an electric-field induced dynamical state in dispersions of highly ... · frequency electric fields...

Colloid Polym Sci (2015) 293:3325–3336DOI 10.1007/s00396-015-3707-4

INVITED ARTICLE

An electric-field induced dynamical state in dispersionsof highly charged colloidal rods: comparison of experimentand theory

K. Kang1 · J. K. G. Dhont1,2

Received: 29 April 2015 / Revised: 8 July 2015 / Accepted: 14 July 2015 / Published online: 9 August 2015© The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract Concentrated dispersions of highly charged rod-like colloids (fd-virus particles) in isotropic-nematic coex-istence exhibit a dynamical state when subjected to low-frequency electric fields [Soft Matter, 2010, 6, 273]. Thisdynamical state consists of nematic domains which persis-tently melt and form on time scales typically of the orderof seconds. The origin of the dynamical state has beenattributed to a field-induced, cyclic dissociation and asso-ciation of condensed ions [Soft Matter, 2014, 10, 1987,Soft Matter, 2015, 11, 2893]. The ionic strength increaseson dissociation of condensed ions, rendering the nematicdomains unstable, while the subsequent decrease of theionic strength due to association of condensed ions leads toa recurrent stabilization of the nematic state. The role of dis-sociation/association of condensed ions in the phase/statebehaviour of charged colloids in electric fields has not beenaddressed before. The electric field strength that is nec-essary to dissociate sufficient condensed ions to render anematic domain unstable, depends critically on the ambientionic strength of the dispersion without the external field,as well as the rod-concentration. The aim of this paper is tocompare experimental results for the location of transitionlines and the dynamics of melting and forming of nematicdomains at various ionic strengths and rod-concentrations

� K. [email protected]

1 Soft Condensed Matter, Forschungszentrum Jülich Instituteof Complex Systems (ICS-3), D-52425 Jülich, Germany

2 Department of Physics, Heinrich-Heine UniversitätDüsseldorf, D-40225 Düsseldorf, Germany

with the ion-dissociation/association model. Phase/state dia-grams in the field-amplitude versus frequency plane attwo different ambient ionic strengths and various rod-concentrations are presented, and compared to the theory.The time scale on which melting and forming of the nematicdomains occurs diverges on approach of the transition linewhere the dynamical state appears. The corresponding crit-ical exponents have been measured by means of imagetime-correlation spectroscopy [Eur. Phys. J. E, 2009, 30,333], and are compared to the theoretical values predictedby the ion-dissociation/association model.

Keywords Colloids · Electric fields · Dynamical state ·Nematic · Fd-virus

Introduction

Phase transitions in colloidal systems can be induced byexternal electric fields due to interactions between polar-ization charges. At high frequencies (in the MHz range),dielectric polarization of the cores of spherical colloids giverise to the formation of strings of particles, and their sub-sequent field-assisted assembly into sheets [1–4]. At thesehigh frequencies, the structure of the diffuse electric doublelayer is not affected by the electric field. In addition, hydro-dynamic interactions between the colloidal particles throughfield-induced electro-osmotic flow ceases to occur at thesehigh frequencies. For lower frequencies, below about a kHz,interactions between colloids are induced through the polar-ization of electric double layers and electro-osmotic flow.These two types of field-induced interactions give rise tophase transitions, dynamical states, and pattern formation.The coupling between polarization of the diffuse electricdouble layers and field-induced electro-osmotic flow is,

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3326 Colloid Polym Sci (2015) 293:3325–3336

for example, responsible for the formation of mesoscopi-cally large zig-zag structures, where electro-osmotic flowpersists along the boundaries of the zig-zag structures [5–7]. The present authors found various phases and statesthat are induced by similarly low-frequency electric fieldsin dispersions of highly charged colloidal rods, at a con-centration where there is isotropic-nematic coexistence inthe absence of the electric field [8]. In particular, a field-induced dynamical state is found where nematic domainspersistently melt and form on a seconds time scale. A mech-anism underlying the existence of this dynamical state isproposed in Ref. [9], where temporal cyclic dissociationand association of condensed ions plays an essential role.The dissociation of condensed ions into the bulk disper-sion leads to a decreased Debye screening length which candestabilize the nematic. After melting of the nematic, re-association of condensed ions occurs, leading to a decreasedDebye length and a re-stabilization of the nematic state.Such a mechanism of dissociation and association of con-densed ions has not been considered before to play a rolein field-induced states. Although the location of state tran-sition lines in the field amplitude versus frequency planefor two rod-concentrations can be described by the ion-dissociation/association model for a fixed ionic strength(as shown in Ref. [9], with a corrigendum in Ref.[10]),the critical test where also the ambient ionic strength isvaried has not been performed yet. According to the ion-dissociation/association model, there should be a strongdependence of the location of transition lines on the ambientionic strength. In addition, we present here a compari-son with experimental results on the dynamics of meltingand forming of nematic domains from Ref. [11], with anemphasis on the critical divergence of the correspondingtime scales on approach of the transition line. The aim ofthis paper is therefore to test the dissociation/associationmodel as presented in Refs. [9, 10] against experimentswhere phase/state diagrams and time scales for melting andforming of nematic domains are measured at various ionicstrengths, as well as various rod-concentrations.

As a model system for rod-like colloids, we use fd-virusparticles. These particles have been used in the past as col-loidal model systems for the study of the isotropic-nematicphase transition under the influence of a magnetic field[12], and later for extensive studies of various liquid crys-talline phases without an external field [13–20], as well asfor single-particle diffusive behaviour [21–23]. In particu-lar, a chiral nematic phase is found for ionic strengths downto 5 mM [13, 14, 18, 19], which is attributed to the helicalcore-structure of the fd-virus. Fd-viruses consist of a DNAstrand with a contour length of 880 nm, which is covered by2700 proteins. The diameter of the core is 6.8 nm, while thecoat proteins renders the fd-viruses relatively stiff, with apersistence length of about 2500 nm. At a pH around 7, the

coat proteins carry 8800 negative elementary charges [24],which leads to strong Manning-ion-condensation, where themajority of the immobile surface charges are neutralized bycondensed ions.

