Shape deformation of a vesicle under axisymmetric non-uniform alternating electric field
Non-uniform fields are commonly used to study vesicle dielectrophoresis and can be used to hitherto relatively unexplored areas of vesicle deformation and electroporation. A common but perplexing problem in vesicle dynamics is the cross over from the entropic to enthalpic (stretching) tension during vesicle deformation. A lucid demonstration of this concept is provided by the study of vesicle deformation and dielectrophoresis under axisymmetric quadrupole electric field. Small deformation theory incorporating the Maxwell stress approach is used (employing area and volume conservation constraints) to estimate the dielectrophoretic velocity. The entropic and enthalpic tensions are implemented to understand vesicle electrohydrodynamics in low and high tension limits. The shapes obtained using the entropic and the enthalpic approaches, show significant differences. A strong dependence of the final vesicle shapes on the ratio of electrical conductivities of the fluids inside and outside the vesicle as well as on the frequency of the applied quadrupole electric field is observed which could be used to estimate electromechanical properties of the vesicle. Moreover, an excess area dependent transition between the entropic and enthalpic regimes is observed. The Maxwell stress approach, used in this work, indicates that Clausius-Mossotti factor obtained by the dipole moment method together with the drag on a rigid sphere explains vesicle dielectrophoresis. Interestingly, the coupling of hydrodynamic and electric stress, important in drops is absent in vesicle dielectrophoresis to linear order.
Non-uniform fields are commonly used to study dielectrophoresis (DEP), which is the movement of an uncharged particle under a spatially non-uniform electric field. DEP results from the interaction of an electric field gradient and an induced dipole in the particle. A particle in a non-uniform field is termed to undergo positive DEP if particles migrate towards a region of high electric field while in negative DEP the particles migrate towards region of lower electric field. An understanding of the DEP behavior of bioparticles Guido et al. (2009); Li et al. (2011) has importance in several biotechnological and biomedical applications Weigl et al. (2003); Khoshmanesh et al. (2011); Jubery et al. (2014); Dey et al. (2015). First stuggested for yeast cells, DEP Pohl and Crane (1971); Crane and Pohl (1972), has subsequently been widely studied for other bioparticles such as RBCs,Gascoyne et al. (1997); Leonard et al. (2008) bacteria,Nakano et al. (2016) DNAs,Chou et al. (2002); Tuukkanen et al. (2005) proteinsNakano et al. (2011) etc. The technique has potential applications in cell manipulation, Gagnon (2011); Cemazar et al. (2013) separation, Kang et al. (2008); Meighan et al. (2009); Lewpiriyawong et al. (2011); Gagnon (2011) sorting, Fiedler et al. (1998); Taff and Voldman (2005); Braschler et al. (2008) to study electrorotation, Reichle et al. (1999); Han et al. (2013) electrofusion, Zimmermann (1982); Cavallaro et al. (2012); Yang et al. (2012) cell-cell interaction Ye et al. (2011) as well as in characterizing their physical properties Patel and Markx (2008) etc as well as for for diagnosis of cancer Alshareef et al. (2013); Gascoyne and Shim (2014); Sonnenberg et al. (2014). On the other hand, single cell studies Jang et al. (2009); Wang et al. (2013); Huang et al. (2014) have important engineering applications in drug delivery, gene introduction, cloning technology etc, apart from the fundamental insights into the mechanism of dielectrophoresis that such studies provide. Excellent controllability, high efficiency and small damage to cells make DEP technique appropriate for contact-free trapping of a cell in a region of low electric field Jang et al. (2009). This has led to research in designing effective non-uniform electric fields by judicious design of electrodes, often in microfluidic/nanofluidic chips by microfabrication techniques Shafiee et al. (2010); Cemazar et al. (2013); Huang et al. (2014) has gained prominence.
