Paramagnetic filaments in a fast precessing field:
Planar versus helical conformations
We examine analytically equilibrium conformations of elastic chains of paramagnetic beads in the presence of a precessing magnetic field. Conformations of these filaments are determined by minimizing their total energy, given in the harmonic approximation by the sum of the bending energy, quadratic in its curvature, and the magnetic dipolar interaction energy, quadratic in the projection of the vector tangent to the filament onto the precession axis. In particular, we analyze two families of open filaments with their ends aligned along the precession axis and described by segments of planar curves and helices. These configurations are characterized in terms of two parameters encoding their features such as their length, separation between their ends, as well as their bending and magnetic moduli, the latter being proportional to the magnitude and precession angle of the magnetic field. Based on energetic arguments, we present the set of parameter values for which each of these families of curves is probable to occur.
pacs:87.16.Ka, 75.75.-c, 87.15.hp
Flexible magnetic filaments can be synthesized by joining ferromagnetic or superparamagnetic beads with elastic linkers Wang et al. (2011); Tierno (2014). The combination of their elastic and magnetic properties gives rise to interesting phenomena, which have been a subject of active research since their introduction more than a decade ago Biswal and Gast (2003); Goubault et al. (2003).
The mechanical properties of magnetic filaments have been extensively characterized. Their Young and bending moduli have been measured directly in bending and compression experiments performed with optical traps Biswal and Gast (2003) and indirectly from measurements of other quantities such as the separation between beads using Bragg diffraction Goubault et al. (2003), or more recently, via their thermal fluctuations Gerbal and Wang (2017). It has been found that the sole magnetic field can drive an Euler buckling instability in a free filament Cēbers (2005a), whose critical value has been determined theoretically and measured experimentally with a good agreement Cēbers and Javaitis (2004a); Gerbal et al. (2015).
This kind of filaments exhibit diverse morphological features Cēbers and Cīrulis (2007), for instance, depending on their length, bending rigidity and magnetic field strength, they may adopt and shapes Goubault et al. (2003); Cēbers (2003) or configurations with more undulations Huang et al. (2016). Configurations of anchored superparamagnetic filaments with a free end may fold into loops, sheets and pillars for different combinations of their bending rigidity and the strength of the magnetic field Wei et al. (2016). In a precessing magnetic field, filaments with loads at their ends can adopt planar or helical configurations by changing the precessing angle and with a time-dependent precession free filaments can assemble into gels of diverse conformations Dempster et al. (2017).
Due to their magnetic features, they have inspired the development of several applications. They can be used as micro-mechanical sensors used in the determination of force-extension laws at the micro-scale Biswal and Gast (2003); Goubault et al. (2003); Koenig et al. (2005). Furthermore, since they possess the interesting feature that their stiffness is tunable Cēbers (2005b), and conformational changes can be controlled through the magnetic field Cēbers and Cīrulis (2007); Huang et al. (2016) or the temperature Cerda et al. (2016), they can also be used as actuators Dempster et al. (2017) or grabbers Martinez-Pedrero et al. (2016).
Their dynamics have also been studied in detail. In a magnetic field rotating on a plane, free filaments rotate rigidly synchronously or asynchronously depending on their length and whether the precession frequency is smaller or bigger than a critical value Biswal and Gast (2004); Cēbers and Javaitis (2004b). In the presence of an alternating magnetic field, magnetic filaments oscillate and displace, so they have been used to design self-propelled swimmers of controllable velocity and displacement direction Dreyfus et al. (2005); Belovs and Cēbers (2006); Roper et al. (2008); Belovs and Cēbers (2013).
Further information about the properties and applications of these magnetic filaments can be found in the reviews Cēbers (2005b); Cēbers and Erglis (2016).
