Charged fibrous viruses (fd) in external electric fields: dynamics and orientational order

We recently found a number of phases and dynamical states that are induced in a concentrated suspension of charged, colloidal rods (fd-viruses) by an alternating external electric field (Kang and Dhont 2008 Eur. Phys. Lett. 84 14005; 2010 Soft Matter 6 273). The various phases and dynamical states are the result of interactions between the charged rods through their polarized electric double layers, polarized layers of condensed ions and/or electro-osmotic flow. At a relatively high frequency, a homogeneous, homeotropically aligned phase is induced (the H-phase). We present a dynamic light-scattering study of the microscopic dynamics of the rods, varying the frequency and field amplitudes along different pathways within this phase. Scattering experiments are performed at very small scattering angles with a home-made vertically mounted dynamic light-scattering setup, where Brownian motion perpendicular to the direction of alignment is probed. The orientational order is measured by means of birefringence experiments. The remarkable finding is that relaxation times and the degree of alignment are independent of the frequency and the amplitude of the applied electric field throughout the entire H-phase. Only within a small region in the neighborhood of the transition line, where the H-phase transforms to an inhomogeneous chiral-nematic phase, is there a frequency and amplitude dependence of relaxation times, which are shown to be the result of the appearance of transient, pre-transitional domains. We also recently identified a non-equilibrium critical point, where a time- and length-scale connected to a dynamical state are shown to diverge (Kang and Dhont 2009 Eur. Phys. J. E 30 333). Approaching this critical point from the side of the H-phase, we find that the light-scattering correlation functions develop a very slowly decaying mode, the origin of which requires further investigation.


Introduction
The dynamical properties of charged colloidal rods in an external electric field are generally affected by interactions between the rods through the field-induced polarization of their double layer and/or their core. At high frequencies (in the MHz range) and relatively high field amplitudes, the cores of the colloids are dielectrically polarized, giving rise to dipolar interactions between the colloids, which can lead to structure formation, such as strings and sheets [1][2][3]. At such high frequencies, electric-double layers around the charged rods (fd-virus particle with a length of L = 880 nm, a bare diameter of D = 6.6 nm, a persistence length at high salt concentration of P = 2200 nm) are not polarized due to the finite mobility of the ions that constitute the double layer. Double-layer polarization is important only for relatively low frequencies. Besides interactions between rods through their polarized double layers, there are two additional field-induced forces that may play a role, which are related to condensed ions and electro-osmotic flow. The origin of the field-induced forces at low frequencies is sketched in figure 1. In the absence of an external field, the electric double layer is cylindrically symmetric, as sketched in figure 1(a). On applying an electric field, the double layer of a rod becomes asymmetric, the layer of condensed ions is polarized and electro-osmotic flow is induced due to electrical body forces on charged volume elements within the double layer, as sketched in figure 1(b). In concentrated suspensions, rods interact with each other due to the polarized double layer, the polarized layer of condensed ions, and hydrodynamically through electroosmotic flow, as sketched in figure 1(c). So far, very little research has been done on phase transitions that can be induced at low frequencies as a result of these field-induced interactions. As far as we know, no research has so far been done on the microscopic dynamics of charged rods in phases that are induced by low-frequency electric fields, which might shed light on the relative importance of the three above-mentioned types of interactions.
We recently found that various phases and dynamical states can be induced in suspensions of charged colloidal rods (fd-viruses), where the concentration of fd-viruses is such that there is an isotropic-nematic coexistence in the absence of the field [4,5]. One of the induced phases, which is of interest in the present paper, is a homogeneous phase with homeotropic alignment of the rods. We referred to this phase as the H-phase. This phase is found for frequencies larger than about 1 kHz, where there is only a weak polarization of the double layer [6], so that the two other forces discussed above might be dominant. Interactions due to electro-osmotic flow In a lowfrequency electric field, the double layer is polarized, the layer of condensed ions is polarized, and electro-osmotic flow is induced. A small charged volume element is shown on which an electric body force acts that leads to electroosmotic flow. (c) Two rods that interact through their polarized double layer and layer of condensed ions, as well as hydrodynamically through the induced electro-osmotic flow.