Giant Unilamellar Vesicles (Liposomes) (GUVs) have emerged as a very reliable bio-memetic system and has been used to understand the DEP response of cells Stoicheva and Hui (1994); Hadady et al. (2015). Unlike biological cells, there are very few experimental Stoicheva and Hui (1994); Froude and Zhu (2009); Kodama et al. (2013) and theoretical Kaler and Jones (1990) investigations on the DEP of vesicles . Korlach et al., Korlach et al. (2005) created a 3D electric field cage to study vesicle deformation and electro-rotation by trapping a vesicle using optical tweezers. Studies on the modification of the electrical properties of GUVs to serve them as test particle for DEP studyDesai et al. (2009), high-frequency DEP response of vesicles to estimate upper and lower crossover frequency at different interior conductivity and membrane electric properties Hadady et al. (2015) and DEP studies on surface-modified liposomes in AC fieldsFroude and Zhu (2009), have also been reported .
A vesicle under non-uniform, axisymmetric quadrupole electric field, not only exhibits dielectrophoresis, but can also deform. Although several experimental and theoretical papers have demonstrated vesicle deformation under ACVlahovska et al. (2009); Yamamoto et al. (2010); Antonova et al. (2010); Peterlin (2010), pulsed DCSalipante and Vlahovska (2014), DC fieldsMcConnell et al. (2013); C. et al. (2015), these fields are mostly uniform. A uniform field leads to prolate and oblate spheroidal (dipolar) deformations, and these have been summarized into a phase diagramDimova et al. (2009); Yamamoto et al. (2010). Moreover non-uniform fields have also shown promise in more efficient electroporation as compared to uniform fields Issadore et al. (2010).
Interestingly, very few such experimental studies on deformation of a vesicle have been conducted Korlach et al. (2005); Issadore et al. (2010) in non-uniform fields whereas there is hardly any theoretical study reported. The experiments Korlach et al. (2005); Issadore et al. (2010) indicate fascinating shapes (prolate, oblate, pear, diamond, square) due to action of quadrupolar and higher order potentials and associated Maxwell stress. The resulting shape is clearly a balance of electric, hydrodynamic and membrane stress. Amongst the different membrane stresses, there is a good understanding of the bending stress as well as the non-uniform tension that arises on account of local membrane incompressibility. However, to describe the uniform tension, two approaches have been used. The entropic approach, wherein, the tension arises due to the thermal undulations of the excess area. The assumption here is that under an external force, the excess area present in the thermal undulations is reduced, leading to a tension (hereafter called as the entropic tension). The membrane is then assumed to have enough excess area not to cause stretching at a molecular level and was employed to describe vesicle deformation under a uniform electric field Vlahovska et al. (2009); Sinha and Thaokar (2017). On the other hand, when a membrane is completely stretched, the uniform tension arises because the excess area of a vesicle can not increase during the shape deformation process (we call this the enthalpic tension). This approach has been used to describe shape deformation for vesicles in shear flowVlahovska (2007),wherein shapes are described by the Legendre mode. On the other hand, the quadrupolar field provides two degrees of freedom for shape deformation, namely the and Legendre modes.
These concepts form the basis for micropipette experiments which were initially proposed by Evans and Rawicz (1990), wherein it was showed that for tensions lower than , the aspiration can be considered entropic and the area change is logarithmic in tension. On the other hand for higher values of tension, a membrane stretches proportional to the tension, and inverse to the area incompressibility modulus (typically of the order ).
Motivated by these issues we ask the following questions,
What is the vesicle deformation in pure quadrupolar field (as well as a mix of uniform and quadrupole fields), and can the prolate, oblate, pear, diamond and square shapes, seen in experiments be explained?
When is the deformation dominated by entropy and enthalpy or how does the deformation differ from the uniform field case?
2 Mathematical formulation
2.1 Model description
The system considered consists of a spherical vesicle of radius , surrounded by a non-conducting bilayer membrane of thickness that has an electrical conductivity () and a finite permittivity (). This bilayer membrane which separates the inner fluid from the suspending medium is characterized by an interfacial tension , bending rigidity and the dilatational viscosity of the membrane, . The Newtonian fluid enclosed within a vesicle has permittivity , conductivity , and viscosity , the suspending Newtonian medium has permittivity , conductivity , and viscosity . Gravity effects are neglected on account of their small size (). We define the ratios of fluid physical properties as . Note that subscript ’in’ and ’ex’ represent quantities associated with the inner and the outer fluid, respectively.
