, the Rouse dynamics of the filaments is considered and the anisotropy of the hydrodynamic friction coefficient neglected. In this case 1Y Y m ¼ K ð6Þ 1 where the friction coefficient can be calculated by the relation 1 = 4kg / (Log(L / a) + c), where the constant c is of order 1. In the case where the transversal force F a is applied to the free end of a filament, the displacement obeys a power law in time pffiffiffi 2 Fa t 3=4 : yðt Þ ¼ ð7Þ Cð7=4Þ
A. Ce¯bers / Current Opinion in Colloid & Interface Science 10 (2005) 167 – 175
This power law was confirmed experimentally by studying the displacement of passive Brownian particles in a network of elastic filaments [32]. Eq. (7) gives the characteristic power law dependence on frequency x 3/4, for the real and imaginary parts of the complex elasticity modulus of the network of elastic filaments [33]. In the case of magnetic filaments [16], the exponent of the power law depends on Cm and approaches 1/2 for large Cm values. An experimental verification of this theoretical prediction should be possible by investigation of the complex shear modulus of magnetorheological suspensions. Such an investigation was carried out recently [34] and the data obtained show the power law dependence of the real part of complex shear modulus with the exponent changing from 0.69 in zero field to 0.18 in H = 29.7 kA/m. The single chain model considered in Ref. [34] predicts an imaginary part of the complex shear modulus which is several orders of magnitude less than that observed in the experiment presumably illustrating that the description of the viscoelastic properties of the magnetorheological suspensions should take into account both the single chain response and the properties of their network. A rather rich behaviour of filaments is observed in a rotating field. Numerical simulations find various regimes [31]. If the magnetoelastic number is large enough, the transition between the different regimes is determined by the Mason number Mn = xs / Cm, where s = 1L 4 / C is the elastic relaxation time. The critical Mason number, 2.37, for the transition to a non-synchronous regime is obtained for the range of magnetoelastic numbers Cm Z [15 : 25] [31]. If the frequency of the rotating field is below this critical value, the filament in the frame of the field has a stationary FS_ like shape. The family of these shapes for different frequencies is shown in Fig. 3. A more general case was considered in Ref. [21] by taking into account the anisotropy of the friction
171
coefficient 1 ik = 1 –D ik + (1 || 1 –)t i t k . Here 1 || / 1 – depends on the axes ratio of the filament and is in the interval [1.5,2]. The dynamics of the filament in this case is determined by 1ik mk ¼ Ki
ð8Þ
Numerical simulation [21] shows that the anisotropy of the friction coefficient increases the phase lag between the filament and the rotating field. Due to this, the critical frequency of the transition to the nonsynchronous regime diminishes in comparison to the isotropic case. An important remark is that, due to the attractive character of the magnetodipolar interactions, the chain of free super-paramagnetic particles in a rotating field has the same stationary shape as a flexible filament. In Ref. [35&] it is illustrated by a comparison of the shapes of chains of free super-paramagnetic particles, found by molecular dynamics calculations, with the shapes found by the flexible magnetic filament model. The critical frequency for the breaking of a chain of N particles obeys the scaling law 64x1 / H 2 = 23.2 / N 2. This corresponds to a value of the Mason number 1.93, which is in the agreement with its the 2.18 value at the breaking point found for the flexible filament model. The curvature elasticity constant of a chain of the super-paramagnetic particles is estimated from the relation C¼
ð l 1Þ 2 4 2 2L a H0 ln a 8
which, apart from the logarithmic factor, is similar to that given above. The magnetoelastic number depends on the number of the particles in the chain as follows Cm ¼ N 2 =6Lnð 2N Þ : The critical condition for breaking in the flexible filament model is determined from the condition that the tension in the filament overcomes the attractive force due to the magnetodipolar forces: K þ Cm 1 3cos2 # > 0
Fig. 3. Steady configurations of the filament in a rotating field. Cm = 25;xs = 10(1); 30(2); 50(3); 60.6(4).
ð9Þ
In the case of a rotating field this condition is fulfilled first in the middle of the filament. The dependence of the number of the particles in an unbroken chain on the frequency, N å x 1/2, fits well with experimental observations [36]. The breaking of chains with a length that is larger than the critical one, and their growth due to coalescence, leads to an interesting dynamic regime which has not been thoroughly studied yet. The behaviour of a chain of magnetic particles under applied shear is another case, which can be compared with that of the flexible magnetic filament. The critical shear rate
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at which a chain with one end fixed to a solid wall breaks obeys the scaling law cc 641 const ¼ 2 H N2 where const = 9.8. This value fits well with the value calculated by the flexible magnetic filament model, 9.15, derived from the condition for breaking at the attached end of the filament. In both cases, at the critical conditions for breaking, the shapes calculated by molecular dynamics and by flexible filament model match each other reasonably well [35&]. These results are of interest for different applications of magnetic particle chains. For example, the FEphesia_ technology for electrophoretic separation of long DNA fragments or certain rare cancer cells by arrays of chains of functionalized magnetic particles [37,38&]. One interesting application of the rotation of magnetic filaments induced by a rotating magnetic field is for mixing in microfluidic systems [39&]. This idea has quite a long history, starting from the late 1970s (see Ref. [40] for earlier references), when several approaches for mixing by rotating particles have been proposed. The efficiency of mixing is determined by the Peclet number Pe = x(2L)2 / D, where D is the diffusion coefficient. In the case where 2L = 20 Am; x = 0.5 Hz and D = 4.10 5 cm2 s, the Peclet number is about 0.3. This estimate correlates with the observed increase of mixing due to rotating magnetic filaments in rotating magnetic field [39&]. In Ref. [39&] it was noticed that the more flexible, polyethylene glycol linked chains, are more efficient as mixers — presumably due to the formation of FU_ like shapes that synchronously rotate with the field. In Ref. [31] it was shown numerically that the FU_ like shape of the filament is unstable in the rotating field and unwinds, forming an FS_ like shape if the frequency of the rotating field is below the critical value. Understanding the formation of FU_ like configurations synchronously rotating with the applied field is possible if some magnetic heterogeneity of the filament is assumed. Numerical experiments confirm this possibility finding that a magnetically heterogeneous filament with FU_ like shape and weaker magnetic properties in the centre rotates synchronously with the field. The observation of a FV_ like shape [39&] may be related to the magnetic repulsion of the tips of the hairpin which is not yet accounted for in the flexible filament model. The observation in Ref. [41] of the rather complicated shapes of magnetic particle chains in a rotating field might therefore be due to a magnetic heterogeneity of the filaments. A homogeneous magnetic filament at the rotating field frequency above the critical point undergoes a periodic back and forth motion that is similar to that observed for rigid rods under the action of a rotating field. For a recent
reference on this subject, see the fine work in Ref. [42], where this kind of motion was observed for borosilicate glass rods under the action of optically created torques. The case of flexible magnetic filament is more complicated due to its periodic bending and straightening in the rotating field, as shown in Fig. 4. More detailed studies of the complex behaviour of the flexible magnetic filament in a rotating field are required. A periodic buckling of the magnetic filament takes place also in the case of shear flow. The dynamics of the elastic filament in shear flow was studied in Ref. [43]. It was found that, after buckling at the transient stage, the filament orients along the velocity vector of the flow. In the field applied in the plane of the shear perpendicularly to the flow magnetic filament has a periodic regime due to the interplay of two mechanisms — the orientation of the filament by the shear flow and the buckling instability due to the magnetic torques [44]. This, for Cm = 25 and dimensionless shear rate cs = 500, is shown in Fig. 5. At present time there are no experimental investigations of this regime. There is a critical value of the magnetoelastic number Cm for each cs value, above which the periodic regime is established. For lower Cm values, a slightly bent configuration oriented along the flow is established [44]. In the case of magnetorheological suspensions, the regime of the periodic breaking and growth of the magnetic particle chains in shear flow should take place. Interesting results in
Fig. 4. Periodic regime of the filament in a rotating field. Cm = 25; xs = 300. t = 0.0104(1); 0.01302(2); 0.0156(3); 0.0182(4); 0.0208(5); 0.0234(6); 0.0260(7); 0.0286(8).