have been investigated by simulations in [7] for two rods with thin double layers. Interactions through the polarization of the layers of condensed ions have very recently been analyzed, for zero frequency, by Manning [8,9]. This analysis suggests that the change in preferred orientation found for fd-suspensions in [10,11] is due to polarization of the layer of condensed ions (although the experiments have been performed with alternating fields). It is not yet clear which of the possible interactions are responsible for the stabilization of the H-phase. It is the purpose of the present paper to characterize the fd-virus dynamics and orientational order in the H-phase, in order to sustain possible future theoretical predictions on the interactions between charged colloidal rods in low-frequency external fields. This paper is organized as follows. We start with a section on the experimental details. In section 2.1, the characteristics of fd-virus particles and their suspensions are presented. Section 2.2 introduces the electrical cell and the dynamic light-scattering setup. Section 2.3 is a summary of the phase/state behavior of fd-virus suspensions in an electric field with varying field amplitude and frequency. The orientational order, along different pathways within the Hphase, is discussed in section 3. The pathways include regions in the phase/state diagram that are in the vicinity of transition lines. In section 4, the microscopic dynamics, as probed with dynamic light scattering, along the same pathways are discussed. Finally, conclusions are given in section 5.

Experimental details
In this section, the properties of fd-virus particle suspensions and their preparation are discussed, we describe the technical details of the electrical sample cell and the small angle dynamic lightscattering setup, and we summarize the phases and dynamical states that can be induced in fd suspensions by means of an electric field.

Charged fibrous virus (fd) suspensions
Bacteriophage fd is a rod-like macromolecule with a length of L = 880 nm, a bare diameter of D = 6.6 nm, a persistence length at high salt concentration of P = 2200 nm and a molecular weight of M = 1.64 × 10 7 g mol −1 . The hydrodynamic properties (and structure) of fd-virus particles are discussed in [12].
The fd-viruses are grown and purified following standard biological protocols [13] using the XL1blue strain of Escherichia coli as the host bacteria. The virus particles are purified by repeated centrifugation (10 5 g for 5 h) and redispersed in TRIS/HCl buffers with varying concentration, depending on the desired final ionic strength. From the dimensions and molecular weight of an fd-virus particle, the volume fraction ϕ of an fd-virus suspension is found to be related to the weight concentration c, as ϕ = 1.10 × 10 −3 c (mg ml −1 ). The overlap concentration of fd-virus suspensions is equal to 0.076 mg ml −1 .
We prepare fd-virus suspensions at a given ionic strength by osmotic equilibration with a TRIS/HCl buffer. The same buffer is used to dilute the suspensions (by typically 20%) to the desired fd concentration. The ionic strength of the osmotic reservoirs is the ionic strength from which the electrostatic Debye-Hückel screening length is calculated. Furthermore, osmotic equilibration ensures that the pH of the suspensions is equal to those of the osmotic reservoir. The TRIS/HCl buffers are prepared by adding a small volume of HCl with a concentration of 1 M to a 20.0 mM TRIS solution till a pH of 8.2 is attained. This buffer is then diluted with deionized water for the preparation of the lower TRIS/HCl buffer concentrations. For buffer concentrations less than about 10 mM, both the ionic strength and the pH are affected by carbon dioxide that dissolves from the air, which in turn affects the surface charges of the fd-virus particles. A detailed account of the effect of dissolved carbon dioxide on the ionic strength and pH as a function of the buffer concentration can be found in [14].
The charge distribution on fd-virus particles has been interpreted on the basis of different models for an fd-virus particle from titration curves. The best fit to titration curves is found when assuming that the charges are located only on the hydrophilic outer region of adsorbed coat proteins and not on the DNA strand of fd itself [15]. The iso-electric point of native fd is equal to pH = 4. In the present study, we assume that the pH is always larger than 6.5, where the fd particles are negatively charged. The charge of an fd-virus particle varies in the range −10 000 Q fd −8500 in the pH range 6.5 pH 8.2 [15]. The buffer capacity of the lowionic-strength TRIS/HCl buffer that we used is still sufficient to keep the pH above 6.5, despite dissolving carbon dioxide from the air [14].

5
In a previous study on the response of fd-virus suspensions to electric fields, it was found that no phase transition could be induced for concentrations outside the isotropic-nematic coexistence region [5]. The fd-concentrations are therefore chosen within the biphasic isotropic-nematic coexistence region, in the absence of an electric field. The buffer concentration is 0.16 mM. The Debye screening length for water at 25 • C, assuming that κ −1 Q (nm) = 0.304/ √ I (M) (where I is the ionic strength corrected for dissolving carbon dioxide [14]), is found to be equal to 27 nm = 4.1D.