To generate a non-uniform electric field, axisymmetric quadrupole electrodes (symmetric about the z axis) are used in this work (Figure1). The geometric center of the axisymmetric electrode setup is the region of minimum electric field, whereas the electric field is maximum at the electrode edges. A spherical coordinate system is assumed such that the origin of the coordinate system is at the geometric center of the electrode system. A time-periodic, non-uniform, axisymmetric, AC electric field is externally applied to a vesicle placed at the center of the electrode system. The applied electric potential is expressed as a sum of uniform and quadrupole electric potentials as , where and are the intensities of the uniform and the quadrupole electric fields, respectively. Here and denote Legendre Polynomials of first and second degree, respectively. The electric field generated due to this applied electric potential can be expressed as . In this work, most of the equations are expressed in their dimensional form (with no over-bar) but the results are primarily presented in non-dimensional form (represented with an over-bar) using appropriate dimensionless parameters.
2.2 Governing equations and boundary conditions
The inner and outer fluids are assumed to be leaky dielectrics. The solution of Laplace equation (where ) in spherical coordinate system results in the electric potential outside () and inside () the vesicle to be of the form,
where coefficients (provided in Appendix-A), are obtained by solving the following electrostatic boundary conditions at the membrane interface () and using orthogonality of Legendre polynomials,
Here and are the transmembrane potentials (Appendix-B) across the membrane associated with the and Legendre modes, respectively. and , which can be modelled as and , are the membrane capacitance and conductance respectively.
The normal and tangential electric fields are obtained from the electric potentials using the definitions , .
Using the Maxwell’s stress tensor where and I is the identity tensor, the normal () and tangential () electric stresses acting at the vesicle surface can be estimated. Here and represents the normal and tangent unit vectors to an undeformed sphere. The non-oscillatory part of the Maxwell stresses can be further expressed asSinha and Thaokar (2017); Thaokar (2016).
where * represents complex conjugate of the respective physical quantity and and are the radial and tangential electric fields respectively. The net normal and tangential electric stresses on the vesicle are . Full expressions for these quantities in a simplified form are provided in the Appendix-C.
The velocity and pressure fields corresponding to each fluid region (inner or outer) are given by the Stokes equation and the continuity equation (). Here, the inertial effects are ignored, thereby addressing small Reynolds number conditions. Assuming axisymmetry and adopting a stream function approach, the stream functions for the outer and inner regions are of the generalized form
where are the Gegenbauer’s function of first kind (Appendix-D). The velocity fields can be expressed in terms of stream functions () as , where and are the normal and tangential velocity components, respectively.
The pressure is governed by the solution of the Laplace equation, . For the outer and inner regions it is considered to be of the form
here and are unknown coefficients to be determined (Appendix-E).
The normal and tangential hydrodynamic stresses at the outer and inner surfaces of the membrane are given by
2.2.3 Membrane mechanics
The surface of a deformed vesicle is described by
where is the radial position of a slightly deformed vesicle surface from it’s center, are deformation amplitudes associated with respective Legendre modes. is obtained by the constraint of volume conservation , to yield
The deviation of the vesicle shape from a sphere (of fixed volume ) is defined by a shape function , therefore the unit normal at is . The constraint of area conservation of the bilayer membrane results in a relation between deformation amplitudes and excess area of the form . This gives the non-dimensional excess area as
Membrane stress: The stress due membrane bending (), due to both uniform () and nonuniform tensions (, ) as well as due to normal and tangential interfacial membrane stresses (, ) are given by
here is the Boussinesq number and is the membrane viscosity associated with dilatational deformation of the membrane. The normal membrane stress component is , the tangential membrane stress component is . Note that the stress associated with the membrane tension has contributions from both uniform tension, which could be entropic or enthalpic and discussed later (equation 26 and Appendix-E) as well as the non-uniform tension which varies with position along the surface and is obtained by applying membrane incompressibility condition (), where is surface gradient operator. This conforms local area conservation and yields the non-uniform tension associated with each mode namely (Appendix-F).
