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Zilin
Wang
^{a},
Hartmut
Kriegs
^{a},
Johan
Buitenhuis
^{a},
Jan K. G.
Dhont
*^{ab} and
Simone
Wiegand
^{a}
^{a}ICS-3 Soft Condensed Matter, Jülich, Germany. E-mail: j.k.g.dhont@fz-juelich.de; s.wiegand@fz-juelich.de
^{b}Institute of Physics, Heinrich-Heine- Universität, D-40225 Düsseldorf, Germany

Received
24th May 2013
, Accepted 18th July 2013

First published on 19th July 2013

The thermal diffusion behavior of dilute solutions of very long and thin, charged colloidal rods (fd-virus particles) is studied using a holographic grating technique. The Soret coefficient of the charged colloids is measured as a function of the Debye screening length, as well as the rod-concentration. The Soret coefficient of the fd-viruses increases monotonically with increasing Debye length, while there is a relatively weak dependence on the rod-concentration when the ionic strength is kept constant. An existing theory for thermal diffusion of charged spheres is extended to describe the thermal diffusion of long and thin charged rods, leading to an expression for the Soret coefficient in terms of the Debye length, the rod-core dimensions, and the surface charge density. The thermal diffusion coefficient of a charged colloidal rod is shown to be accurately represented, for arbitrary Debye lengths, by a superposition of spherical beads with the same diameter of the rod and the same surface charge density. The experimental Soret coefficients are compared with this and other theories, and are contrasted against the thermal diffusion behaviour of charged colloidal spheres.

In many previous studies, attempts have been made for mixtures of non-polar liquids to relate the Soret coefficient to the mass of the molecules, their moment of inertia, and the viscosity and thermal expansion coefficient.^{1} In aqueous systems, where the situation is more complicated, hydrogen bonds and the charge effect are of significant importance.^{3,4}

With the advent of new experimental techniques in the last few years, it became possible to investigate the thermal diffusion behaviour of relatively slowly diffusing macromolecules. A number of experimental studies have been performed on charged macromolecules, such as DNA, ionic surfactants, surface modified polystyrene spheres and silica colloids.^{2,5–8} For charged silica Ludox particles it was found that the Soret coefficient increases with increasing Debye length and slightly drops at large Debye lengths, of the order of the core-radius of the colloids. For all investigated concentrations at room temperature, a negative Soret coefficient has been observed for this system.^{2} Another study reveals an increasing linear dependence of the Soret coefficient with the Debye length for carboxyl-modified polystyrene beads.^{5} Here, the Debye length is always very small as compared to the particle radius. Apart from colloids, a study of micellar solutions with the ionic surfactant sodium dedecyl sulphate also shows that raising the Debye length leads to an increase of the Soret coefficient.^{7}

To gain a better understanding of the microscopic mechanism of the thermal diffusion process of macromolecules, several theoretical approaches have been developed. The first theory was published by Ruckenstein in 1981,^{9} where a connection between thermophoresis of solid colloidal particles and the Marangoni effect is made. Later, models for single spherical particles have been derived in terms of surface potential, independently by Morozov and Piazza.^{6,10} A few years later, Bringuier and Bourdon proposed an expression for S_{T} in terms of the total internal energy of a particle, based on the kinetic theory of Brownian motion.^{11} Independently, Fayolle et al.^{12} and later Duhr and Braun^{13} derived an expression for the Soret coefficient for charged colloids with thin double layers, which was subsequently generalized by Dhont and Briels^{14} to arbitrary Debye lengths. All these theoretical approaches apply only to spherical colloids, while little is known about non-spherical colloids. The present paper is devoted to thermal diffusion of very long and thin, charged colloidal rods.

Fd-virus suspensions are widely used as model systems for colloidal rods. Suspensions of fd-virus particles have been shown to exhibit several liquid-crystalline phases (for an overview see ref. 15), and have been used for studies on the response of rod-like colloids to shear flow^{16,17} and electric fields.^{18} Their suspensions are stable up to 65 °C,^{19} and are highly monodisperse. The wild type fd-virus has a molecular weight of 1.64 × 10^{7} g mol^{−1}, a contour length L of 880 nm, a radius a of 3.4 nm, and a persistence length L_{P} of 2.2 μm. The ratio of the persistence length to contour length L_{P}/L = 2.5 indicates that these rods are semi-flexible, which leads to a deviation of isotropic–nematic coexistence concentrations as predicted by Onsager for stiff rods.^{15,20} Above pH 4 these particles are negatively charged and interact via a combination of electrostatic repulsion and hard-core interactions. The net surface charge can be increased or decreased by increasing or decreasing the solution pH, respectively.^{21}

In the present paper we use fd-virus suspensions to investigate the thermal diffusion behaviour of very long and thin, charged colloidal rods. The paper is structured as follows. First, we describe the experimental details. In the subsequent section we extend the Dhont–Briels model^{14} for spherical colloids to rod-shaped particles. In the results and discussion section we present experimental results for thermal diffusion coefficients and mass diffusion coefficient, where both the ionic strength and concentration are independently and systematically varied. The experimental results are compared to our theory and other theories, and also to former experimental results for spherical colloids. The main conclusions and remarks with respect to possible future work close the paper.