The electrical cell and small angle vertical dynamic light-scattering setup
A home-made optically transparent electrical cell is used to facilitate dynamic light scattering, imaging through a microscope and birefringence measurements. The sample is contained between two horizontal parallel ITO glass plates, with a spacer of 1.4 mm thickness. The conductive ITO coatings are connected to a function generator. The dynamical light-scattering setup is vertically mounted, where the direction of incidence of a 5 mW He-Ne laser is along the electric field, vertical to the two plates. Scattered light is collected through a fiber that is connected to an avalanche photo diode detector. The setup is used in VV mode. The VHscattered intensities from fd-virus suspensions are too low to measure reliable correlation functions.
Due to refraction of scattered light from the sample-glass and glass-air interfaces, the true scattering angle s is not equal to the goniometer angle g (see figure 2(a)). The ratio of the true scattering wave vector q to the apparent scattering wave vector q g as calculated from the goniometer angle is given in figure 2(b) as a function of the goniometer angle. As can be seen from this plot, the scattering wave vector is significantly affected by interface refraction only for scattering angles larger than about 20 • .
At small scattering angles, the scattering vector is essentially perpendicular to the direction of the external electric field, as sketched in figure 2(c). For larger scattering angles, there is a small contribution to the scattering wave vector that is parallel to the external field. The relative values of the perpendicular q ⊥ and parallel q components to the total scattering wave vector are plotted in figure 2(d). As can be seen, for up to about a 15 • scattering angle, only the dynamics related to motion perpendicular to the electric field are probed, whereas for larger scattering angles also a small contribution from motion parallel to the electric field is involved.
An essential part of the vertical dynamical light-scattering setup involves a lens (with focal length 75 mm) that ensures that only light that is scattered from the bulk part of the sample is probed. Figures 3(a) and (b) show the normalized intensity correlation functions for a dilute silica-sphere suspension as measured without and with this lens, respectively. As can be seen, the long-time behavior of the correlation functions (also shown in the insets) is quite different. Without the lens, scattering by particles near the wall leads to spurious slow relaxation. The dynamics of particles near the wall are severely slowed down by hydrodynamic interactions with the wall. The single-exponential behavior of the correlation function that is expected for dilute suspensions of spheres in bulk is indeed found for the setup with the lens. This can be seen from the solid line in figure 3(b), which is a fit to a single exponential decay.

Electric phase/state diagrams of fd-suspensions
In two earlier publications [4,5], a number of driven phases and dynamical states in fdvirus suspensions were found, which are induced by means of an external electric field. The to the N -phase, which is a coexistence between nematic domains and chiral-nematic domains. The chiral-nematic regions extend over much larger regions as compared to the non-chiral nematic domains. Two images for the N -phase are shown in figure 4, one at low frequency (on the left side) and the other at high frequency (on the right side). As can be seen, the pitch is markedly larger at high frequencies, near the H-to-N transition line, as compared to the pitch at low frequency. On further increasing the field amplitude, the N-domains disconnect from each other and become significantly smaller within a small field amplitude range, which we refer to as the N D -phase, where the subscript 'D' stands for 'disconnected'. The difference between the typical morphologies of the N -and N D -phases is shown in the two left images in figure 4. The chiral texture melts at a certain higher field amplitude, and at the same time the N-domains melt and reform. This dynamical state is referred to as the D s -state, a snapshot of which is given in the top of figure 4. The subscript 's' stands for 'slow'. On further increasing the field amplitude, the dynamics of melting and formation become faster, referred to as the D f -state, where the subscript 'f' stands for 'fast'.
Of particular interest in the present study is the uniform phase at relatively high frequencies, where the fd-virus particles are homeotropically aligned, parallel to the external field. This homogeneous phase is referred to as the H-phase (where 'H' stands for 'homeotropic'). The corresponding morphology is given in the image in the right side of figure 4. In the present paper, we study the microscopic dynamics and the degree of alignment of the fd-rods in this H-phase, varying both the frequency and the field amplitude.