This paper is organized as follows. In “The electricphase/state diagram” section, we briefly present the generalfeatures of the phase/state diagram. The theory that semi-quantitatively describes the location of the transition line inthe field-amplitude versus frequency plane to the dynamicalstate, as well as the dynamical features of melting and form-ing of nematic domains, is summarized in “The ion-dissoci-ation/association model” section. A necessary ingredient ofthe theory is the lower isotropic-nematic binodal concentra-tion, which is addressed in the “Binodal and spinodal con-centrations” section. In the “Comparison to experiments”section, we present experimental phase/state diagrams fortwo buffer concentrations of 0.16 and 0.0032 mM, cor-responding to two different ionic strengths, and for vari-ous fd-concentrations. The location of the transition lineto the dynamical state is compared to the prediction bythe dissociation/association model in “The location of theN∗-to-D transition line” section. The dynamics of melt-ing and forming of nematic domains is discussed in“The dynamical behavior on approach of the N∗-to-Dtransition line” section.

The electric phase/state diagram

The phase/state diagram in the field strength versus fre-quency plane is given in Fig. 1, for a fd-concentrationof 2.0 mg/ml. The fd-dispersion is dialyzed againsta TRIS/HCl buffer solution with a concentration of0.032 mM, with a pH of 5.8. The following phases and statesare found (for a more detailed description of the variousphases and states, see Ref. [8]):

Fig. 1 The phase/state diagram for a fd-concentration of 2.0 mg/mland a TRIS/HCl-buffer concentration of 0.032 mM. The variousphases are discussed in the main text

Colloid Polym Sci (2015) 293:3325–3336 3327

(i) The N-phase is a coexistence between isotropic and(non-chiral) nematic regions. This is the phase thatexists without the external electric field, and persistto be stable up to a finite electric field strength. Asmentioned in the introduction, at buffer concentra-tions larger than 5 mM, the nematic is chiral, dueto the helical structure of the core of fd-virus parti-cles. For the low ambient ionic strength used here, thelong-ranged electrostatic repulsions render the aver-age distance between fd-rods sufficiently large, sothat the helical structure of the cores is screened. TheDebye length at this buffer concentration is equal to54 nm (where the solution of carbon dioxide from theair is taken into account [25]).

(ii) For frequencies below about 300 Hz, the nematicdomains become chiral nematic, the N∗-phase,on increasing the field amplitude. The N-to-N∗transition line is indicated in blue in Fig. 1. This tran-sition is most probably due to the increase of theionic strength resulting from the overall dissociationof condensed ions. The chiral nematic state is alsoseen without the electric field but for larger ambientionic strengths.

(iii) At frequencies larger than about 300 Hz, a uniformphase is found, where the rods are aligned alongthe electric field, perpendicular to the electrodes.This phase is named the H -phase, standing for thehomeotropic alignment of the rods. The frequency atwhich the polarization of the electric double layerand the layer of condensed ions ceases to occur canbe estimated to be around the frequency where theN∗-to-H transition line is located (the green linein Fig. 1). The H -phase is therefore stabilized byhydrodynamic interactions through the field-inducedelectro-osmotic flow.

(iv) At sufficiently high electric field strengths (above thered line in Fig. 1), theN∗- andH -phase transform to adynamical state, the D-state, where nematic domainspersistently melt and form.

There are two additional gradual transitions which arenot indicated Fig. 1. Within the N∗-phase, at low fieldstrengths, the nematic domains form a seemingly intercon-nected structure. On increasing the electric field strength,the interconnectivity is gradually lost. There is thus agradual transition within the N∗-phase where the chiralnematic texture becomes disconnected on increasing thefield strength [8]. Close to the N∗-to-D transition line thedynamics of melting is very slow, while far from the tran-sition line, the time scale on which melting and formingof domains occurs levels off to about 1.5 s. There is thusa gradual transition from very slow to fast dynamics ofmelting and forming of nematic domains.

It is the dynamical state that is of interest in this paper.In the next section, the microscopic origin of this state isdiscussed, first on an intuitive level, followed by a semi-quantitative analysis.

The ion-dissociation/association model

The microscopic origin for the existence of the D-stateis attributed to the cyclic dissociation and association ofcondensed ions. Figure 2, which is taken from Ref. [9],illustrates the mechanism through which such a cyclicdissociation and association is kinetically induced. Firstconsider a nematic domain with its director aligned alongthe direction of the electric field (stage (I) in Fig. 2). InFig. 2, only two rods out of the entire domain are depictedfor clarity. The domain orientation along the field direction