A. Ce¯bers / Current Opinion in Colloid & Interface Science 10 (2005) 167 – 175
Fig. 5. Periodic regime of the filament in a shear flow. Dimensionless time t = 0.038(1); 0.045(2); 0.052(3); 0.056(4); 0.059(5); 0.063(6); 0.066(7); 0.073(8). Velocity of a shear flow is up on the right and down on the left. Magnetic field is from left to right.
the investigation of the rheological behaviour of suspensions of magnetic filaments are expected in the future.
4. Magnetic filaments as an active colloidal system There is a lot of interest in the mechanisms of the selfpropulsion of different microorganisms [45]. The energy for the motion is supplied by the chemical reactions arising from their metabolism. Magnetic filaments give an opportunity to create self-propelling micro machines with an energy supplied by an external field [46&&]. The creation of different micro-devices for the transport of liquids and other purposes is an active field of research (see Refs. [47,48] for further references). The self-propulsion of magnetic filament is illustrated numerically in Ref. [46&&] on the basis of the model (1). For this, the angle b, which the AC magnetic field makes with the x axis, is taken to change according to a harmonic law b = b 0 sin(xt). An important issue, which should be accounted for in the numerical model of self-propulsion in an AC field, is the anisotropy of the friction coefficient [49]. To break the symmetry of the filament, different possibilities can be considered. A simple one is to consider the heterogeneous deformations of the hairpin formed by the buckling instability of the filament in the field. In this case, the symmetry breaking is due to the hairpin shape. Another possibility is to consider the magnetic filament with some load attached to the one of its ends. In the case of the hairpin, some attention should be paid to the stabilization of the hairpin shape to obtain a permanent self-propulsion in the AC field. In Ref. [46&&] this is achieved by introducing magnetic heterogeneity in the filament. The magnetic torque in this case is given by the relation Y ð l 1Þ 2 2 2 T ¼ a H0 f ðl Þsinð2ð# bÞÞb 8ð l þ 1Þ Y
173
Here f(l) = 1 1 / ((l l 0) / l*)2 describes some heterogeneity with a characteristic width l* localized at l 0. If such an assumption is not introduced in the numerical model, the hairpin, after some transient period, unwinds so that the necessary symmetry breaking for self-propulsion disappears. Stabilization of the hairpin by magnetic heterogeneity occurs due to the tendency to keep the part of the filament with the weaker magnetic properties perpendicular to the alternating direction of the magnetic field. It should be noted that the hairpin with magnetic heterogeneity is just one of the possible schemes, which can be proposed for the creation of a self-propelling colloidal structures. The analysis of their efficiency is an interesting problem to be studied in the future. Heterogeneous deformations of the arms of a hairpin induced by the magnetic field of alternating direction are shown in Fig. 6. The magnetic field induces heterogeneous bending of hairpin arms, moving in the half-period shown from up to down. The scaling of the velocity of the hairpin with Lx / 2 depends on the ratio of the hairpin length to the elastic deformation penetration length l(x) = (x1 / C)1/4 and has a maximum for the parameters given at L / l(x) = 5.5. This value is close to the 4.07 found earlier for one oscillating arm [50]. If the amplitude b 0 of the angle b variation is small, the velocity of self-propulsion is proportional to b 02, as is characteristic for different mechanisms of self-propulsion. A simple model that qualitatively describes the selfpropelling motion can be formulated on the basis of the models proposed earlier for self-propulsion caused by oscillating flagella of microorganisms [51]. In the model considered in Ref. [51], the volume torque on the flagella is created by molecular motors. Their frequency of oscillation is determined by the collective properties of the system of molecular motors. In the case of the magnetic filament, torque is created – and thus energy supplied – by the external magnetic field. Considering each arm of the hairpin as a load for the other and assuming that they are unclamped at the connection point, the following boundary problem can be formulated for the displacement of the filament yt ¼ yxxx þ 2Cmyxx yxx ð1Þ ¼ 0; yxxx ð1Þ þ 2Cmðyx ð1Þ bÞ ¼ 0; yxx ð0Þ ¼ 0 and 10 yt ð0Þ ¼ yxxx ð0Þ þ 2Cmðyx ð0Þ bÞ 18 L Here 1 0 is the hydrodynamic friction coefficient of the load. The resulting velocity of the self-propelling hairpin can be calculated according to the relation
1 1þ 0 1jj L
! ¼
18 1 1jj
!
xs 2 b 2 0
Z
1
A2 ax dx 0
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Fig. 6. Deformation of the hairpin arms for a half-period in a self-propelling motion at the time intervals 1.74I10 4. Cm = 50;xs = 1000;b 0 = k/6.
where the amplitude and phase of the hairpin arm deformation are introduced: y(x) = A sin(xt a(x)). Numerical results [46&&] agree well, qualitatively and quantitatively, with this simple model. The numerical analysis makes it possible to estimate the force developed by the magnetic filament. The characteristic force can be estimated as C / L 2, where at Cm = 50 and xs = 1000, found numerically for b 02 = 0.25, is about 3.3. At realistic values of the physical properties of the filament, C = 10 12 dyn cm2 and L = 50 Am, this yields a value for the force of about 1.3 pN. The characteristic velocity of swimming motion in the water in this case is about 20 Am/s. The requirements for the AC magnetic field necessary for the creation of such motion in a colloidal swimmer are quite reasonable, and in the case of the magnetic particles with l = 1.7 and a = 0.4 Am the parameters chosen are achieved in magnetic fields of about 23 Oe and frequency 12 Hz. This is the same order of the magnitude as the force created by molecular motors. Due to this, interesting opportunities arise for the development of new micromanipulation tools inside cells. In this connection it is interesting to note that the force developed by the molecular motors was recently determined from separation of magnetically interacting endosomes in the cell [52&]. The results reported in this section show that magnetic beads can be used, not only as passive tools for micromanipulation of such objects as macromolecules, for which is already a well developed technique [53], but also to create an active micro-device which can be manipulated externally. In the future some rather exciting investigations of these new possibilities are expected.