The solid lines in the phase/state diagrams in figure 4 refer to sharp phase/statetransition lines, whereas the dotted lines refer to gradual transitions (the N -to-N D and D s -to-D f transitions).
There is a special point where, in the phase/state diagram in figure 4, various phase/statetransition lines meet. It resembles a critical point in equilibrium phase diagrams. On approaching this point from the side of the dynamical state, like for equilibrium critical points, a divergent length-and timescale can be identified. As shown in [16], the size of the melting and forming nematic domains in the dynamical state, and the timescale on which melting/forming evolves, both diverge on approaching this non-equilibrium critical point. As will be seen in the present study, there are also signatures for critical behavior on approaching the critical point from the side of the H-phase.

Electric birefringence
Electric birefringence measurements are performed for an fd concentration of 2.0 mg ml −1 along various pathways within the H-phase, where either the field amplitude is varied at fixed 9 frequency, or the frequency is varied at a fixed field amplitude, in the electric phase/state diagram. The following pathways have been probed: • pathway (i): varying the field amplitude at a fixed frequency of ν = 2 kHz; • pathway (ii): varying the frequency at a fixed field amplitude of E = 2.9 V mm −1 ; • pathway (iii): varying the frequency at a fixed field amplitude of E = 3.5 V mm −1 . The field amplitude for pathway (iii) is the critical field amplitude, that is, the field amplitude where the non-equilibrium point in the phase/state diagram in figure 4 is located [16]. On lowering the frequency along pathway (iii), the critical point is thus approached. Note that the critical field amplitude in the phase diagram in figure 4 is a bit higher than 3.5 V mm −1 . As discussed in [5], this is related to the finite measuring time for the data points in the phase diagram in combination with critical slowing down of dynamics near the critical point. Accurate values for the critical amplitude and frequency have been determined in [16]. On lowering the frequency along pathway (ii), the off-critical part of the H-to-N transition line is approached. The H-to-N transition line is approached along pathway (i) on lowering the field amplitude.
The He-Ne laser beam of the vertically mounted, null-optical-train birefringence setup makes an angle of 20 • with the direction of the external electric field. Since in the H-phase the rods are aligned along the external field, no birefringence would be detected in the case of a normal incidence onto the sample cell. A description of the experimental setup and a detailed analysis of the interpretation of such birefringence measurements are given in [5]. In a null-optical-train birefringence setup, the transmitted intensity is measured as a function of the orientation of a polarizer. The analyzer angle where the minimum transmitted intensity is found, relative to that of the pure buffer, is equal to half the birefringence retardation. The retardation is in turn proportional to the orientational order parameter, where the proportionality constant depends on the angle of incidence of the laser beam [5].
The left column of plots in figures 5(a)-(c) gives birefringence data for the three pathways (i)-(iii), as defined above, respectively. These plots are transmitted intensities (relative to the maximum transmitted intensity) as a function of the analyzer angle α in degrees. The filled circles are data for the pure buffer and the open circles are data for the fd suspension (curves are consecutively vertically shifted by 0.0005 for clarity). The right column in figure 5 shows the analyzer angle where the minimum intensity is found, relative to the pure buffer, as a function of the distance from the transition lines: E for pathway (i) and ν for pathways (ii) and (iii). As can be seen, the analyzer angle where the minimum transmitted intensity is found is surprisingly found to be equal to 1.4 • (to within an experimental error of 0.3 • ), independent of the field amplitude and frequency. The corresponding value for the orientational order parameter is 0.38 ± 0.08 and is the same throughout the H-phase.
The independence of the degree of alignment from both the field amplitude and the frequency could be explained as follows. The ions within the diffuse double layer (about 15% of the total charge) and the condensed ions (about 85%) have a finite diffusivity. At sufficiently high frequencies, therefore, the polarization of the double layer and the layer of condensed ions become less important. For frequencies above a few kHz range, the ions in the diffuse double layer and in the layer of condensed ions cannot follow the external field anymore. The establishment of electro-osmotic flow, on the contrary, is very fast (on the ns time scale), so that electro-osmotic flow remains active also at high frequencies. Since polarization would lead to a field-amplitude-dependent response, we anticipate that electro-osmotic flow gives rise to the stability of the H-phase. As shown in [5], a significantly aligned state cannot be induced at lower concentrations in the isotropic state. This implies that the alignment in the H-phase is due to rod-rod interactions induced by the external field and not due to single-particle alignment.