Fig. 2 A sketch of themicroscopic origin of thedynamical state, where nematicdomains melt and form. Thedifferent stages during a cycle ofmelting-and-forming of adomain are discussed in themain text


is a consequence of single-particle torques resulting fromfield-induced polarization of the double-layer and the layerof condensed ions. In such an orientation, their is a rela-tively large amount of excess condensed (positive) ions atthe top side of the rod (indicated in red), while there is adepleted region at the bottom part (indicated in blue). Theexcess of condensed ions at the top creates an electric fieldthat pushes the ions out-of the condensed layer into bulksolution (indicated by the arrows in stage (I)). The oppositehappens at the bottom part of the rod. As the concentrationof ions in the condensed layer is much larger than in thediffuse double layer, the net result is a dissociation of con-densed ions. After a time comparable to the time neededfor the released ions to diffuse into the bulk solvent, theionic strength will increase. This will decrease the Debyelength (the extent of the diffuse double layer is indicatedin Fig. 2 by the dotted blue lines around the core of therods). Hence, in stage (II), the rods carry less condensedions, while the Debye length is smaller. A reduction of theDebye length corresponds to a reduction of the effectiveconcentration (the effective concentration will be quantifiedin the subsequent paragraph). When the effective concentra-tion becomes less than the lower isotropic-nematic binodal,the nematic domain becomes unstable and melts, which isaccompanied by a de-alignment, as depicted in Fig. 2 instage (III). Association of condensed ions occurs as the rodstake orientations towards directions perpendicular to theelectric field. This leads in turn to a decreased ambient ionicstrength, and thereby to an increase of the Debye length(as depicted in stage (IV)). The accompanied increase ofthe effective concentration renders the nematic phase stableagain. The resulting nematic domain aligns along the elec-tric field direction due to single-particle torques (see stage(V)), during which polarization takes place leading to stage(I), after which the cycle subsequently repeats itself.

The above-mentioned time-dependent “effective concen-tration” is to be understood as follows. For suspensions ofvery long and thin colloidal rods with hard-core interac-tions, Onsager showed that the location of isotropic-nematicbinodal- and spinodal concentrations depends on the dimen-sionless concentration (L/d) ϕ, with L the length of therod, d the core diameter, and ϕ = (π/4) d2 L ρ the volumefraction (the fraction of the total volume occupied by thecores of the rods), with ρ the number concentration of rods[26, 27]. In case the rods are charged, the same Onsagertheory can be employed, except that the core thickness islarger due to the additional repulsive electrostatic interac-tions. This defines an effective, time-dependent diameterdeff , and thereby an effective dimensionless concentration(L/deff ) ϕeff . The following expression for the effectivediameter can be derived [9, 10],

deff = κ−1[lnKQ + γE

], (1)

where κ−1 is the Debye length and γE = 0.5772 · · · isEuler’s constant, and where,

KQ = 2π exp{κ d}(1 + 12κ d

)2lB

κ L2

(N0 − Nc,0

)2,

with d, as before, the core diameter, lB is the Bjerrumlength, N0 is the number of immobile charges chemicallyattached to the surface of a rod, and Nc,0 the number ofcondensed ions of a rod in the absence of an electric field.Considerations concerning the quantification of an effec-tive diameter can also be found in Refs. [26–30]. Note thatthe total rod-surface charge that is relevant for the ion-concentrations within the diffuse double layer is equal to−e ( N0 − Nc,0

)(with e the elementary charge), which the

total immobile surface-charge plus the total charge withinthe layer of condensed ions.

In order to describe the dynamics of melting and formingof nematic domains, an equation of motion for the orien-tational order parameter tensor S should be derived. Thistensor is defined as the ensemble average of the dyadic prod-uct of the unit vector û that specifies the orientation of arod,

S(t) ≡ < û û > (t) . (2)The largest eigenvalue λ of the orientational order

parameter tensor quantifies the degree of alignment of therods. In the isotropic phase λ = 1/3 while in a per-fectly aligned state λ = 1. The concentration dependenceof the order parameter (without the electric field) is mostconveniently understood on the basis of the bifurcationdiagram given in Fig. 3. The values of the effective concen-tration (L/deff ) ϕeff where the isotropic-nematic binodalsand spinodals are located, according to Onsager [26, 27],are given on the lower axis of the bifurcation diagram.Here, C(−)bin and C

(+)bin are the lower and upper binodal con-

centrations. For initial overall concentrations in betweenthe two binodal concentrations, the equilibrium state is acoexistence between an isotropic and nematic phase. Forconcentrations below C(−)bin , the isotropic phase is stable,and above C(+)bin the nematic phase is stable. The spinodalconcentrations C(±)spin relate to the stability of the uniformisotropic and nematic state. The spinodal concentrationC

(+)spin marks the concentration where a uniform isotropic

state becomes unstable against the uniform nematic stateupon increasing the concentration. The vertical dashedarrow in Fig. 3 depicts the temporal increase of the orien-tational order parameter towards the nematic branch (thesolid line in blue). On lowering the concentration of a uni-form nematic below the spinodal concentration C(−)spin, thenematic becomes unstable against the isotropic state. Theorder parameter of the uniform nematic state now decreases


Fig. 3 A sketch of the bifurcation diagram for the isotropic-nematicphase transition, where the orientational order parameter λ is plottedagainst the effective concentration. The location of binodals, spinodals,and stability curves are given, which bound regions where the isotropicand nematic state are either unstable or meta-stable. The Onsager val-ues of the effective concentration for the location of binodals andspinodals are given in the lower part of the figure. The red, closed curveis a sketch of the limit cycle, indicating cyclic melting and forming ofnematic domains

towards 1/3. We thus find that the uniform nematic is unsta-ble for effective concentrations (L/deff ) ϕeff < C

(−)spin,

while the uniform isotropic state is meta-stable for concen-trations (L/deff ) ϕeff < C

(+)spin, as indicated in Fig. 3. The

red, closed curve depicts the limit cycle corresponding tothe alternating crossing of the lower binodal in the dynami-cal state under the action of an electric field. Hence, meltingof the nematic state occurs from the unstable state, throughspinodal decomposition. The isotropic state grows from themeta-stable state, through nucleation and growth. Nucle-ation times in the present case are probably small, as thereis some reminiscent alignment after melting.