In general, most of the previous works consider magnetic filaments with one or both free ends and in precessing fields at a fixed angle, typically precessing on a plane. In this paper we examine open superparamagnetic filaments in a magnetic field precessing at different constant angles, and with their ends held along the precession axis at a certain distance. Molecular dynamics simulations suggest that under these conditions filaments exhibit different behavior depending on the value of the precessing angle relative to a critical angle: if it is smaller, filaments bend but remain on a plane, whereas if it is larger, filaments explore the ambient space adopting helical structures Dempster et al. (2017). Here we present a detailed analytic description of these two families of filaments. We determine their equilibrium configurations by minimizing their ascribed total energy, which at quadratic order has two contributions, the bending energy and the magnetic energy due to dipolar interactions between the beads Cēbers (2003); Goubault et al. (2003). In principle, the behavior of these magnetic filaments depends on their intrinsic properties: length and bending modulus; as well as of their extrinsic properties: separation between their ends and magnetic modulus, which depends on the magnetic field parameters (magnitude and precession angle) and can be positive, negative or even vanish. However, it is possible to characterize their equilibrium configurations in terms of just two parameters capturing all of their characteristics: the boundary separation and the ratio of the magnetic to bending moduli, both scaled with powers of the total length of the filament so as to adimensionalize them. We discuss the forces required to hold the boundaries of the filaments and the behavior of their total energy as a function of these two parameters. Although both families are critical points of the total energy regardless of the precession angle, by comparing their total energies we investigate their plausibility in each precession regime, determining the parameter values for which each family is more likely to take place.
This paper is organized as follows. We begin in Sec. II with the framework that we employ to describe superparamagnetic filaments, to this end we define their energy and we express the corresponding Euler-Lagrange (EL) equations, that their equilibrium configurations must satisfy, in terms of the stresses on the filaments. In Sect. III we specialize this framework to the case of planar curves, which in Sec. IV is applied to examine the family of vertical planar filaments (their ends are fixed and aligned with the precession axis) in the perturbative and non-linear regimes. In Sec. V we do the respective analysis of the family of helices. In Sec. VI we compare the total energy of both families of filaments with same parameters to assess their possible physical realization. We close with our conclusions and discussion of future work in Sec. VII. Some derivations and calculations used or discussed in the main text are presented in the appendices.
Ii Energy and stresses
The magnetic filament is described by the curve , parametrized by arc length in three-dimensional Euclidean space and passing trough the center of the beads. Geometric quantities of the curve are expressed in terms of the Frenet-Serret (FS) frame adapted to the curve, denoted by , see Fig. 1.
The rotation of the FS frame along the curve is given by the FS formula
where is the Darboux vector; and are the FS curvature and torsion, quantifying how the curve bends in the osculating and normal planes, respectively Kreyszig (1991).
We consider paramagnetic filaments in the presence of a magnetic field precessing at an angle about a direction we choose as the axis, see Fig. 2(a).
The total energy density of the paramagnetic filament is the sum of the bending and magnetic energies ,111We disregard the weight of the filament so we do not include gravitational effects; nor we include the magnetic dipole induced by the neighbors, whose influence would rescale the magnetic modulus Zhang and Widom (1995); Cēbers (2003) with quadratic in the curvature Kratky and Porod (1949); Kamien (2002), and given by the time-averaged dipolar interactions between nearest neighbors induced by the magnetic field (a derivation of is presented in Appendix A) Goubault et al. (2003); Cēbers (2003),
where is the projection of the tangent vector onto the precession axis; is the bending modulus (with units of force times squared length); is the magnetic modulus (with units of force) defined by
with the vacuum permeability, the magnitude of the magnetic dipoles, the separation between their centers and the precession angle.
As shown below, configurations of the filaments depend sensitively on the sign of , determined in turn by . vanishes at the so-called “magic” angle , so in this case the leading order of the magnetic energy will be the quadrupolar term, which is of short range so filaments behave mostly as elastic curves Osterman et al. (2009). In the regime , which will be termed as regime , see Fig. 2(b), the magnetic modulus is positive, (the magnetic dipolar interactions are attractive), and from Eq. (2) we see that in order to minimize the filaments will tend to align with the precession axis to maximize Martin et al. (2000). By contrast in the regime , termed as regime , the magnetic modulus becomes negative, (the magnetic dipolar interactions are repulsive), and is minimized when vanishes, so the filaments will tend to lie on the plane orthogonal to the precession axis Martin et al. (2000).