2.2.4 Boundary conditions
The balance of membrane and fluid stresses and continuity of their velocity fields across the membrane interface are given by Taylor expanding the following boundary conditions at the undeformed membrane surface
(b1) Normal stress balance: , where [[.]] represents the difference in properties of outer and inner fluid across the interface
(b2) Tangential velocity continuity:
(b3) Membrane incompressibility condition:
(b4) Tangential stress balance:
(b5) Kinematic condition:
The above boundary conditions (b1-b5) are solved using the orthogonality condition for Legendre polynomials. The boundary conditions are integrated at each order of the Legendre polynomials in order to get the unknown constants associated with the velocity and pressure fields for both the inner and the outer fluids (Appendix-G).
3 Entropic and enthalpic tension approach for uniform tension
In the present work, the electric field induced shape deformations are estimated by two different ways of describing the uniform tension in the membrane. These are based on the two regimes for tension as given by Evans and Rawicz (1990). In the first case, when the induced tension is low, the tension is estimated by using the entropic theory. At low tension a membrane is in a highly fluctuating state where in the excess area of a vesicle is present in various deformation modes. Therefore an applied stress leads to a change in area that is described by the amplitudes of modes (equation16) due to straightening of the fluctuations (wiggles), resulting in membrane tension that is given by
Therefore, the uniform tension in the membrane because of excess area is
where is the initial tension in the membrane. Here, contains contribution from all the three modes ( and ).
In the second case, when the induced tension is high, a vesicle is said to be in the enthalpic regime. In this state, the deformation modes are related by the constraint of the vesicle area remaining constant during deformation (equation16), , leading to
By substituting the solutions from kinematic conditions (equation 24) for the enthalpic tension, , can be determined and is provided in Appendix-E.
The overall shapes of a vesicle obtained in the entropic and the enthalpic cases are described by the positions of their interface, that is . Here deformation mode deforms a vesicle into a prolate/oblate ellipsoids, the mode makes the shapes non-axisymmetric about the z-axis, while the mode imparts higher order shapes (square, diamond, pear etc).
4 Dimensionless parameters
All length scales are non dimensionalized by the radius of vesicle , the potential by either or , the tension by , and the velocity and the stress by and , respectively. A dimensionless factor is introduced, which compares the relative strength of the applied uniform and quadrupole electric field, such that represents a pure quadrupole field. Another dimensionless quantity which compares the relative strength of the shape deforming electric stress and shape resisting bending stress is the capillary number . There are several time scales in the problem, the hydrodynamic time scale , the charge relaxation time scale of outer fluid , the Maxwell Wagner relaxation time and the membrane charging time , where . Here we use for non-dimensionalizing the time and represents the non-dimensional frequency. For simplicity we present results for the case , which is valid for low conductivity fluids.
The non-dimensional equations for the time evolution of the various deformation modes are given by,
Rate of shape deformation
where constants terms ( and ) are parts of the normal and tangential electric stresses, respectively. These are lengthy expressions and therefore not provided in the manuscript.
5 Results and discussion
Although the results are presented in non-dimensional parameters (represented by an over-bar) it is important to mention the typical experimental parameters of relevanceFroude and Zhu (2009); Korlach et al. (2005). These are mostly borrowed from the values reported in Froude and Zhu (2009); Korlach et al. (2005) and Issadore et al. (2010), and have been used to determine the range of non-dimensional parameters used in this work. An isolated spherical vesicle of , motivated by typical size of a biological cell, membrane conductivity , membrane permittivity , initial membrane tension , and membrane thickness , containing a fluid of is assumed to be suspended in a medium of , . Here is the permittivity of free space.
The axisymmetric quadrupole electrodes in experiments typically consist of two end cap electrodes maintained at a certain voltage and a ring electrode which is often ground. An AC electric field can be generated by a peak-to-peak voltage of = 40 applied to the end-cap (live) and the ring electrode ( = 0 ) is grounded. An additional 10 potential can be superimposed "between" two end cap electrodes to produce a simultaneous uniform and quadrupole field between the electrodes. The frequency can be varied from around 500 Hz to more than 10 . A separation of and , can be maintained between the electrodes (see figure1), where are the distances of the end caps and the ring respectively, from the center of quadrupole system. These trap parameters are similar to the experimental work of Froude and Zhu (2009) except while their electrode arrangement is 2D planar quadrupole, the present work deals with axisymmetric quadrupolar system. Similar to Froude and Zhu (2009) the conductivity ratio is considered, and additionally, is also used to investigate the case.