5. Conclusions The investigation of the flexible magnetic filaments recently created in different labs is just at its beginning. Nevertheless, results obtained by different groups show that they have interesting possibilities for application in biotechnology, microfluidics and nanotechnology. A unifying approach for the description of their behaviour in various static and dynamic conditions is possible on the basis of an extended Kirchhoff model of elastic rod.
References and recommended readings [1] B. Alberts, et al., Molecular biology of the cell, Third edition, Garland Publishing, Inc., New York, 1994. [2] T. Matsunga, Y. Okamura, Genes and proteins involved in bacterial magnetic particle formation, Trends Microbiol 11 (2003) 536 – 541. [3] V.P. Shcherbakov, et al., Elastic stability of chains of magnetosomes in magnetotactic bacteria, Eur Biophys J 26 (1997) 319 – 326. [4] GR Bossis (Ed.), Electrorheological Fluids and Magnetorheological Suspensions, World Scientific Publishing Company, 2002. [5 ] C. Goubault, et al., Flexible magnetic filaments as micromechanical && sensors, Phys Rev Lett 91 (2003) 260802. In this paper new objects – flexible magnetic filaments – chains of superparamagnetic particles irreversibly linked by polymer are obtained for the first time (see also Ref. [8&&]) and their main physical properties — curvature elasticity and anisotropy of magnetic susceptibility are described. [6 ] L. Cohen-Tannoudji, et al., Polymer bridging probed by magnetic && colloids, Phys Rev Lett 94 (2005) 038301. The physical mechanisms of irreversible linking of super-paramagnetic particle chains in polymer solution are described.
& &&
of special interest. of outstanding interest.
A. Ce¯bers / Current Opinion in Colloid & Interface Science 10 (2005) 167 – 175 [7] C. Goubault, et al., Self-assembled magnetic nanowires made irreversible by polymer bridging, Langmuir 21 (2005) 3725 – 3729. [8] S.L. Biswal, A.P. Gast, Mechanics of semiflexible chains formed by && poly(ethylene glycol)-linked paramagnetic particles, Phys Rev E 68 (2003) 021402. Flexible magnetic filaments by using covalent streptavidin – biotin covalent bounds between functionalized super-paramagnetic particles are obtained and their physical properties by using laser tweezers are measured. [9] I.A. Banerjee, et al., Magnetic nanotube fabrication by using bacterial magnetic nanocrystals, Adv Mater 17 (2005) 1128 – 1131. [10] A. Cebers, Dynamics of a chain of magnetic particles connected with && elastic linkers, J Phys Condens Matter 15 (2003) S1335 – S1344. The mathematical model of flexible magnetic filament based on the extended Kirchhoff model of elastic rod, which takes into account the orientating effect of the field on the filament is proposed. [11] A. Cebers, Dynamics of elongated magnetic droplets and elastic rods in magnetic field, JMMM 289 (2005) 335 – 338. [12] V.P. Shcherbakov, M. Winklhofer, Bending of magnetic filaments under a magnetic field, Phys Rev E 70 (2004) 061803. [13] Z. Dogic, et al., Elongation and fluctuations of semiflexible polymers in a nematic solvent, Phys Rev Lett 92 (2004) 125503. [14] V. Berejnov, et al., Lyotropic ferronematics: magnetic orientational transition in the discotic phase, Europhys Lett 41 (1998) 507 – 512. [15] K.I. Morozov, Nature of ferronematic alignment in a magnetic field, Phys Rev E 66 (2002) 011704. [16] A. Ce¯bers, I. Javaitis, Bending of flexible magnetic rods, Phys Rev E 70 (2004) 011404. [17] P. Martin, A.D. Mehta, A.J. Hudspeth, Negative hair-bundle stiffness betrays a mechanism for mechanical amplification by the hair cell, PNAS 97 (2000) 12026 – 12031. [18] O. Roure, et al., Force mapping in epithelial cell migration, PNAS 102 (2005) 2390 – 2395. [19] W. Wen, et al., Aggregation kinetics and stability of structures formed by magnetic microspheres, Phys Rev E 59 (1999) R4758 – R4761. [20] F. Kun, et al., Breakup of dipolar rings under a perpendicular magnetic field, Phys Rev E 64 (2001) 061503. [21] A. Cebers, I. Javaitis, Dynamics of flexible magnetic filaments, Magnetohydrodynamics 40 (2004) 345 – 350. [22] A.P. Philipse, D. Maas, Magnetic colloids from magnetotactic bacteria:chain formation and colloidal stability, Langmuir 18 (2002) 9977 – 9984. [23] H. Yamakawa, W.H. Stockmayer, Statistical mechanics of wormlike chains: II. Excluded volume effects, J Chem Phys 57 (1972) 2843–2854. [24] T. Tlusty, S.A. Safran, Defect-induced phase separation in dipolar fluids, Science 290 (2000) 1328 – 1331. [25] K.I. Morozov, M.I. Shliomis, Ferrofluids: flexibility of magnetic particle chains, J Phys Condens Matter 16 (2004) 3807 – 3818. [26] M. Klokkenburg, et al., Direct imaging of zero-field dipolar structures in colloidal dispersions of synthetic magnetite, J Am Chem Soc 126 (2005) 16706 – 16707. [27] K. Butter, et al., Direct observation of dipolar chains in ferrofluids in & zero field using cryogenic electron microscopy, J Phys Condens Matter 15 (2003) S1451 – S1470. Direct images of fluctuating chains of ferromagnetic nanoparticles are obtained by using cryogenic electron microscopy. [28] A. Snezhko, I.S. Aranson, W.K. Kwok, Structure formation in electromagnetically driven granular media, Phys Rev Lett 94 (2005) 108002. [29] E. Janiaud, et al., Magnetic forces in 2D foams, JMMM 289 (2005) 215 – 218. [30] S.L. Biswal, A.P. Gast, Rotational dynamics of semiflexible paramagnetic particle chains, Phys Rev E 69 (2004) 041406.