To reduce the material parameters space we rescale all quantities by the bending modulus and denote the rescaled quantity by an overbar. In particular, the rescaled quantity possesses units of inverse squared length, so the inverse of its square root, , provides the characteristic length scale at which buckling occurs. The dimensionless parameter is known as the magnetoelastic parameter. This parameter quantifies the ratio of magnetic to bending energies: bending and magnetic energy scale as and , so their ratio scale as . Below, we use as a parameter to characterize the conformations of the filaments. Typical experimental values of these paramagnetic filaments222For beads with diameter , magnetic susceptibility in a magnetic field , the magnitude of the induced dipole is . are , , , so , and , Biswal and Gast (2003); Goubault et al. (2003); Biswal and Gast (2004).
The total bending and magnetic energies of the filament are given by the line integrals of the corresponding energy densities
Thus the total energy is , but since the filament is inextensible, we consider the effective energy
where is a Lagrange multiplier fixing total length which acts as an intrinsic line tension.
The change of the energy under a deformation of the curve is given by Capovilla et al. (2002); Guven and Vázquez-Montejo (2012); Guven et al. (2014)
In the first term, which represents the response of the energy to a deformation in the bulk, is the force vector, given by the sum of the bending and magnetic forces, , defined by Langer and Singer (1996); Capovilla et al. (2002); Guven and Vázquez-Montejo (2012); Guven et al. (2014); Dempster et al. (2017)
where and . is the force exerted by the line element at on the neighboring line element at , so () represents compression (tension). From the force balance at the boundaries follows that is the external force on the filament Langer and Singer (1996).
We see in Eq. (7) that the magnitudes of the bending and magnetic forces scale as and . Thus, the magnetoelastic parameter also quantifies the ratio of magnetic to bending forces, , Cēbers (2003). If the filaments are immersed in a medium of viscosity , we have from the balance of bending and viscous forces that the characteristic bending relaxation time is , Powers (2010), whereas the characteristic magnetic relaxation time is Dempster et al. (2017). The ratio of bending to magnetic relaxation times is the magnetoelastic parameter . For a filament of length , bending and magnetic moduli and , in water () we have and . Therefore, in order to be legitimate, the use of the time-averaged magnetic energy density given in Eq. (2) is justified if the precessing period is less than a millisecond, (frequency Dempster et al. (2017)), so that the characteristic relaxation times are much larger, .
The second term in Eq. (6) contains quantities arising after integration by parts and is given by the total derivative of
so it represents the change of the energy due to boundary deformations.
Stationarity of the energy implies that in equilibrium the force vector is conserved along the filament, , a consequence of the translational invariance of the total energy. By contrast, the torque vector, , with , is not conserved: ,333In this expression we have used the identity , Capovilla et al. (2002). while the first term vanishes in equilibrium, the second term, representing a torque per unit length due to the magnetic field, , does not vanish in general. However, the component of the torque along the precession axis, , is conserved on account of the rotational symmetry of the energy about such direction: , which vanishes in equilibrium.
Spanning the derivative of in terms of the two normals as ,444The projection onto the tangent vanishes identically due to the reparametrization invariance of the energy Capovilla et al. (2002). so the normal projections of the conservation law provide the Euler-Lagrange (EL) equations satisfied by equilibrium configurations, which read Dempster et al. (2017)
In solving these equations, the Lagrange multiplier is determined from boundary or periodicity conditions.
The squared magnitude of the force vector
is constant on account of the conservation law of . This constant corresponds to the first Casimir of the Euclidean group and provides a first integral of the EL equations.555EL Eq. (9b) can be written as . Thus, the scalar quantity , corresponding to the second Casimir in the case of Euler Elastica, is not conserved in equilibrium because the magnetic field breaks the rotational invariance of the energy and introduces a source of stresses. Below we analyze solutions of two families of curves satisfying these equations with their ends held along the precession axis: curves lying on a plane passing through the precession axis and helices whose axis is parallel to the precession axis.