A vesicle, initially positioned at the center of electrode system, deforms under applied electric field. Additionally, it undergoes dielectrophoresis for a non-zero value of the parameter . Therefore, the study is divided into three parts. The first and second parts address the problem of vesicle deformation under pure quadrupole field () as well as under simultaneous uniform and quadrupole fields (non-zero ), respectively. Both entropic and enthalpic tension approaches are used in the deformation studies and a variety of shapes are reported. A qualitative comparison with the vesicle shapes reported in recent experimental papers Korlach et al. (2005); Issadore et al. (2010) is also made. Lastly, the dielectrophoretic motion of a vesicle under non-uniform electric field is presented when a combination of uniform and quadrupole electric field is applied (non-zero ) and the details are provided in the supplementary material.
5.1 Transmembrane potential
The transmembrane potential (TMP) generated by a uniform electric field is equal and opposite at the north and south poles of the vesicle. On the contrary the transmembrane potential due to the quadrupole field is symmetric about the equator and therefore identical at the north and south poles. Therefore for membranes with a finite resting potential (in the absence of field), while uniform field would lead to a difference in poration tendency at the north or south pole, a quadrupole field will cause symmetric poration at the two poles. The variation of the amplitude of the transmembrane potential (Appendix B: equation 37 and 38), plotted in figure 2 for , shows that the TMP falls over a frequency equivalent to the reciprocal of the capacitor charging time for and for ) for both uniform and quadrupolar parts of the applied fields. The transmembrane potential due to quadrupolar part is greater than that due to the uniform part.
Increasing the membrane conductance from 0 to 0.5 reduces the transmembrane potential slightly for . However, the reduction is precipitous for . The transmembrane potential in the low frequency limit in a non-conducting membrane is a result of the charges built on the membrane to reduce the normal electric field in the outer region to zero. This potential is independent of the conductivity ratio of the two fluids for a non-conducting membrane. In a conducting membrane, the outer electric field need not be zero, since the membrane can allow current to flow through it. This leads to a drop in the transmembrane potential. A reduction in means a lower electrical conductivity of the inner fluid for the same conductivity of the outer fluid (used in non-dimensionalization). This leads to lower transmembrane current (ohmic) which results in a very small transmembrane potential being able to drive ohmic current through the conducting membrane when .
5.2 Deformation in pure quadrupole electric field
A vesicle deforms in an applied electric field on account of the Maxwell stress, proportional to the square of electric field, acting on it. Thus while a uniform electric field () produces shapes described by the mode, when a pure quadrupolar () field is applied, the Maxwell stresses and thereby the resulting shapes are expected to have the modes. Atleast two experimental results are reported in the literature on vesicle deformation in multipolar fields Korlach et al. (2005); Issadore et al. (2010). Unlike uniform electric fields which typically result in prolate or oblate spheroids, higher order multipolar fields (e.g. quadrupole and octupole) lead to interesting shapes such as square, diamond and even hexagonal. We therefore understand the deformation of a vesicle in quadrupole potentials using the Maxwell stress approach.
To determine realistic parameters for quadrupolar fields that can cause deformation, one can consider around potential applied between the end cap electrodes and the ring electrodes, with and =10 . The applied potential generates an axisymmetric quadrupole electric field of strength . A vesicle of size and excess area can be considered to be at the center where the DEP force on the vesicle is zero on account of zero electric field, thereby preventing its translation. This allows a systematic analysis of shape deformations due to non-zero electric stress at the vesicle surface. In calculations presented , that gives a capillary number of . These numbers serve as a reference for the choice of non-dimensional parameters used in the present analysis.
Figure 3 shows the evolution of amplitude of deformation modes, and , with time for very high and low at an intermediate frequency () in both entropic and enthalpic regimes. For both and are positive, while for , and are negative. The deformation mode is not admitted in the pure quadrupole case (). Thus starting with an initial spherical shape, both the enthalpic and entropic regimes show that a final, non-spherical, equilibrium shape is reached. The deformations in the enthalpic regime are higher than those in the entropic regime.