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[31] A. Cebers, I. Javaitis, Dynamics of a flexible magnetic chain in a rotating magnetic field, Phys Rev E 69 (2004) 021404. [32] A. Caspi, R. Granek, M. Elbaum, Diffusion and directed motion in cellular transport, Phys Rev E 66 (2002) 011916. [33] Y. Tseng, et al., Local dynamics and viscoelastic properties of cell biological systems, Curr Opin Colloid Interface Sci 7 (2002) 210 – 217. [34] J. De Vicente, et al., A slender-body micromechanical model for viscoelasticity of magnetic colloids: comparison with preliminary experimental data, J Colloid Interface Sci 282 (2005) 193 – 201. [35] I. Drikis, A. Cebers, Molecular dynamics simulation of the chain of & magnetic particles and flexible filament model, Magnetohydrodynamics 40 (2004) 351 – 358. Confirmation by direct molecular dynamics simulation that steady configurations of the chains of free super-paramagnetic particles can be described by the model of flexible magnetic filament introduced in Ref. [10&&]. [36] S. Melle, J.E. Martin, Chain model of a magnetorheological suspension in a rotating field, J Chem Phys 118 (2003) 9875 – 9881. [37] N. Minc, et al., Quantitative microfluidic separation of DNA in selfassembled magnetic matrixes, Anal Chem 76 (2004) 3770 – 3776. [38] N. Minc, J.-L. Viovy, Microfluidique et applications biologiques: & enjeux et tendencies, C R – Acad Sci, Phys 5 (2004) 565 – 575. Different important and interesting applications of the flexible magnetic filaments are described — producing of the sieves for the fractionation of long DNA fragments, separation of rare cancer cells. [39] S.L. Biswal, A.P. Gast, Micromixing with linked chains of para& magnetic particles, Anal Chem 76 (2004) 6448 – 6455. Experimental evidence of the possibility of using of the flexible magnetic filaments for mixing in microfluidics is given. [40] A. Ce¯bers, E. Lemaire, L. Lobry, Internal rotations in dielectric suspensions, Magn Gidrodin 36 (2000) 347 – 364. [41] A.K. Vuppu, A.A. Garcia, M.A. Hayes, Video microscopy of dynamically aggregated paramagnetic particle chains in an applied rotating magnetic field, Langmuir 19 (2003) 8646 – 8653. [42] W.A. Shelton, K.D. Bonin, T.G. Walker, Nonlinear motion of optically torqued nanorods, Phys Rev E 71 (2005) 036204. [43] E.L. Becker, M.J. Shelley, Instability of elastic filaments in shear flow yields first-normal-stress differences, Phys Rev Lett 87 (2001) 198301. [44] A. Cebers, Flexible magnetic filaments, Proceedings of Moscow International Symposium on Magnetism, Moscow June 25 – 30, 2005. [45] D. Bray, Cell Movements, Second edition, Garland, 2001. [46] A. Cebers, Flexible magnetic swimmer, Magnetohydrodynamics 41 && (2005) 63 – 72. By direct numerical calculation based on flexible magnetic filament model [10&&] the possibility of self-propulsion of filaments in AC magnetic field is shown. [47] H.A. Stone, A.D. Stroock, A. Ajdari, Engineering flows in small devices, Annu Rev Fluid Mech 36 (2004) 381 – 411. [48] L.E. Becker, S.A. Koehler, H.A. Stone, On self-propulsion of micromachines at low Reynolds number: Purcell’s three-link swimmer, J Fluid Mech 490 (2003) 15 – 35. [49] H.A. Stone, A.D.T. Samuel, Propulsion of microorganisms by surface distortions, Phys Rev Lett 77 (1996) 4102 – 4104. [50] C.H. Wiggins, R.E. Goldstein, Flexive and propulsive dynamics of elastica at low Reynolds number, Phys Rev Lett 80 (1998) 3879 – 3882. [51] S. Camalet, F. Julicher, J. Prost, Self-organized beating and swimming of internally driven filaments, Phys Rev Lett 82 (1999) 1590 – 1593. [52] C. Riviere, et al., Passive versus active local microrheology in mamma& lian cells and amoebae, Magnetohydrodynamics 40 (2004) 321–335. The application of magnetic endosomes for microrheology in cells is shown. [53] Strick, et al., Stretching of macromolecules and proteins, Rep Prog Phys 66 (2003) 1 – 45.
Flexible magnetic filaments A. Ce¯bers * Institute of Physics, Salaspils-1, LV-2169, Latvia Available online 18 August 2005
Abstract Flexible magnetic filaments, created recently in different labs, possess interesting physical properties such as novel buckling instabilities. These magnetic filaments may be used as sensors in different conditions or as mixers for microfluidics applications. This paper reviews recent achievements in this new field, based on the extension of the Kirchhoff model of elastic rods by including the orientating effect of the magnetic field upon the filaments. As an illustration of the various intriguing properties revealed by the model, self-propelling under the action of an AC magnetic field is discussed. D 2005 Elsevier Ltd. All rights reserved. Keywords: Magnetic filaments; Curvature elasticity; Buckling instability; Mixing; Self-propulsion
1. Introduction Filamentary objects are ubiquitous in nature carrying out several important functions inside the cell [1]. Magnetic filaments, composed of chains of ferromagnetic particles, are used by the magnetotactic bacteria for navigation purposes [2,3]. Chains of super-paramagnetic particles are mesoscopic elements, which determine the properties of the magnetorheological suspensions [4]. Magnetic filaments with interesting properties were created recently [5&&, 6&&,7,8&&] by linking micron size super-paramagnetic particles with some polymer. As a result, objects with a flexibility that is a billion times greater than that of different common materials are obtained.