Iii Planar cuves
Let us consider curves on a plane, say -, so the embedding functions are and the tangent vector is . Since the curve lies on a plane, it has vanishing torsion, , and the EL equation associated with deformations along reduces to
whereas the EL corresponding to deformations along is satisfied identically, because vanishes on account of the orthogonality of the binormal vector to the plane of the curve, i.e. . The force vector, defined in Eq. (7), lies on the osculating plane of the curve
Projecting onto the FS basis we obtain
Differentiating Eq. (13a) with respect to and using the FS formula we obtain Eq. (13b), whereas differentiation of Eq. (13b) reproduces the EL Eq. (11). Therefore Eq. (13a) provides a second integral of the EL Eq. (11), which permit us to express the difference between the bending and magnetic energy densities as the sum of the tangential component of the force and the constant . Moreover, this relation can be used to eliminate the curvature in favor of the projections of the tangent vector, for instance, the total energy density can be recast as
In terms of the tangent and normal vectors are and , whereas the FS curvature is . Expressing Eq. (13a) in terms of , it reduces to a quadrature for :
where we have defined
which in the mechanical analogy would represent the corresponding equation of motion, see Appendix B.
We consider filaments with their boundaries fixed, but not the tangents. Thus the variation vanishes at the boundaries (which we set at ), i.e. , and from Eq. (8) we have that the stationarity of the energy at the boundaries, , imply the vanishing of the curvature at those points. Therefore the appropriate boundary conditions (BC) for equilibrium configurations is
In consequence, the intrinsic torque vanishes at the ends and only the torque coupling position and force contributes. Furthermore, the quadrature implies that the maximum value of the angle, say , occurs at the boundaries, i.e. . Thus at the turning points the “kinetic” energy vanishes and the “potential” energy is equal to the “total energy” Audoly and Pomeau (2010), which determines the Lagrange multiplier in terms of the angle :
Once has been determined as a function of , the coordinates are obtained by integrating the tangential components
In the next section we apply these results to the case of filaments aligned with the precession axis.
Iv Vertical filaments
Here we consider a curve resulting from a deformation of a straight filament lying along the precession axis, chosen as the axis, such that the end points remain along this axis (see Fig. 3). In consequence the force is also along the precession axis: and . Thus, the potential reduces to , which has period and left-right symmetry .
We set the mid point of the curve at from where arc length is measured, being positive (negative) above (below) the axis, i.e. with . We denote the height of the boundary by so that the height difference is . We characterize the curves by the height difference rescaled with the total length
To gain some insight about how the magnetic field modifies the behavior of the filaments, we first solve the quadrature (15) in the regime of small deviations from a vertical straight line.
iv.1 Perturbative regime
We consider a small perturbation of a straight line with and we expand the constants perturbatively as , .666 and are constants, however, we are interested in determining the corrections as functions of a small parameter, determined below, required by deviations from a straight line. At quadratic order, the quadrature describes a harmonic motion
Only for 777If , then and the curve is a straight line. the quadratic potential is positive and bounded solutions are possible, given by
If the filament develops half periods,888For instance the filament shown in Fig. 3 completes only one half-period (). the wave number is given by
We see that the magnetic field modifies the minimum force required to trigger an Euler buckling instability. Furthermore, unlike the purely elastic case where the force on the filaments is always compressive, the magnetic contribution enables the force to be either tensile or compressive depending on the value of the magnetoelastic parameter relative to the squared number of half-periods: for (precession regime or ) the force is positive, , so the filament is under compression, whereas for (precession regime ) the force becomes negative, , and the filament is under tension. In the particular case with , free filaments with are possible Cēbers (2005a). The coordinates can be obtained by integrating Eq. (20), obtaining
Evaluating the second expression at the boundaries we determine the amplitude in terms of the scaled height difference defined in (21):
The total energy of the filament is , where is the scaled energy of the original straight vertical state and the second order correction is
Since increases linearly with the magnitude of the force it can be either positive or negative.