Figures 4 and 5 show the variation of the deformation ( and ) and the final equilibrium shapes, respectively. The variation is shown with frequency, at different conductivity ratios, in the entropic and enthalpic regimes. Unlike the case of uniform fields, where prolate and oblate spheroids imply that deformation suffices to explain the shape of the vesicles, in quadrupole fields, occurrence of and amplitudes suggest that the actual shape should be discussed (figure 5). A distinctive feature of pure quadrupole fields is the observation of oblate deformation for the () mode even at very low frequencies in agreement with similar results for a drop Deshmukh and Thaokar (2012) (Figure 6a). This is essentially due to the fact that pure normal stresses in quadrupole field favor oblate shapes even in the absence tangential stresses Deshmukh and Thaokar (2012), unlike uniform fields wherein vesicles are always prolate in the low frequency regime.
In the entropic regime, in the intermediate frequency range, , a vesicle behaves like a drop. At higher inner conductivity () the deformation is predominantly prolate for all frequencies (see figure 6b), whereas for , oblate deformations are observed at low and intermediate frequencies (see figure 6c). A feature of these plots (compared to uniform field which admits only the mode) is the different qualitative behaviour of the and amplitudes of the , modes with frequency. In the high frequency regime, a nearly spherical shape is seen since the Maxwell stresses become small although do not disappear. Thus unlike a drop even at high frequencies, a vesicle does show non-zero, although small, tangential and normal electric fields and stresses, on account of the capacitance of the membrane.
In the enthalpic regime, at low and intermediate frequencies, the variation of the shapes with frequency is qualitatively similar to that observed in the entropic case. However, unlike the entropic case, low deformation (that is nearly spherical shape) is not observed at very high frequencies, since the modes have to conserve the total area and volume simultaneously. The distribution of this area between and is clearly seen in the figures 4,5. A variety of shapes are therefore observed which are a competition between minimization of the bending, tension and the electrostatic energy. The role of electrostatics is indicated by prolate shapes for and oblate shapes at intermediate frequencies for and also by the fact that the the shapes depend upon the conductivity ratio. Both the entropic and enthalpic regimes show significant deviation from ellipsoidal shapes, exhibiting rhomboidal, cuboidal shapes on account of . The transition frequencies can be used to estimate electromechanical properties of a vesicle.
Figure 5a shows the stable shapes for high and low in the entropic regime. It should be noted that while the shapes in the pure entropic regime are obtained by solving the dynamical equations for and , calculations in the enthalpic regime lead to different nature of the equations. The steady state solution to the evolution equations (33-31) for and in the enthalpic regime gives multiple roots (as pairs in ) of which only one root (and thereby shape) is expected to be stable. It should be noted that each root corresponds to a different shape. To find the stable root, the eigenvalues of the linearized coefficient matrix resulting from the the two non-linear differential equations for and (33-31) are determined for each of the roots, and the root is deemed stable if the eigenvalues are negative (and unstable if at least one of the eigen value is positive). This is also demonstrated by solving the evolution equations (33-31) with respect to time, with the initial shape ( and values) being a perturbation around the stable or unstable roots. When the initial condition corresponds to the unstable roots, the system is seen to evolve to the stable roots (and thereby shapes). Figure 7 shows these plots for and respectively. A systematic evolution to the stable shapes is seen, with an interesting period of relative quiescence before the transition.
Figure 9 shows the contribution of different stabilizing membrane forces (bending, uniform and nonuniform tension) as well as deforming electric forces and the resulting hydrodynamic forces. The figure 9(a,b) shows that the deforming normal electric stresses are balanced by the uniform tension in both the entropic regimes at all frequencies and enthalpic regime at intermediate and low frequencies (only shown in figure). At high frequencies, in the enthalpic regime though, a small but non-trivial contribution of the bending and nonuniform tension generated normal forces is observed. This is really due to the low absolute value of the normal stresses at high frequencies. The contribution of bending forces is found to increase further and equal to that due to the nonuniform tension at small capillary numbers (figure 9(e,f)).
The tangential electric stresses are balanced by the non uniform tension and the tangential hydrodynamic stress in both the entropic and enthalpic regimes at intermediate and high frequencies (only shown in figure 9(c,d)). At very low frequencies, the tangential electric stress tends to zero, and while in the entropic regime, the small tangential hydrodynamic stress is balanced by the non uniform tension, in the enthalpic regime, the values of tangential stress is even lower, and a balance of all the three stresses, hydrodynamic, electrical and non uniform tension is observed (figure 9(g,h)).