In order to quantify the dynamics of melting and form-ing of nematic domains in the dynamical state, accordingto the above discussion, we need two equations of motionfor the orientational order parameter tensor (2): one equa-tion of motion for spinodal melting of the nematic statewhen (L/deff ) ϕeff < C

(−)bin , and one for nucleation and

growth of the nematic from an (near-) isotropic state when(L/deff ) ϕeff > C

(−)bin .

An equation of motion for spinodal decomposition ofa nematic state can be derived from the Smoluchowskiequation, through a Ginzburg-Landau expansion upto fourthorder in the orientational order parameter. Such a Ginzburg-Landau expansion can only be employed to describe thekinetics of an initially unstable state, and therefore describesmelting of the nematic, as discussed above. For frequenciesof the external field that are sufficiently large that during a

cycle of the field the configuration of the rods is essentiallyunchanged (for the fd-suspensions under consideration, thisfrequency is about 50–100 Hz), an equation for the ori-entational order parameter tensor can be derived from theSmoluchowski equation [9, 10]. This equation of motioncan be written as a sum of various contributions,

∂ S∂ τ

= id + Q,hc + twist + pol + torque , (3)with the dimensionless time variable,

τ = Dr t ,where Dr is the free rotational diffusion coefficient. Thevarious contributions are as follows. First of all, id is thecontribution from free diffusion,

id = 6[1

3Î − S

].

The second contribution Q,hc stems from interac-tions, unperturbed by the external field, with an effectivehard-core diameter that accounts for the above discussedelectrostatic interactions, as indicated by the subscript “Q”,

Q,hc = 92

L

deffϕeff {S · S − SS : S } .

The third contribution twist is the twist contribution,

twist = − 92

[5

4− ln 2

]1

κ deff

L

deffϕeff {S · S − S S : S } .

This contribution describes the effect, referred to as“the twist effect” [28, 29], that there is a preference fora non-parallel, twisted orientation of two rods due to theenergetically unfavorable overlap of diffuse double lay-ers in parallel orientation. The contribution pol is thecontribution due to interactions from polarization charges,

pol = 760

[KE

KQ

]2 1κ deff

L

deffϕeff h(�) E40

(S : Ê0Ê0

)F(S, Ê0) ,

while torque accounts for single-particle torques withwhich the external field acts on polarization charges,

torque = 180

L

lBF̃ I (�) E20 F(S, Ê0) ,

where E0 is the dimensionless external field strength (withβ = 1/kBT , and E0 the external field strength),E0 = β e L E0 , (4)and F(S, Ê0) is an abbreviation for,

F(S, Ê0) ≡ 32S · Ê0Ê0 + 3

2Ê0Ê0 · S + S · S · Ê0Ê0

+Ê0Ê0 · S · S − 2 S · Ê0Ê0 · S − 3 SS : Ê0Ê0,with Ê0 is the unit vector in the direction of the external


field. The frequency-dependent functions appearing in theabove equations are,

h(�) =[1

�

sin{�}+sinh{�}[cos{2�}+cosh{2�}]2

]2 [1+ 4

3�4+ 2

5�8

],

I (�) = 12�3

sinh{2�} − sin{2�}cosh{2�} + cos{2�} ,

where [31],

� =√

ω L2

8Deff,

is a dimensionless frequency, with ω the frequency of theexternal field, and with Deff the effective translational dif-fusion coefficient of the condensed ions, which is equalto,

Deff = D [ 1 + 2 κc a K(κ a) ] , (5)where D is the free translational diffusion coefficient ofcondensed ions. The second term within the square brack-ets accounts for the repulsive interactions between thecondensed ions, with,

κc = 2 lBd L

Nc ,

the inverse “condensate length.” The frequency-dependentfunctions h(�) and I (�) are essentially zero for � > 3, forwhich the polarization of the layer of condensed ions ceasesto occur. At that frequency, the H -phase becomes the stablephase, as discussed in “The electric phase/state diagram”section. The constants KE and F̃ are equal to [9, 10, 31],

KE = π exp{κ d}2 (1 + κ a)2 (1 + 2 kc a K(κ a))2

lB

κ L2N2c ,

F̃ = V (κca) [W(κca, κa) + 1 ]×

{2 [1 + κc a B(κ a)]2 − κc a [ 1 + κc a B(κ a) ]

},

with a = d/2 the hard-core radius, and where V and Wstand for,

V (κca) = κc a( 1 + κc a B(κ a))2

,

W(κca, κa) = − 2 κc a K(κ a)1 + 2 κc a K(κ a) ,

with (K0 is the modified Bessel function of the second kindof zeroth order),

K(κ a) ≡ 12π

∫ 2π

0dϕ K0

(κ a

√2 (1 − cosϕ)

),

B(κ a) ≡ 1π

∫ 2π

0dϕ cos{ϕ}K0

(κ a

√2(1−cosϕ)

).

An equation of motion for the nucleation and growthof a nematic domain from the isotropic state requires thesolution of the Smoluchowski equation including all ordersof the orientational order parameter, as well as the spatial

dependence of the orientational order parameter tensor. Thisis a problem that is probably too complicated to allow foran analytical treatment. We therefore adopt the exponentialgrowth that is found in simulations [32],

∂ S∂ τ

= S̄ − ST + pol + torque , (6)

where S̄ is the order parameter tensor of the nematic phasein equilibrium, without the electric field, and where T isthe time scale on which the internal orientational order ofdomains increases. The last term is responsible for the ori-entation of the nematic director towards the electric-fielddirection, which is an essential ingredient for the existenceof the dynamical state.

We note that both equations of motion (3, 6) neglect thefinite size of nematic domains. Spatial variation of the ori-entational order parameter is not considered here. This isprobably a reasonable approximation in view of the quitefuzzy interface between the nematic and isotropic regions,as evidenced from microscopy images. On the other hand,the dynamics of a given domain might be affected byadjacent domains.