Let us now look at the stability of these states. To lowest order, the differential operator of the second variation of the energy, (derived in Appendix C, Eq. (78)), reads
The two trivial zero modes (with vanishing eigenvalues), , with constant, are associated to the translational invariance of the energy: at lowest order they correspond to infinitesimal vertical and horizontal translations respectively. To analyze the eigenmodes of we use the basis , which in order to preserve the periodicity of the original curves should have wave numbers , , so the corresponding eigenvalues for each case are
The two non-trivial zero modes with correspond to infinitesimal rotations in the plane, but for finite rotations such eigenmodes will not be zero modes, because the energy is only invariant under rotations about the precession axis (in the - plane). The eigenvalues corresponding to states with are shown in Fig. 4. We see that, like in the purely elastic case, only the eigenvalues of the ground state are all positive, so it is the only stable state in the perturbative regime. Therefore, any excited state would decay recursively to the next intermediate state with the more negative eigenvalue until the ground state is reached, Guven et al. (2012). As we will see below, comparison of the total energy of successive states leads suggests that the state is still the ground state in the non-linear regime.
iv.2 Non-linear regime
Now, we describe the behavior of the filaments in the non-linear regime, i.e. deformations far from straight configurations under the influence of a magnetic field (for comparison purposes, the case of elastic curves, is reviewed in Appendix D). If (), the quadrature (15) can be recast as
Integrating the quadrature twice, we obtain the coordinates in terms of elliptic functions in each regime,
(details are provided in Appendix E for the interested reader):
where the constants and are defined by
, and are the sine, cosine, and delta Jacobi elliptic functions; is the Jacobi amplitude; is the incomplete elliptic integral of the third kind Abramowitz and Stegun (1965); Gradshteyn and Ryzhik (2007). Recall is the buckling characteristic length.
Like in the perturbative case, if the filament possesses half periods, the wave number is given by
where is the complete elliptic integral of the first kind Abramowitz and Stegun (1965); Gradshteyn and Ryzhik (2007).
The last relation permits us to express constants and , defined in Eq. (33), in terms of the modulus and the magnetoelastic parameter .
The FS curvature of the filament is given by
where is the maximum value of the curvature. The condition (18) of vanishing curvature at the boundaries determines, .
Evaluating expressions (32) for at the boundaries and using the identities and , we get the following equation for the scaled boundary separation , defined in Eq. (21)
To determine , these equations are solved numerically for given values of , , and .999Alternatively, one could extend the method employed in Ref. Hu et al. (2013) for the purely elastic case, in which case would be expanded as a series in and and the coefficients would be determined from Eq. (36). This completes the determination of all parameters of the curve. States with in regime are plotted for different values of and in Figs. (5) and (6). Corresponding states with in regime are plotted in Figs (7) and (8). In these sequences we choose initial states with different boundary angles , specifically (top rows with ), (middle rows with ) and (bottom rows with ).101010For smaller separations , the boundaries, and also upper and bottom segments of the filaments for strong magnetic fields, get close and in consequence our nearest neighbors approximation is no longer valid. In both regimes we observe that for relative small absolute values of the magnetoelastic parameter, , the filaments behave mainly as elastic curves (shown with dashed black lines in the plots with ), adopting and shapes for and , respectively. As is increased, we observe deviations from elastic behavior depending on the regime: in regime , at , filaments begin to elongate along the precession axis and squeezing inwards along the orthogonal direction, and for a large value, , they form thin vertical hairpins connected by straight segments aligned with the precession axis; in regime the converse behavior is observed, at they begin to stretch outwards and orthogonally to the precession axis, and at they are mostly straightened and with the filament’s horizontal extremum farthest from the precession axis.
The magnitude of the forces in these planar curves is given by111111It can be positive or negative depending on the sign of the constant satisfying Eq. (36).
is plotted for states in Fig 9. For vertical curves with , the force is linear in the magnetoelastic parameter, , as found in the perturbative analysis. We see that in regime is positive for all values of and , indicating that the filaments are under compression, as is usual for elastic curves bent under compression. By contrast, there are regions in regime where becomes negative in which case filaments are under tension, reflecting the fact that they tend to lie orthogonally to the precession axis.