5.2.1 Entropic (Ca dependent) and Enthalpic tension vs Excess area
Figure 8 presents variation of entropic and enthalpic tension with excess area at three different frequencies. Figure 8a shows that in the fluctuation dominated regime, the tension varies exponentially with the excess area, in agreement with the prediction of Evans and Rawicz (1990). Figure 8b shows that a vesicle in the enthalpic regime, exhibits a decrease in tension with an increase in the excess area ( where ), for the same and different frequencies, as well as for different capillary numbers for a given frequency. The tension increases with capillary number, and decreases with the frequency. The experimental work by Evans and Rawicz (1990) indicates that when a vesicle is deflated from a spherical shape due to application of an external force, the maximum tension could not be more than 0.5 in the low tension regime. In the high tension regime, the maximum allowable vesicle tension is known to be typically around , thereby limiting the capillary number. The entropic and enthalpic tensions as a function of the capillary number are plotted in figure 8c. The figure shows that for a given Ca, the excess area dependent enthalpic tension is higher for smaller excess area, but is always lower than the entropic tension. The entropic tension increases weakly with the capillary number in the low capillary number limit, and shows a scaling at around Ca 50. The enthalpic tensions scales as . in the high capillary number limit. It is therefore proposed that a transition from the entropic to enthalpic regime can be expected to occur around Ca 100. The dimensional value of tension around this regime is of the order of 0.5 mN/m.
5.3 Deformation in mixed field
To generate a mixed field, non-zero values of and are required yielding a finite value of as discussed earlier for the case of dielectrophoresis. Using similar electrical parameters as for dielectrophoresis, yield a capillary number, and .
Figures 10 and 11 show the variation of the deformation amplitudes with frequency in the entropic and enthalpic regimes. A clear presence of the amplitude is seen indicating that asymmetric shapes are admitted. Depending upon the frequency regime and the conductivity ratio, the three modes have varying magnitudes. The dominance of mode () in the deformation increases as the increases, conforming to the known results of spheroidal deformation in a uniform electric field. Figures 12 corroborates these findings, wherein the shapes of the vesicles are shown and asymmetry is seen to increase as takes intermediate values.
In the enthalpic regime, figure 13 shows that for asymmetric modes are stable only at high frequencies. On the contrary, for , asymmetric shapes are seen only at while near quadrupole and near uniform fields admit symmetric shapes at all frequencies.
5.4 Comments on dielectrophoresis in quadrupole field and the Maxwell stress approach
An important question of relevance is the stability (with respect to position) of the vesicle in side the quadrupole field. Vesicles are known to undergo dielectrophoresis in non-uniform electric fields. Typically calculation of dielectrophoretic velocity has two parts, the total electrostatic force acting on the vesicle and the drag force on a moving vesicle. Dielectrophoresis has been studied using a rigorous Maxwell stress approach Sauer (1983); Rosales and Lim (2005); Kumar and Hesketh (2012); Jarro et al. (2007); Kurgan (2011); Wang and Gascoyne (1997) by integrating the electric stress tensor over the spherical surface of a particle in a leaky media under a slightly nonuniform electric field. On the other hand the more popular dipole moment methodKaler and Jones (1990); Morgan and Green (2003) considers the force exerted by the gradient of the applied field on the polarization vector induced in a spherical particle assuming it is subjected to the electric field at the center of mass. The dipole moment method, although not exact, is more commonly used due to its applicability to arbitrary fields. On the contrary, for composite, concentric spherical systems gets complicated and the Maxwell stress method might be more straightforward, ass used in this work.