An oscillatory state is only found when dissocia-tion/association of condensed ions is included. In Ref. [9],the following semi-empirical equation of motion for thenumber Nc of condensed ions is proposed,

d Nc

dτ= ± Cd

{N2c −N2lim

}( z2lBL [1+2 κc a K(κ a)]

)2

× E20(Ê0Ê0 :

[S(t) − αthr Î

] )I (�) , (7)

where Cd is the “effective dissociation constant.” Dissoci-ation occurs only when there is sufficient polarization ofthe layer of condensed ions along the long axis of the rod,which requires a minimum component of the orientationof a rod along the external field. The number αthr thusspecifies the minimum value of the orientation along thefield direction upon which dissociation can occur. When(S : Ê0Ê0) > αthr dissociation occurs (and the “− ” inEq. 7 applies), whenever the actual number of condensedions is larger than the limiting number of condensed ionsNlim, which is given by,

Nlim = αlim Nc, 0αlim + E20

(Ê0Ê0 : S

)I (�)

when, (S : Ê0Ê0) > αthr . (8)This is the limiting, time averaged number of condensed

ions in the stationary state when a rod with a fixed orienta-tion is subjected to the external field for a long time. When,on the other hand, (S : Ê0Ê0) < αthr (and the “+ ” applies),association of condensed ions occurs, and,

Nlim = Nc,0 , when, (S : Ê0Ê0) < αthr ,


where, as before, Nc,0 is the number of condensed ions inthe absence of the external field (note that Nlim ≤ Nc,0).

It takes some time before the ion-concentration withinthe bulk of the solvent is affected by the dissociation orassociation of condensed ions. Ions that dissociate from thecondensed layer must diffuse over distances of the order ofa rod length, in order to change the bulk ionic strength. Sim-ilarly, it takes some time for ions to diffuse from the bulkto the condensed layer as association occurs. The changeof the bulk concentration of ions at time t is thus approx-imately proportional to the number Nc = Nc, 0 − Nc ofreleased ions at an earlier time t − τdif , where τdif is thetime required for ions to diffuse over distances of the orderof a rod length. The time-dependent (inverse) Debye lengthat time t is therefore taken equal to,1

κ(t) =√

β e2[2 c0 + ρ̄ Nc(t − τdif )

]

�, (9)

where ρ̄ is the number density of rods, and c0 is the ambientionic strength,

c0 = 12

∑

α

cα z2α ,

where the summation ranges over all species of ions inbulk solution, and cα is the number concentration of speciesα (with valency zα), in the absence of the electric field.Therefore, the effective diameter in Eq. 1 becomes time-dependent,

deff (t)

d= 1

κ(t) d

[ln{KQ(κ ≡ κ(t))} + γE

],

where the interaction strength KQ is evaluated with aninverse Debye length equal to κ(t). This time dependencequantifies the variation of the effective concentration upondissociation/association of condensed ions, which is at theorigin of the dynamical state.

Binodal and spinodal concentrations

As a last step, we have to specify the lower binodal con-centration C(−)bin in order to decide whether Eq. 3 or 6should be used to describe the dynamics of the orderparameter. It turns out that the external electric field has avery minor effect on the concentration where the isotropic-nematic phase transition line is located. The location ofthe lower binodal concentration, including the twist effect,can therefore simply be obtained from the correspondingthe dimensionless concentration (L/d) ϕ = 3.290 · · · forhard-spheres, as predicted by Onsager [26, 27]. This relieson the fact that in the equation of motion (3), the charge-charge interactions as well as the twist effect have the same

1Note that there is a misprint in Eq. (48) in Ref.[9]

functional dependence on the orientational order parametertensor. The lower binodal concentration is thus set by,[

L

dϕ

]

Onsager

= 3.290 · · ·

= Ldeff

ϕeff

{1−

[5

4−ln 2

]1

κ deff

}. (10)

The same procedure can be used to obtain the locationof the upper binodal and two spinodal concentrations. TheOnsager value of (L/d)ϕ for the upper binodal is 4.191, andfor the lower and upper spinodal 3.556 and 4, respectively.The location of binodals and spinodals depend both on theDebye screening length, as well as the effective number ofsurface charges N0 − Nc,0 (as before, N0 is the total num-ber of immobile bare charges on a single rod, and Nc,0 isthe number of condensed ions in the absence of the externalelectric field). The plots in Fig. 4 give the binodal and spin-odal concentrations as a function of the Debye length forseveral values of the effective number of charges.

The predicted isotropic-nematic coexistence regions forN0 − Nc,0 = 500 (a number that is found from fits tothe location of the N∗-to-D transition line, as discussedin the next section) are 1.68 < [f d] < 2.14 mg/ml and0.66 < [f d] < 0.84 mg/ml for buffer concentrations of

Fig. 4 The lower a and upper b binodal and spinodal concentrations,as a function of the Debye length for several values of the number ofeffective charges N0−Nc,0. Dashed lines are spinodals and solid linesare binodals


0.16 and 0.032 mM, respectively. The experimental binodalconcentrations are 1.5 ± 0.2 and 3.4 ± 0.5 mg/ml for the0.16 mM buffer, and 0.8 ± 0.2 and 1.5 ± 0.4 mg/ml forthe 0.032 mM buffer. The lower binodal concentrations arein very good agreement, while the experimental upper bin-odal concentrations are slightly higher than the theoreticallypredicted values.