2. Statics of magnetic filaments Colloidal chemistry aspects of the synthesis of magnetic filaments, which are super-paramagnetic particle chains irreversibly linked by some polymer (PAA [5&&,6&&,7] or covalent biotin – streptovidin binding [8&&]) are considered in
* Tel.: +371 9195961; fax: +371 7901214. E-mail address: [email protected]. 1359-0294/$ - see front matter D 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.cocis.2005.07.002
Refs. [6&&,7]. It is believed that the irreversible linking of the suspension of the super-paramagnetic particles is driven by depletion interactions in the polymer solution when the incubation time is short enough [7]. Rather exotic magnetic filaments are created by incorporating 50 nm size magnetosomes, produced by magnetotactic bacteria, in peptidenanotube template [9]. Magnetic filament can be considered as a Kirchhoff elastic rod taking into account the orientating magnetic field effect on the filament [10&&]. Therefore its energy reads Z Z 1 1 ð l 1Þ 2 2 2 Y Y 2 h t dl a H0 dl E¼ C 2 R2 8ð l þ 1Þ Z Kðl Þdl ð1Þ Here R is the radius of the curvature of the centre line of a filament, C is the curvature elasticity constant which can be expressed through the elastic properties of linkers [5&&], a is the radius of the circular cross section of the filament with total length 2L, H 0 is the applied magnetic field strength and l is the effective magnetic permeability of the filament. The Lagrange multiplier K, which has the meaning of compressional stress, accounts for local inextensibility of the filament. Different properties of the magnetic filaments were investigated utilizing this model. Its important feature lies in the appearance of volume torques due to the
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orienting effect of the applied field. The torque balance reads Y
h Yi Y dM Y þ t xF þ T ¼ 0 dl Y
2
ð2Þ Y
Þ 2 2 Here T ¼ 8ðl1 ðlþ1Þ a H0 sinð2#Þb Yis the torque per unit length due to the applied field, F is the stress in the Y filament, b ¼ ½Y t xnY is the binormal to the filament, and Y t ¼ ðcos#; sin#Þ is the tangent vector given by the angle #, which it makes with the field. The torque balance equation, accounting for the momentum stresses M = C / R [10&&] in the filament and Y the condition of its mechanical equilibrium dF =dl ¼ 0, && && predicts the buckling instability [5 ,10 ,11] for a filament orientated perpendicularly to the applied field. The critical ðl1Þ2 2 2 2 value of the magnetoelastic number Cm ¼ a L H0 8 ð lþ1 ÞC Y Y for the free F ¼ 0 and unclamped M ¼ 0 ends of the filament reads CmðnoÞ ¼ ð2n þ 1Þ2 k2 =8ð# ¼ k=2þ eðoÞ sin ðð2n þ 1Þkl=ð 2LÞÞ; odd modeÞ and CmðneÞ ¼ n2 k2 =2ð# ¼ k=2 þ eðeÞ cosðnkl=ð 2LÞÞ; even modeÞ. The lowest critical value, Cmn(e) = 0, corresponds to the rotation of the filament as whole in the direction of the magnetic field. The lowest critical value, k2 / 8, for no uniform deformation of the filament corresponds to the first odd mode (n = 0). The last condition corresponds to the appearance of the steady nontrivial solutions ofpthe Eq. ffiffiffiffiffiffiffiffiffi ffi (2) found in Ref. [12] 2L / l m = k, where lm ¼ L= 2Cm is the characteristic magnetic length of the filament. Among the other problems solved in Ref. [12], the bending of the filament containing the magnetic tip under the action of transversal force should be mentioned. The relation of these problems to the magneticsense organs of homing pigeons may be pointed out [12]. Development of the first odd mode leads to the formation of hairpins, which have been observed experimentally recently [5&&]. In the case of a filament with infinite length, the hairpin shape can be calculated theoretically [5&&,11]. This, for the maximal radius of the curvature of the filament, gives pffiffiffiffiffiffiffiffiffiffi L ¼ 2Cm : ð3Þ R max
Eq. (3), and the value of the curvature measured for the filaments that are linked with a polyacrylic acid [5&&] allows one to estimate the value of the magnetoelastic number, which is proportional to the square of the field strength. According to the experimental data [5&&], Cm = 12 at a quite moderate field 4k Oe is already well above the critical value for buckling instability. The family of hairpins of filaments with finite length calculated numerically [10&&,11] is shown in Fig. 1. The curvature at the tip of the hairpin obeys the theoretical relation given by Eq. (3) if the Cm number is large enough. Numerical studies reveal interesting properties of the filaments, namely, that the hairpin shape of the filament also exists if the lengths of their arms are
Fig. 1. Family of hairpins. Cm = 1.5(1); 3(2); 5(3); 12(4).
different. If the length of the shorter arm is small enough, the hairpin relaxes to a straight configuration. At the corresponding initial perturbation, shapes with several hairpins can be established as long living transient states, which relax to a final straight configuration along the direction of the field. Shapes with several buckles, observed in the experiments [5&&], may arise due to some stabilization of these transitory configurations due to the interaction with a solid wall. A similar behaviour was found recently for actin filaments in a nematic solvent (solution of fd viruses) [13] — namely, hairpins with unequal lengths of the arms relax to a straight configuration. The relations given above allow one to estimate the characteristic interaction strength k BTC of the filament with background with the director a nematic R 2 nYn : 12 kB T C tY Y nn dl. The critical length of the shortest arm required for the straightening of the hairpin can be estimated to be equal to the minimal curvature radius given by Eq. (3) (Fig. 1e [13]). Then the characteristic correlation pffiffiffiffiffiffiffiffiffi length of the filament in a nematic background, k ¼ lp =C, is about 5 Am. This value is in reasonable agreement with the picture shown in Fig. 1a [13]. Since the persistence length of the actin filament l p = C / k BT is 16 Am, the characteristic interaction strength of the actin filament with a nematic background can be estimated as 2.510 11 erg/cm, which corresponds to the weak anchoring of the nematic background to the actin filament characterized by the anchoring parameter x = k BTC / K ; 1.310 3. Its value allows one to presume that the anchoring of actin filaments to the nematic background is caused by steric interactions. The interaction energy of magnetic filaments with a nematic background is an important issue for the ferronematics — nematic liquid crystals with incorporated chains of ferromagnetic particles [14,15].
A. Ce¯bers / Current Opinion in Colloid & Interface Science 10 (2005) 167 – 175
169
Fig. 2. Bending of magnetic filament. Transversal force on the right end of filament is applied in y axis direction. Points — experimental data [8&&]. Dashed line — Cm = 0; F = 0.15.