Based on the above measurements of electric birefringence, we can conclude that the highfrequency uniformly aligned homeotropic H-phase is governed by different interactions among fd-virus particles, which is not by the deformation of the double layer, but by the electroosmotic flow that acts on the small volume element in the solvent and double layers. This is quite substantially different for low-frequency interactions, which are mainly determined by the direct deformation of the double layers in various phases/states. Further investigations of the clear distinction between the possible mechanisms of these frequency-response interactions are a follow-up subject to study in both theory and experiments.

Diffusion perpendicular to the director: q ⊥ E
As discussed in section 2.2, for the very small scattering angles that can be accessed by our vertically mounted dynamic light-scattering setup, diffusion perpendicular to the direction of alignment in the H-phase is probed. In this section, we discuss the dynamics related to displacements perpendicular to the direction of alignment. Dynamic light-scattering experiments at scattering angles less than 12 • along the same pathways (i)-(iii) as those defined in the previous section on birefringence will be performed.
Dynamic light scattering measures the intensity auto-correlation function C(t), the relaxation rate of which characterizes the dynamics of the rods. For the system under consideration of strongly interacting rods, the intensity auto-correlation function is generally a complicated function of time. Correlation functions are commonly highly non-singleexponential. The correlation functions obtained in our experiments are found to be described by a single, stretched exponential function, where A is the dynamical contrast, ⊥ is the relaxation rate and β is the stretching coefficient. The need for a stretched exponential function is due to the many different relaxation modes that contribute due to the strong interactions between the fd-virus particles. Due to the finite size of the scattering volume, typical values for the dynamical contrast are around 0.5. For larger frequencies, far away from transition lines, there is a very small amplitude contribution to the correlation functions of a fast relaxing mode, which is probably due to diffusion parallel to the director. The relative amplitude of this contribution is less than about 1% at very small scattering angles, and is too small to affect the fitting values of the parameters in equation (1). However, the amplitude of this fast mode is more pronounced at larger scattering angles. The contribution of the fast mode is more pronounced at higher fd concentrations and/or lower ionic strengths, which will be discussed in a separate paper.
Contrary to the off-critical path (ii), for which typical correlation functions are shown in figure 6(a), the correlation functions along pathway (iii) develop a very slow relaxing mode on approaching the critical point. This is shown in figure 6(b), where three correlation functions for various distances from the critical point are shown. The solid lines are fits to a stretched exponential in equation (1), excluding the long times. Relaxation of the initial part is seen to become faster on approaching the critical point, due to pre-transitional domains as discussed above, while a very slow decaying tail develops with increasing amplitude and relaxation time near the critical point. This speeding up of the initial relaxation and slowing down of the long-time decay on approaching the critical point lead to a crossing of the two correlations for ν = 30 and 100 Hz, as indicated by the circle in figure 6(b). In addition, the size of the pre-transitional domains is much larger as compared to the off-critical pathway. The identification of the point in the phase/state diagram where several transition lines meet as a critical point therefore seems meaningful on approaching that point not only from the dynamical state [16] but also from the side of the H-phase. The detailed features of the critical behavior on approaching from the side of the H-phase require further study and are outside the scope of the present study.
Relaxation rates along the pathways (i) and (ii) are given as a function of q 2 in figures 7(a) and (b). For pathway (i) in figure 7(a), the relaxation rate is seen to be independent of the field amplitude, right down to the H-to-N transition line. For pathway (ii), however, the relaxation rates are strongly increasing on approaching the H-to-N transition line, as can be seen from figure 7(b). More than about 100-200 Hz away from the H-to-N transition line, well within the H-phase, the relaxation rates become independent of the frequency and are equal to those measured for pathway (i) (data not shown). The same features are exhibited along pathway (iii). Since the inverse relaxation rates vary linearly with q 2 , we can define an effective diffusion coefficient D ⊥ = ⊥ /q 2 . The diffusion coefficients are plotted in figure 8 for the off-critical  figure 8, in between the images for the N -and H-phase. Near the critical point the size of these domains is much larger than the domains observed along the off-critical pathway. The domains appear and disappear on a timescale of a few minutes. The local nematic director within these domains is not oriented along the electric field. Dynamic light scattering therefore also probes diffusion of rods within the pre-transitional domains in directions parallel to the director. The increase in the measured diffusion coefficient on approaching the transition line is therefore attributed to diffusion parallel to the director within the pre-transitional domains. Since diffusion parallel to the director is faster, the measured diffusion coefficient increases as more of the nematic domains are formed near the transition line. The fact that, within the region where pre-transitional domains are present, the apparent diffusion coefficient along pathway (iii) is larger as compared to pathway (ii) indicates that there are relatively more rods oriented perpendicular to the field along the critical pathway. That is, domains have a preferential orientation perpendicular to the field near the critical point.