Comparison to experiments

In a comparison to experiments, the volume fraction ϕ mustbe calculated from the weight concentration of fd-virus par-ticles. From the length, thickness, and molecular weight ofa fd-virus particle, the relation between the hard-core vol-ume fraction and the weight concentration [f d] in units ofmg/ml is found to be: ϕ = 0.0011 × [f d]. In addition,the dimensionless electric field amplitude (4) is calculatedfrom E0 = 0.096 × 0.035 × E0 [V/mm], where the fac-tor 0.096 accounts for the reduction of the applied fieldstrength E0 due to the dielectric polarization of the ITO-water interface. The data that will be shown are corrected forthe decrease of the electric field strength in the bulk of thesuspensions due to the partial buildup of electric double lay-ers at the electrodes, also known as “electrode polarization”[8]. Electrode polarization is essentially absent for frequen-cies ν = ω/2π larger than about 60 and 120 Hz for thebuffer concentrations 0.16 and 0.032 mM, respectively.

There are a number of parameters in the dissocia-tion/association model of which the numerical values haveto specified. Some of these parameters can be specifiedindependently, while others have to be adjusted to fitexperimental data. Table 1 shows the parameters that aredetermined independently for the two buffer concentra-tions of 0.032 and 0.16 mM. The pH and ionic strengthc0 are calculated from the buffer concentrations, includ-ing the contribution from carbon dioxide that dissolvesfrom the air, as discussed in Ref. [25]. The Debye lengthκ−1 (at 25 ◦C) is calculated from the ionic strength:κ−1 [nm] = 0.304/√c0 [M]. The number N0 of immo-bile charges on a fd-virus particle depends on the pHthrough the dissociation/association of surface groups onthe coat proteins, which can be obtained from Ref. [24]. Theorientational order parameter within the full nematic state ismeasured as a function of the buffer concentration in Ref.

Table 1 Independent numerical parameter values for the two bufferconcentrations

Conc. (mM) pH c0 (mM) κ−1 (nm) N0 S̄

0.032 5.8 0.032 53.7 7800 0.93

0.16 6.9 0.143 25.4 8700 0.93

[25], down to a buffer concentration of 0.16 mM. Extrap-olation of the scalar order parameter to the upper-binodalconcentration gives the scalar orientational order parame-ter that is relevant in Eq. 6. The scalar order parameter isessentially the same for the two buffer concentrations, andis equal to 0.93 ± 0.02 (a plot of the scalar order parameterversus the buffer concentration from the data in Ref. [25]shows that the difference of the order parameter for the twobuffer concentrations is less than 0.01). The order parametertensor S̄ in Eq. 6 is thus equal to 0.93 × Ê0Ê0.

The location of the N∗-to-D transition line

The location of the N∗-to-D transition line is indepen-dent of the numerical values of the dynamical parametersT (which sets the nematic growth rate from the meta-stable state), Cd (the effective dissociation constant), τdif(the time condensed ions need to diffusive into the bulk),and αthr (that sets the degree of alignment beyond whichdissociation occurs). The three remaining parameters thatdetermine the location of N∗-to-D transition lines are (i)the number Nc,0 of condensed ions in equilibrium, with-out the electric field, (ii) αmin in Eq. 8, which determinesthe remaining limiting number of condensed ions on a rodwith a fixed orientation as a function of the electric fieldamplitude and frequency, and (iii) the diffusion coefficientof the condensed ions in Eq. 5. The experimental phase-state diagrams for the buffer concentrations [b] = 0.16 and0.032 mM are given in Figs. 5 and 6, respectively. Thesolid red lines are the theoretical N∗-to-D transition linesfor the values of Nc,0, αlim, and D/D0 as given in Table 2(with D0 = 2.0 × 10−9 m2/s the typical value of diffusioncoefficients of ions in solution). The reduction of the barediffusion coefficient is due to additional friction of ions withthe core of the rod, which is independent of ion concen-tration. The grey areas in these plots indicate the range ofvalidity of the present theory: the frequency should be largerthan about 50–100 Hz due to the assumed constant orienta-tion of the rods during a cycle of the external electric field,and should not be larger than about 300 Hz where polar-ization is essentially absent and hydrodynamic interactionsthrough electro-osmotic flow become significant.

In particular for the lower ionic strength, there are twocounter balancing effects on the location of the N∗-to-D transition line as the fd-concentration is increased. Fora given frequency, the transition shifts to a higher fieldamplitude on increasing the fd-concentration, since theincrease of the ionic strength due to the dissociation ofcondensed ions must be larger, as the concentration is fur-ther away from the lower-binodal concentration. On theother hand, the required number of released condensedions per individual rod becomes less on increasing the fd-concentration, as there are more rods per unit volume at


Fig. 5 The phase/state diagrams for the buffer concentration of0.16 mM for several fd-concentrations, as indicated in the figures. Thered lines are the calculated N∗-to-D transition lines, the thin blacklines are guides-to-the-eye, and the vertical red line in the secondfigure indicates the location of the critical point. The red and blueareas are outside the validity of the theory

a higher concentration, which lowers the field amplitudewhere the transition occurs. As can be seen from Figs. 5 and6, the field amplitude where the transition occurs increaseswith increasing fd-concentration. The above-mentioned for-mer mechanism wins over the latter, which is reproduced bythe theory.

Since the bare charge for the two different values of thepH for the two buffer concentrations of 0.032 and 0.16 mMis equal to 7800 and 8700, respectively [24], the equilibriumnumber of condensed ions, in the absence of the externalelectric field, is found to be equal to N0 − Nc,0 = 500 forboth buffer concentrations (see Table 2). The correspondingequilibrium line-charge density is about a factor of two lessthan the classical predicted value from condensation the-ory of e/lB ≈ 1100 (where e is the elementary charge andlB = 0.75 nm is the Bjerrum length) [33, 34]. The parameterαmin, on the contrary, is a strong function of the buffer con-centration, as can be seen from Table 2. This implies thatthe limiting number of condensed ions on a rod with a fixedorientation under the action of an electric field stronglydepends on the bulk ionic strength (see Eq. 8), in contrast tothe number of condensed ions in the absence of the electricfield.