The bending and magnetic energy densities in terms of arc length read
The bending energy is positive, increasing with (); the magnetic energy is positive in regime and negative in regime , thus the total energy density , can be positive or negative in regime , but it is strictly positive in regime . is shown with a color scale for states in Figs. (5)-(8). For filaments in regime , we see that initially is concentrated in the extremum and low in the boundaries, but as is increased, the high-energy regions migrates towards the hairpins near the boundaries and low-energy regions move to the extremum where straight segments (minimizing both energies) are developed. In regime , high-energy regions always occur at the extremum where the curvature concentrates, whereas low-energy regions correspond to the straight segments near to the boundaries. Moreover, the former regions become more localized and the latter regions more spread as is increased.
In the calculation of the total energy, although integration of is simple, integration of is rather complicated because it involves . However, we can integrate expression (14) for the total energy density, where was replaced in favor of by means of the quadrature, (13a), obtaining the following expressions of the total energy for each case (details are presented in Appendix E):
where the constant is defined by
The total energy of states are plotted in Figs. 10(a) and 10(b). As found in the perturbative regime, regardless of , straight lines with have scaled total energy . We see that is negative almost everywhere (except in a small fringe of values in the vicinity of ) in regime and it is positive everywhere in regime . Values of and for which are shown with a solid black line, and the energies of elastic curves () are shown with a dashed black line.
The total energy of the filaments increases as augments for any value of and . To show this, in Fig. 10(c), we plot the energy difference between states and , , where we see that everywhere on the parameter space , result that can be verified for states with higher , i.e, . Thus is the ground state among planar configurations for all parameter values. However, as we will see below this does not hold in general when non-planar configurations are considered, in particular the energy may be lowered for some values of and if filaments adopt helical configurations, which we examine in the next section.
Here we demonstrate that helices are also critical points of the total energy. Recall that a helix is characterized by its radius and pitch , with the pitch angle defined by (see Fig. 11).
The helix can be parametrized in cylindrical coordinates by the azimuthal angle as,
A helical segment is specified by the total azimuthal angle . We consider helices completing full turns, so and the distance between the end points is . Arc length is proportional to , , so total length is proportional to , . From these relations follows that . Inverting to get and in terms of , , and , we get
The FS basis adapted to the helix is
whereas the FS curvature and torsion are given by
The sign of the torsion determines the chirality of the helix, ()
corresponds to right (left) handed helices. The degenerate cases and correspond to circles on the plane - and to vertical lines, respectively. For helices, the Darboux vector is along the helical axis .
Since and are constant and , the EL Eq. (9a) is satisfied if is constant, taking the value
and the EL Eq. (9b) vanishes identically. Hence, helices satisfying Eq. (45) minimize the total energy . Using these expressions for , , and in Eq. (7) for , we find that the scaled force required to hold the helix is linear in the separation of the end points and directed along the helical axis,
The magnitude of the force is plotted for states as a function of and in Fig. 12. In this plot the line over () represents the scaled force required in the Euler buckling instability of a straight line, with the elastic term four times larger as compared with the scaled force required in the planar case, . Like the case of planar curves, the force can be tensile or compressive depending on value of the magnetoelastic parameter relative to the total azimuthal angle: if () the magnitude of the axial force is positive (negative), (), and the helix is under compression (tension). For circles with () and configurations with , there is no vertical force, .
The difference of the force between successive states and ,
is independent of and positive for any value of , and it increases with .
The torque vector has two components, one introduced by the magnetic field and another one of elastic character
The magnitude of the azimuthal torque is linear in the magnetoelastic parameter, so it vanishes for elastic curves with and its direction is reversed when changing from regime to regime . For a given value of it vanishes for circles and lines with , respectively, and is maximum for helices of maximum torsion with . The magnitude of the axial torque is proportional to the total azimuthal angle and increases as the boundary points are approached, so it is maximal for circles and vanishing for vertical filaments.
The total scaled energy of the helices is harmonic in the separation of the end points Dempster et al. (2017)