Diffusion can be superdiffusive (in the sense that β > 1) or sub-diffusive (β < 1), depending on the scattering wave vector q and the distance from the H-to-N transition line. As can be seen from the left plot in figure 9, diffusion along pathway (i) is always sub-diffusive. For very small scattering wave vectors, normal diffusion (where β = 1) is approached. The behavior of β for pathways (ii) and (iii) is very similar, so that we show only data on the former in the two plots on the right in figure 9. As can be seen, diffusion is sub-diffusive well within the H-phase. The stretching coefficient as a function of q is the same as that for pathway (i) to within experimental error. The stretching coefficients thus seem to be the same throughout the H-phase, and only depend on the scattering wave vector. Diffusion is superdiffusive only within the region where pre-transitional domains are present. This is probably related to the strong inhomogeneities that exist in this region of the phase/state diagram. The plot in figure 9 on the right shows the variation in β at a fixed q (corresponding to a scattering angle of 10 • ) as a function of the distance from the H-to-N transition line. The value of β seems to become equal to 2 for small wave vectors at the transition line. An understanding of the behavior of the stretching exponent requires a microscopic theory for the rod dynamics within the H-phase, which doesn't yet exist.
We can conclude that the degree of orientational order and the microscopic dynamics corresponding to displacements perpendicular to the electric field are essentially independent of frequency and field amplitude well within the H-phase, away from the H-to-N transition line. There are significant frequency dependencies, only close to the H-to-N transition line, due to the existence of pre-transitional nematic domains, which have a typical lifetime of a few minutes. Due to these 'anomalous' domains, diffusion can be strongly superdiffusive, especially for small wave vectors. Within the H-phase, perpendicular diffusion is sub-diffusive. Furthermore, on approaching the critical point from the H-phase, a slow decaying relaxation process is observed. The critical features of this point on approaching from the dynamical state as described in [16] are thus also present on approaching from the H-phase.

Conclusions
A number of phases and dynamical states are induced in a concentrated suspension of charged, colloidal rods (fd-viruses) by an external electric field. We performed electric birefringence and small-angle dynamic light-scattering experiments along different pathways within one of the phases, the H-phase, which is a homogeneous phase with homeotropic alignment. The H-phase exists for frequencies larger than about 1 kHz. Since at frequencies higher than a few kHz the diffuse double layers and the layers of condensed ions of the colloidal rods are only weakly polarized, it is not yet clear by which mechanism the H-phase is stabilized. Since electro-osmotic flow is established on a much smaller timescale (of the order of 1 ns), the fieldindependent response within the H-phase might be due to rod-rod interactions induced by this solvent flow.
The key finding of this paper is that diffusion in the direction perpendicular to the electric field (which is also the direction of orientation of the rods) and the degree of orientational order in the H-phase are independent of the applied frequency and field amplitude. This suggests that the rod-rod interactions that stabilize the H-phase are not due to charge polarization but are due to hydrodynamic interactions through electro-osmotic flow. As yet there is no theory that could explain these observations. Only quite near to the H-to-N transition line (within a frequency interval of about 100-200 Hz) we found an increase in perpendicular diffusion coefficients. This is due to the existence of pre-transitional nematic domains with a director that is not parallel to the electric field. The lifetime of these domains is typically a few minutes and is visualized by means of polarization microscopy.
There is a point in the phase/state diagram where several transition lines meet. This point has been shown [16] to exhibit a diverging time-and length-scale on approaching from the side of the dynamical state. We have observed an extremely slow mode on approaching this non-equilibrium critical point also from the side of the H-phase. In addition, the size of the pretransitional domains is much larger than for an off-critical approach of the H-to-N transition. The characterization of the critical behavior within the H-phase needs further study.