Table 2 Fitted numerical parameter values for the two bufferconcentrations

Conc. (mM) Nc,0 αlim D/D0 αthr T Cd τdif

0.032 7300 1.0 × 10−5 1/350 − − − −0.16 8200 2.2 × 10−4 1/350 1/2 65 105 0.039 s

The dynamical behavior on approach of the N∗-to-Dtransition line

The remaining parameters that affect the dynamicalbehaviour are (i) the growth time T in Eq. 6 of the isotropicmeta-stable state towards the nematic state, (ii) the effectivedissociation constant Cd of condensed ions in Eq. 7, (iii) the

Fig. 6 The same as in Fig. 5, but now for a buffer concentration of0.032 mM


time τdif that determines how long ions need to diffuse intothe bulk in order to affect the overall ionic strength, and (iv)the threshold value αthr that sets the degree of alignmentalong the electric field beyond which dissociation occur.The critical exponents that quantify the divergence of thecharacteristic melting/forming time τ of nematic domainson approach of the N∗-to-D transition line turn out to bequite independent of the precise values of Cd , τdif , andαthr . Reasonable agreement with experiments is found forthe parameters given in Table 2. The tabulated time for τdifcorresponds to the time required for ions in solution (witha diffusion coefficient of 2 × 10−9 m2/s) to diffuse over adistance of ten rod lengths, which seems a reasonable dis-tance in order that dissociated condensed ions affect thebulk ionic strength. The dimensionless, numerical value ofT was adjusted such that the limiting characteristic timeof approximately 1.5 s far away from the transition line isreproduced by the theory. The tabulated value of T corre-sponds to a real time of 65/Dr = 3.3 s for the growth rate ofa nematic domain, which is of the order of magnitude seenexperimentally in fd-virus suspensions.

Figure 7 gives a comparison between the present the-ory with the above mentioned numerical values of theparameters and the experiments from Ref. [11], where thedivergence of the characteristic time on approach of theN∗-to-D transition line has been measured. Figure 7a is aschematic overview of the experimentally found behaviourof the characteristic time [11]: on approach of the crit-ical point by lowering the field-amplitude at the criticalfrequency a power-law divergence is found with a criticalexponent of μE = 1.39 ± 0.18 (see the data points in Fig.7b), while an exponent of μν = 0.65 ± 0.15 is found onapproach of the critical point on increasing the frequency(see Fig. 7c), and a logarithmic divergence is found for

an off-critical approach of the N∗-to-D transition line (asshown in Fig. 7d). The critical exponent μEτ is defined as,

τ − τb ∼(

E

Ec− 1

)−μEτ, (11)

for the critical divergence of the characteristic time τ onlowering the field amplitude at the critical frequency νc (Fig.7b), and similarly,

τ − τb ∼(1 − ν

νc

)−μντ, (12)

on increasing the frequency at the critical field amplitudeEc (Fig. 7c). Here, τb is the non-critical background valueof the characteristic time of about 1.5 s far away from thecritical point. The theory also predicts a power-law diver-gence, with exponents equal to μEτ = 0.89 (instead of theexperimental value 1.39 ± 0.18), and μντ = 0.91 (insteadof 0.65 ± 0.15). It was not possible to find parameters suchthat the experimentally found critical exponents are accu-rately reproduced by the theory. The differences between thetheoretical predictions and the experimental results for thevalues of the critical exponents is most probably due to thefact that interactions between rods due to electro-osmoticflow come into play. A power-law divergence is also pre-dicted by theory for off-critical approaches of the D-to-N∗transition line, with a similar exponent 0.98 as for the crit-ical approaches. The experiments reveal, on the contrary, alogarithmic divergence,

τ − τb ∼ 10log(

E

Etrans− 1

), (13)

where Etrans is the location of the transition line for a fixedfrequency of 150 Hz. This discrepancy might be due to theinteractions between neighboring nematic domains.

Fig. 7 The divergence of the characteristic time for melting and form-ing of nematic domains in the dynamical state. a A schematic of theexperimentally found divergence of the characteristic time on approachof the N∗-to-D transition line: a power law divergence is found onapproach of the critical point (with an exponent of 1.39 ± 0.18 onlowering the amplitude, and 0.65± 0.15 on increasing the frequency),while a logarithmic divergence is found for an off-critical approach.b The divergence as a function of the electric-field amplitude at the

critical frequency νc, and c as a function of the frequency at the criti-cal field amplitude Ec. d The divergence for the off-critical approachat a fixed frequency of 150 Hz, where Etrans is field strength at thetransition line. The data points are taken from Ref. [11], for an fd-concentration of 2.0 mg/ml, with a buffer concentration of 0.16 mM.The dashed lines are the predictions by theory. The numbers in thefigures are values for the critical exponents


In order to reproduce the experimentally found criti-cal exponents accurately, the theory should be improved toinclude inter-rod interactions due to field-induced electro-osmotic flow (for the critical approach), the existence ofdomain boundaries, and interactions between domains (forthe off-critical approach).

Summary and conclusions

The phase/state behavior of concentrated suspensions ofhighly charged rods (fd-virus particles) have been investi-gated experimentally in Ref. [8], where a dynamical state(the D-state) is identified in which nematic domains per-sistently melt and form. The time scale on which meltingand forming of the domains occurs is about 1.5 s faraway from the transition line where the quiescent chiral-nematic/isotropic coexistent state (the N∗-phase) trans-forms to the dynamical state. On approach of the N∗-to-Dtransition line, the characteristic time for domain meltingand forming is found experimentally to diverge [11]. Theorigin of the dynamical state is attributed to cyclic disso-ciation and association of condensed ions which leads toan alternating increase and decrease of the Debye screeninglength [9] (with a corrigendum in Ref. [10]). This in turnleads to an alternating, time-dependent effective thicknessof the rods, such that the lower isotropic-nematic binodalis alternatingly crossed.This leads to the observed cyclicmelting and forming of nematic domains.