Direct measurements of the elastic properties of magnetic filaments were carried out recently by studying their bending deformations due to the force applied by laser tweezers [8&&]. Filaments of super-paramagnetic particles linked by polyethylene glycol with different molecular weights were studied. The value of the elasticity constant C was found to depend on the type of experiment carried out — bending at applied transversal force, buckling at the compressional force or relaxation to straight configuration. In Ref. [16] by numerical calculation it was shown that the model of the magnetic filament (1) describes the set of experiments in Ref. [8&&] reasonably well. The shape of the filament is compared to the experimental data for Cm = 0.9 and F = 0.45 (the force applied transversally to the filament is scaled with C / L 2) in Fig. 2. An interesting property of the magnetic filaments is that they can be used as force sensors, since their stiffness depends on the magnetic field. The displacement of the filament y(x) can be found from the torque balance (Eq. (2)) Y at F ¼ Fa eYy , where F a is the force applied transversally to the field direction, and satisfying the boundary conditions y(0) = 0; y x (0) = 0; y xx (1) = 0 Fa y¼ 2Cm
pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ! sh 2Cm sh 2Cmð1 xÞ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi x : 2Cmch 2Cm
The stiffness of the filament k = F a / y(1) in dimensionless units is given by k¼
2Cm pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 1 tanh 2Cm = 2Cm
and can be changed due to the applied field by several orders of magnitude. The value of k for a 10 Am long filament with a curvature elasticity constant C = 10 12 dyn cm2 [8&&] at zero field is 3 pN/Am, which is of interest for different experimental techniques using force clamp and mapping. One such example is the measurement of the stiffness of the stereocilia [17], or the deformation of a substrate caused by the migration of cells [18]. Another important issue is the buckling instability of the filament caused by the applied compressional stress studied in Refs. [8&&,16]. The torques balance (Eq. (2)) accounting for the constraint of the filament end to the direction of the applied force gives the following equation for the critical force [16]: tg
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fc 2Cm ¼ Fc 2Cm :
ð4Þ
The following boundary conditions are applied: y( 1) = y x ( 1) = 0 and y(1) = y xx (1) = 0. The clamp of the fixed end of the filament in Ref. [8&&] is carried out by two polystyrene spheres held by the laser tweezers. According to Eq. (4) the critical force for the first mode of the buckling instability is F c = 2Cm + (2.247)2. Non-symmetrical buckled shapes calculated numerically [16], match those found in the experiment reasonably well. Since the buckling instability at compressional stress is studied in Ref. [8&&] only for a fixed value of the magnetic field, there is, at the present time, no experimental confirmation of the dependence of the critical force on the strength of the magnetic field. An interesting buckling instability for a magnetic filament of a different kind — a chain of ferromagnetic
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particles, is reported in Ref. [3]. The curvature elasticity of these objects can be determined by magnetodipolar interactions alone [11]. In considering the interaction between the nearest neighbours in the chain of the ferromagnetic particles [11], for the curvature elasticity constant, the value m 2 / 2d 2 is obtained. Here m is the magnetic moment of the particle, but d is its diameter. This relation predicts the number of the particles in the chain N at which the formation of a closed ring with the radius R is energetically advantageous:
2m2 1 m2 2k <0: þ 3 4 d2 R d
ð5Þ
In this case, the gain in the energy of magnetodipolar interactions due to the approaching ends of the filament is larger than the loss due to the bending of the chain. Eq. (5) shows that this will take place when N > k2 / 2, close to the exact result N = 4 when the energy of the closed ring structure becomes less than that of the straight chain [19]. In the general case, the effective curvature elasticity constant of the chain of ferromagnetic particles depends on the orientation of the dipolar moment with respect to its axis, due to the anisotropy of the magnetodipolar interactions. This anisotropy of the curvature elasticity is observed by breaking the ring of ferromagnetic particles in the magnetic field perpendicular to its plane when the field strength increases above a critical value [20]. Investigation of the interesting issue of the mechanics of the flexible magnetic filaments with curvature elasticity depending on the orientation of filament and anisotropic hydrodynamic friction is under way at present [21]. The value of the curvature elasticity constant allows one to predict the critical compressional stress at which the chain buckles: K = k2C / (2L)2. A numerical estimate for the magnetosome of ten magnetite particles with a size 0.1 Am, which corresponds to the real case of magnetotactic bacteria [3], yields a critical compressional stress value of about 3.4 pN. How the properties of the magnetic filaments are used for magnetic field sensing by the magnetotactic organisms is an active field of research [3,12]. It should be noted that the closed structures of ferromagnetic particle chains are not necessarily ideal rings. Closed structures with several kinks are observed for the chains of the ferromagnetic particles extracted from magnetotactic bacteria [22]. Sometimes, short chains are attached to the kinks forming branched filaments [22]. This correlates with the fact, arising from the elasticity theory of the Kirchhoff rod, that the elastic energy of the ring E r = 19.7392C / 2L is larger than the energy of a loop with a fixed point and kink angle 81- — E l = 14.05555C / 2L [23]. The phase separation in systems of filaments with branching points is considered from a general point of view in Ref. [24]. The curvature elasticity constant due to magnetodipolar interactions allows one to estimate the persistence length of the chain of magnetic particles l p = kd / 2, where k = m 2 /
d 3k BT is the magnetodipolar interaction parameter. Correction to this first order term for the persistence length of the chain of dipolar particles by slowly converging asymptotic series with respect to the magnetodipolar interaction parameter k is given in Ref. [25]. k, for d = 21 nm Fe3O4 particles, equals to 16. This correlates with the experimental observation by cryogen transmission electron microscopy of rather long chains containing about 8 colloidal Fe3O4 particles [26]. The values of the magnetodipolar interaction parameter for colloids of Fe0.75C0.25 particles with sizes 6.9 and 8.2 nm and thickness of surfactant layer 7 nm are 4.9 and 10.54, respectively. This corresponds to the persistence length of chains containing 2 –3 and 5 – 6 particles, which is in good agreement with the cryogenic electron micrographs in Ref. [27&]. An interesting example of magnetic particle chain formation is given in Ref. [28], where the chaining of 90 Am Ni particles in the horizontal plane is observed under the action of a vertical AC magnetic field. This somewhat unexpected result can be explained by the orientation of the easy magnetization axes of the ferromagnetic particle in the direction perpendicular to the field if the characteristic particle reorientation time is greater than the period of AC magnetic field. An important feature of the flexible magnetic filament model (1) lies in the appearance of normal stress due to the action of the magnetic torques. This was confirmed recently [29] by the observation of the dependence of the Plateau angle in 2D magnetic foam on the strength of the magnetic field. It may be described within the limit of a small deformation of the foam by taking into account in the condition of the force balance acting on the vertex, in addition to the tangential forces due to the surface tension also the normal stresses caused by the action of the magnetic torques on the Plateau borders of the foam.