The advantage of Maxwell stress method over the dipole moment method is best demonstrated in the case of liquid drops, wherein non uniform electric field has been extensively studied to understand their translation, deformation, levitation, breakup etc Feng (1996); Im and Kang (2003); Kim et al. (2007); Thaokar (2012); Deshmukh and Thaokar (2012, 2013); Mandal et al. (2016, 2017). The analysis shows that the tangential electric stresses in a leaky dielectric system (both the drop and the fluid medium in which it is suspended are leaky dielectrics), leads to circulation inside the drop, as well as in the outer fluid medium. This alters the drag on the particle. Therefore to obtain the dielectrophoretic velocity, the problem has to be solved using the Maxwell stress approach. A simplified approach, where the DEP force is calculated from the dipole moment method, and the drag say the Hadamard-Rybczynski equation, can lead to erroneous results.
Similar to a drop interface, the vesicle interface, which is typically a bilayer membrane, is deformable as discussed by Powers (2010) and demonstrated in this work. There is no apriori reason to not expect this coupling in a vesicle (or a biological cell) as well, since the electric stresses at the interface can in-principle drive fluid motion in the inner and outer side of the vesicle. Thus although the scientific community working on dielectrophoresis of cells and vesicles has been using the net dielectrophoretic force calculated by the dipole moment method, and the net drag as that given by assumption of rigid body hydrodynaimcs, this assumption of a rigid body drag cannot be apriori assumed, and if true, should be shown rigorously.
It is therefore important to self consistently solve the electrohydrodynamics problem using the Maxwell-stress and low Re approach. A self consistent calculation, presented in the supplementary material, on the axisymmetric quadrupole electric field yields the following results,
The dipole moment method and the Maxwell stress method to describe the dielectrophoresis of vesicles are indeed identical for Quadrupole fields and yield the same dielectrophoretic force.
The correct hydrodynamics in such a case, specifically the drag on a vesicle, is found to obey the drag on a rigid sphere (Stokes drag) thereby vindicating the often used, but not explicitly proved, assumption, typically used in the literature.
6 Concluding remarks
A systematic analysis of vesicle dielectrophoresis and deformation in non-uniform AC electric field is presented. The deformation of a vesicle in quadrupole field shows a variety of shapes such as cuboid and rhomboid, significantly different that the spheroids seen in uniform field, and these shapes depend upon the regime, entropic or enthalpic, as well as the conductivity ratio and the applied frequency.
It would be appropriate to compare the experimental results in Issadore et al. (2010) and Korlach et al. (2005) with the analysis conducted in this work while accounting for the planar(quadrupole) and 3D (octupole) fields in their set-ups respectively as against the axisymmetric quadrupole field in the present case. The squaring of shapes is clearly observed in the experiments of Issadore et al. (2010) for planar quadrupole. Using their experimental parameters it is seen that two shapes from the enthalpic theory presented in this work (figure 14) are identical to their experimental shapes (figures 6(c,e) of Issadore et al. (2010)).
Similarly as predicted in this work, intermediate frequency squaring and high frequency near spherical vesicles can be seen in figures 3(b) and 3(c), respectively in Korlach et al. (2005).
The electrical parameters as well as the quadrupole electrode design suggested in this work, should allow the method to be used for understanding electrodeformation of vesicles and biological cells in non uniform fields that are more commonly used in experiments and applications. The method shows that in the entropic regime, the vesicles admit higher order shapes, indicating influence of quadrupole field. In the enthalpic regime, the electric field as well as the frequency and the conductivity ratio determines the final shape of the vesicle with a given excess area and high nonlinearity in the shapes is observed. When a uniform electric field is employed in the enthalpic regime, the shape is prolate or oblate spheroid that satisfies the excess area constraint, and therefore an interplay of different shape modes cannot be investigated. Thus quadrupole field is the simplest axisymmetric configuration that explores the competition of and deformation in determining the final shape.
It should be noted that though highly non linear shapes at high capillary numbers are presented in this work, the electrostatics and hydrodynamics are solved on a sphere though. However, it should be mentioned that the membrane deformation leads to non-linear equations due to area incompressibility conditions even at linear order in deformation. Thus although the drag on the non-spherical shapes could be different, this should only lead to slower dynamics, while keeping the shapes similar to what are predicted in the present work.
This work using the more rigorous Maxwell stress tensor method, additionally corroborates that the dipole moment method with Stokes drag for a rigid sphere suffices to estimate the dielectrophoretic velocity of a vesicle in quadrupole field.
Authors would like to acknowledge the Department of Science and Technology, India, for financial support.
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