There are three coupled equations of motion involvedin the theory for the dynamical state: (A) an equation ofmotion for melting of the unstable nematic, (B) for the meta-stable growth of the (near-) isotropic state, and (C) for thenumber of condensed ions. The a priori unknown variableswhich enter these equations of motion are the number ofcondensed ions [in (A), (B), and (C)], the free diffusioncoefficient of the condensed ions [in (A)], the time constantfor the growth of the nematic phase [in (B)], the limitingnumber of condensed ions after a rod with fixed orientationis subjected for a long time to the oscillating field, the effec-tive condensed-ion dissociation constant, and the thresholdvalue for the orientation of a rod along the external fieldbeyond which dissociation occurs [in (C)]. It turns out thatthe numerical results are relatively insensitive to the effec-tive condensed-ion dissociation constant and the thresholdvalue of orientation beyond which dissociation is possible.Independent of the three equations of motion, the time thatis required for ions to diffuse from (or to) the layer of con-densed ions into (or from) the bulk solution is an additionalvariable.

A few approximations are made in the theory: (i) dur-ing a cycle of the external electric field, the configurationof the rods is assumed to remain essentially unchanged,

(ii) the finite size of domains is not considered, that is,the theory applies to a homogeneous system, (iii) the rod-rod interactions through field-induced electro-osmotic floware neglected, and (iv) a semi-empirical expression for thefield-induced dissociation/association of condensed ions isproposed. The first assumption (i) limits the validity of thetheory to frequencies larger than approximately 100 Hz forthe fd-virus particles under consideration, while assump-tion (iii) limits the validity to frequencies lower than thecritical frequency. Above the critical frequency polariza-tion is essentially absent. The homeotropic H -phase thatis formed at these higher frequencies is stabilized dueto hydrodynamic interactions mediated by field-inducedelectro-osmotic flow.

Experimental phase/state diagrams for various fd-concentrations and ionic strengths are presented, and thelocation of the D-to-N∗ transition line is compared with theabove mentioned theory [9, 10]. In view of the approximatenature of the theory, the location of the D-to-N∗ transitionlines in the field-amplitude versus frequency plane are rea-sonably well reproduced within the frequency range wherethe theory applies. The location of the transition lines asa function of fd-concentration and the buffer concentration(or ionic strength) are captured by the theory. Especiallythe subtle dependence of the location of the transition lineson the fd-concentration is captured, where there are twocounter acting effects: on increasing the fd-concentration,the location of the transition line shifts to higher fieldstrengths as the difference in concentration with the lower-binodal concentration increases, but at the same time thefield strength tends to decrease because the number of ionsthat should be released per rod becomes less as more rodsare present. The critical divergence of the characteristic timefor melting and forming of domains is also found by the the-ory. However, the theoretical values for critical exponents onapproach of the critical point are somewhat different fromthose found experimentally. Moreover, for an off-criticalapproach of the transition line, a power-law divergenceis predicted by the theory while a logarithmic divergenceis observed experimentally. The former difference is mostprobably due to the increasing importance of interactionsdue to electro-osmotic flow, while the latter might be due tothe interactions between domains, which are not included inthe theory.

The most significant challenge to improve the the-ory is a quantitative description of field-induced dissocia-tion/association of condensed ions. As mentioned above, thesemi-empirical equation of motion for the number of con-densed ions contains as many as three a priori unknownvariables. The solution of the non-linear Poisson-Boltzmannequation for rod-like colloids in an oscillatory external elec-tric field would not only more precisely quantify an equationof motion for the number of condensed ions but also reduce


the number of unknown variables. Secondly, the presenttheory assumes a homogeneous nematic, and thus neglectsthe finite size of the nematic domains. The analysis ofthe Smoluchowski equation could possibly be extended toinclude spatial inhomogeneities of the orientational orderparameter.

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Kyongok Kang had a PhD inPhysics at Kent State Univer-sity, and a postdoctoral fellow-ship in the Institute of Com-plex Systems at the ResearchCenter in Juelich, Germany.Since 2007, she is a per-manent scientist. She inves-tigates both non-equilibriumphase transitions and equi-librium orientation dynamicsof systems of charged chi-ral rods. She developed instru-mentation to investigate non-equilibrium phases/states, byimage-time correlation, small

angle dynamic light scattering, and electric birefringence.

Jan Dhont was appointed asan associated professor at thevan ’t Hoff laboratory, UtrechtUniversity, in the group ofProf. H.N.W. Lekkerkerkerin 1987. In 2000, he wasappointed as a director at theResearch Center of Juelich,and in 2001 as a full profes-sor at the Heinrich-Heine-Uni-versity in Duesseldorf. Hiscurrent research interests areconcerned with the non-equi-librium behavior of soft-mattersystems in external fields, suchas flow, confinement, and elec-tric fields.

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An electric-field induced dynamical state in dispersions of highly charged colloidal rods: comparison of experiment and theoryAbstractIntroductionThe electric phase/state diagramThe ion-dissociation/association modelBinodal and spinodal concentrationsComparison to experimentsThe location of the N*-to-D transition lineThe dynamical behavior on approach of the N*-to-D transition line

Summary and conclusionsOpen Access References

an electric-field induced dynamical state in dispersions of highly ... · frequency electric fields...

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