3. Dynamics of magnetic filaments Magnetic filaments have interesting dynamic properties, which were studied by applying AC magnetic fields [30,31]. In the simplest case, the Rouse dynamics of the filaments is considered and the anisotropy of the hydrodynamic friction coefficient neglected. In this case 1Y Y m ¼ K ð6Þ 1 where the friction coefficient can be calculated by the relation 1 = 4kg / (Log(L / a) + c), where the constant c is of order 1. In the case where the transversal force F a is applied to the free end of a filament, the displacement obeys a power law in time pffiffiffi 2 Fa t 3=4 : yðt Þ ¼ ð7Þ Cð7=4Þ
A. Ce¯bers / Current Opinion in Colloid & Interface Science 10 (2005) 167 – 175
This power law was confirmed experimentally by studying the displacement of passive Brownian particles in a network of elastic filaments [32]. Eq. (7) gives the characteristic power law dependence on frequency x 3/4, for the real and imaginary parts of the complex elasticity modulus of the network of elastic filaments [33]. In the case of magnetic filaments [16], the exponent of the power law depends on Cm and approaches 1/2 for large Cm values. An experimental verification of this theoretical prediction should be possible by investigation of the complex shear modulus of magnetorheological suspensions. Such an investigation was carried out recently [34] and the data obtained show the power law dependence of the real part of complex shear modulus with the exponent changing from 0.69 in zero field to 0.18 in H = 29.7 kA/m. The single chain model considered in Ref. [34] predicts an imaginary part of the complex shear modulus which is several orders of magnitude less than that observed in the experiment presumably illustrating that the description of the viscoelastic properties of the magnetorheological suspensions should take into account both the single chain response and the properties of their network. A rather rich behaviour of filaments is observed in a rotating field. Numerical simulations find various regimes [31]. If the magnetoelastic number is large enough, the transition between the different regimes is determined by the Mason number Mn = xs / Cm, where s = 1L 4 / C is the elastic relaxation time. The critical Mason number, 2.37, for the transition to a non-synchronous regime is obtained for the range of magnetoelastic numbers Cm Z [15 : 25] [31]. If the frequency of the rotating field is below this critical value, the filament in the frame of the field has a stationary FS_ like shape. The family of these shapes for different frequencies is shown in Fig. 3. A more general case was considered in Ref. [21] by taking into account the anisotropy of the friction
171
coefficient 1 ik = 1 –D ik + (1 || 1 –)t i t k . Here 1 || / 1 – depends on the axes ratio of the filament and is in the interval [1.5,2]. The dynamics of the filament in this case is determined by 1ik mk ¼ Ki
ð8Þ
Numerical simulation [21] shows that the anisotropy of the friction coefficient increases the phase lag between the filament and the rotating field. Due to this, the critical frequency of the transition to the nonsynchronous regime diminishes in comparison to the isotropic case. An important remark is that, due to the attractive character of the magnetodipolar interactions, the chain of free super-paramagnetic particles in a rotating field has the same stationary shape as a flexible filament. In Ref. [35&] it is illustrated by a comparison of the shapes of chains of free super-paramagnetic particles, found by molecular dynamics calculations, with the shapes found by the flexible magnetic filament model. The critical frequency for the breaking of a chain of N particles obeys the scaling law 64x1 / H 2 = 23.2 / N 2. This corresponds to a value of the Mason number 1.93, which is in the agreement with its the 2.18 value at the breaking point found for the flexible filament model. The curvature elasticity constant of a chain of the super-paramagnetic particles is estimated from the relation C¼
ð l 1Þ 2 4 2 2L a H0 ln a 8
which, apart from the logarithmic factor, is similar to that given above. The magnetoelastic number depends on the number of the particles in the chain as follows Cm ¼ N 2 =6Lnð 2N Þ : The critical condition for breaking in the flexible filament model is determined from the condition that the tension in the filament overcomes the attractive force due to the magnetodipolar forces: K þ Cm 1 3cos2 # > 0
Fig. 3. Steady configurations of the filament in a rotating field. Cm = 25;xs = 10(1); 30(2); 50(3); 60.6(4).
ð9Þ
In the case of a rotating field this condition is fulfilled first in the middle of the filament. The dependence of the number of the particles in an unbroken chain on the frequency, N å x 1/2, fits well with experimental observations [36]. The breaking of chains with a length that is larger than the critical one, and their growth due to coalescence, leads to an interesting dynamic regime which has not been thoroughly studied yet. The behaviour of a chain of magnetic particles under applied shear is another case, which can be compared with that of the flexible magnetic filament. The critical shear rate
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at which a chain with one end fixed to a solid wall breaks obeys the scaling law cc 641 const ¼ 2 H N2 where const = 9.8. This value fits well with the value calculated by the flexible magnetic filament model, 9.15, derived from the condition for breaking at the attached end of the filament. In both cases, at the critical conditions for breaking, the shapes calculated by molecular dynamics and by flexible filament model match each other reasonably well [35&]. These results are of interest for different applications of magnetic particle chains. For example, the FEphesia_ technology for electrophoretic separation of long DNA fragments or certain rare cancer cells by arrays of chains of functionalized magnetic particles [37,38&]. One interesting application of the rotation of magnetic filaments induced by a rotating magnetic field is for mixing in microfluidic systems [39&]. This idea has quite a long history, starting from the late 1970s (see Ref. [40] for earlier references), when several approaches for mixing by rotating particles have been proposed. The efficiency of mixing is determined by the Peclet number Pe = x(2L)2 / D, where D is the diffusion coefficient. In the case where 2L = 20 Am; x = 0.5 Hz and D = 4.10 5 cm2 s, the Peclet number is about 0.3. This estimate correlates with the observed increase of mixing due to rotating magnetic filaments in rotating magnetic field [39&]. In Ref. [39&] it was noticed that the more flexible, polyethylene glycol linked chains, are more efficient as mixers — presumably due to the formation of FU_ like shapes that synchronously rotate with the field. In Ref. [31] it was shown numerically that the FU_ like shape of the filament is unstable in the rotating field and unwinds, forming an FS_ like shape if the frequency of the rotating field is below the critical value. Understanding the formation of FU_ like configurations synchronously rotating with the applied field is possible if some magnetic heterogeneity of the filament is assumed. Numerical experiments confirm this possibility finding that a magnetically heterogeneous filament with FU_ like shape and weaker magnetic properties in the centre rotates synchronously with the field. The observation of a FV_ like shape [39&] may be related to the magnetic repulsion of the tips of the hairpin which is not yet accounted for in the flexible filament model. The observation in Ref. [41] of the rather complicated shapes of magnetic particle chains in a rotating field might therefore be due to a magnetic heterogeneity of the filaments. A homogeneous magnetic filament at the rotating field frequency above the critical point undergoes a periodic back and forth motion that is similar to that observed for rigid rods under the action of a rotating field. For a recent
reference on this subject, see the fine work in Ref. [42], where this kind of motion was observed for borosilicate glass rods under the action of optically created torques. The case of flexible magnetic filament is more complicated due to its periodic bending and straightening in the rotating field, as shown in Fig. 4. More detailed studies of the complex behaviour of the flexible magnetic filament in a rotating field are required. A periodic buckling of the magnetic filament takes place also in the case of shear flow. The dynamics of the elastic filament in shear flow was studied in Ref. [43]. It was found that, after buckling at the transient stage, the filament orients along the velocity vector of the flow. In the field applied in the plane of the shear perpendicularly to the flow magnetic filament has a periodic regime due to the interplay of two mechanisms — the orientation of the filament by the shear flow and the buckling instability due to the magnetic torques [44]. This, for Cm = 25 and dimensionless shear rate cs = 500, is shown in Fig. 5. At present time there are no experimental investigations of this regime. There is a critical value of the magnetoelastic number Cm for each cs value, above which the periodic regime is established. For lower Cm values, a slightly bent configuration oriented along the flow is established [44]. In the case of magnetorheological suspensions, the regime of the periodic breaking and growth of the magnetic particle chains in shear flow should take place. Interesting results in
Fig. 4. Periodic regime of the filament in a rotating field. Cm = 25; xs = 300. t = 0.0104(1); 0.01302(2); 0.0156(3); 0.0182(4); 0.0208(5); 0.0234(6); 0.0260(7); 0.0286(8).
A. Ce¯bers / Current Opinion in Colloid & Interface Science 10 (2005) 167 – 175
Fig. 5. Periodic regime of the filament in a shear flow. Dimensionless time t = 0.038(1); 0.045(2); 0.052(3); 0.056(4); 0.059(5); 0.063(6); 0.066(7); 0.073(8). Velocity of a shear flow is up on the right and down on the left. Magnetic field is from left to right.
the investigation of the rheological behaviour of suspensions of magnetic filaments are expected in the future.
4. Magnetic filaments as an active colloidal system There is a lot of interest in the mechanisms of the selfpropulsion of different microorganisms [45]. The energy for the motion is supplied by the chemical reactions arising from their metabolism. Magnetic filaments give an opportunity to create self-propelling micro machines with an energy supplied by an external field [46&&]. The creation of different micro-devices for the transport of liquids and other purposes is an active field of research (see Refs. [47,48] for further references). The self-propulsion of magnetic filament is illustrated numerically in Ref. [46&&] on the basis of the model (1). For this, the angle b, which the AC magnetic field makes with the x axis, is taken to change according to a harmonic law b = b 0 sin(xt). An important issue, which should be accounted for in the numerical model of self-propulsion in an AC field, is the anisotropy of the friction coefficient [49]. To break the symmetry of the filament, different possibilities can be considered. A simple one is to consider the heterogeneous deformations of the hairpin formed by the buckling instability of the filament in the field. In this case, the symmetry breaking is due to the hairpin shape. Another possibility is to consider the magnetic filament with some load attached to the one of its ends. In the case of the hairpin, some attention should be paid to the stabilization of the hairpin shape to obtain a permanent self-propulsion in the AC field. In Ref. [46&&] this is achieved by introducing magnetic heterogeneity in the filament. The magnetic torque in this case is given by the relation Y ð l 1Þ 2 2 2 T ¼ a H0 f ðl Þsinð2ð# bÞÞb 8ð l þ 1Þ Y
173
Here f(l) = 1 1 / ((l l 0) / l*)2 describes some heterogeneity with a characteristic width l* localized at l 0. If such an assumption is not introduced in the numerical model, the hairpin, after some transient period, unwinds so that the necessary symmetry breaking for self-propulsion disappears. Stabilization of the hairpin by magnetic heterogeneity occurs due to the tendency to keep the part of the filament with the weaker magnetic properties perpendicular to the alternating direction of the magnetic field. It should be noted that the hairpin with magnetic heterogeneity is just one of the possible schemes, which can be proposed for the creation of a self-propelling colloidal structures. The analysis of their efficiency is an interesting problem to be studied in the future. Heterogeneous deformations of the arms of a hairpin induced by the magnetic field of alternating direction are shown in Fig. 6. The magnetic field induces heterogeneous bending of hairpin arms, moving in the half-period shown from up to down. The scaling of the velocity of the hairpin with Lx / 2 depends on the ratio of the hairpin length to the elastic deformation penetration length l(x) = (x1 / C)1/4 and has a maximum for the parameters given at L / l(x) = 5.5. This value is close to the 4.07 found earlier for one oscillating arm [50]. If the amplitude b 0 of the angle b variation is small, the velocity of self-propulsion is proportional to b 02, as is characteristic for different mechanisms of self-propulsion. A simple model that qualitatively describes the selfpropelling motion can be formulated on the basis of the models proposed earlier for self-propulsion caused by oscillating flagella of microorganisms [51]. In the model considered in Ref. [51], the volume torque on the flagella is created by molecular motors. Their frequency of oscillation is determined by the collective properties of the system of molecular motors. In the case of the magnetic filament, torque is created – and thus energy supplied – by the external magnetic field. Considering each arm of the hairpin as a load for the other and assuming that they are unclamped at the connection point, the following boundary problem can be formulated for the displacement of the filament yt ¼ yxxx þ 2Cmyxx yxx ð1Þ ¼ 0; yxxx ð1Þ þ 2Cmðyx ð1Þ bÞ ¼ 0; yxx ð0Þ ¼ 0 and 10 yt ð0Þ ¼ yxxx ð0Þ þ 2Cmðyx ð0Þ bÞ 18 L Here 1 0 is the hydrodynamic friction coefficient of the load. The resulting velocity of the self-propelling hairpin can be calculated according to the relation
1 1þ 0 1jj L
! ¼
18 1 1jj
!
xs 2 b 2 0
Z
1
A2 ax dx 0
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A. Ce¯bers / Current Opinion in Colloid & Interface Science 10 (2005) 167 – 175
Fig. 6. Deformation of the hairpin arms for a half-period in a self-propelling motion at the time intervals 1.74I10 4. Cm = 50;xs = 1000;b 0 = k/6.
where the amplitude and phase of the hairpin arm deformation are introduced: y(x) = A sin(xt a(x)). Numerical results [46&&] agree well, qualitatively and quantitatively, with this simple model. The numerical analysis makes it possible to estimate the force developed by the magnetic filament. The characteristic force can be estimated as C / L 2
5. Conclusions The investigation of the flexible magnetic filaments recently created in different labs is just at its beginning. Nevertheless, results obtained by different groups show that they have interesting possibilities for application in biotechnology, microfluidics and nanotechnology. A unifying approach for the description of their behaviour in various static and dynamic conditions is possible on the basis of an extended Kirchhoff model of elastic rod.
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& &&
of special interest. of outstanding interest.
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