The normalized heterodyne scattered intensity ζ^{th}_{het}(t), assuming an ideal excitation with a step function, is given by,

(1) |

(2) |

(3) |

Fig. 1 The alternative path to move the colloid from a box with temperature T over a distance dz to the neighbouring box with temperature T + dT. W^{(dl)}(T) is the reversible work that is required to charge the colloid. The colloid is the blue rod, while the red dashed lines are used to indicate the presence of the double layer. |

For the calculation of the temperature derivative of the reversible energy to charge the colloidal particle, analytical results can be obtained within the Debye–Hückel approximation. The work required to add a charge dq to the surface of the colloid, homogeneously distributed over its surface, is equal to Ψ_{s}(q)dq, where Ψ_{s}(q) is the potential at the surface of the colloid with the total charge equal to q. Since within the Debye–Hückel approximation the surface potential is proportional to the charge, the work to charge the colloid up to a total charge equal to Q is given by

(4) |

This result is valid for arbitrary double-layer thickness.^{25} The relationship between the surface potential and the total charge for a spherical colloid reads

(5) |

(6) |

Combining the above results it is found that the thermal diffusion coefficient can be written in terms of the surface charge density σ and the Bjerrum length l_{B} as

(7) |

(8) |

(9) |

(10) |

For the cylindrical colloid there are two limiting situations, where either κa_{c} ≪ 1 or κa_{c} ≫ 1. In the first case of very thick double layers compared to the cylindrical core radius, the core of the rod may be considered as a string of L/2a_{c} beads of radii a_{c}. The extended double layer structure of each bead is essentially unaffected by the presence of the relatively small excluded volume of the neighbouring beads, so that within the linearized Poisson–Boltzmann approach the structure of the double layer of the cylinder is well approximated by a sum of the spherical double layers of the beads. Except for the diffusion coefficient D_{0}, which is different for a sphere and a rod, the thermal diffusion coefficient of the rod is now a sum of the thermal diffusion coefficients of the beads, that is,

(11) |

The expression for the thermal diffusion coefficient in the other limiting case of thin double layers is given by eqn (9). May be surprisingly, this expression predicts, like for the thick double layers, that the thermal diffusion coefficient of a rod is simply a superposition of the thermal diffusion coefficients of the beads (with radii a_{c}). This can be analytically verified by substitution of the two leading terms in an asymptotic expansion of the two Bessel functions. Numerical results are given in Fig. 2. Fig. 2a and b show that for κa_{c} ≳ 2 the result in eqn (9) is quite accurately approximated by a representation of 2a_{c}/L beads with each having a thermal diffusion coefficient given in eqn (8) with a = a_{c}, where there is an increasing accuracy for thinner double layers. The thermal diffusion coefficient D_{T} in dimensionless form is plotted in Fig. 2c for a cylinder and the corresponding bead model, with the value dlnε/dlnT = −1.43 for water. As can be seen, the approximation of the thermal diffusion coefficient of a rod by that of a string of spheres becomes more accurate on decreasing the Debye length.

Fig. 2 The dimensionless quantities Ã = CA, = CB and _{T} = CD_{T}, where C = T16πεa/βD_{0}ρQ^{2} as a function of κa (with a = a_{c} for the rods), where Q is the monomer charge. |

The conclusion from the above analysis is that both for thick and thin double layers, the thermal diffusion coefficient of a rod-like colloid can be accurately approximated by L/2a_{c} times the thermal diffusion coefficient (7) and (8) of a spherical colloid with radius a = a_{c}, with the same surface charge density as the rod. From the above analysis we thus find that the Soret coefficient S^{(rod)}_{T} = D^{(rod)}/ρD^{(rod)}_{0} is equal to

(12) |

This superposition of thermal diffusion coefficients of spherical beads to approximate the thermal diffusion coefficient of a rod is valid for arbitrary Debye lengths, and will be used in the sequel for a comparison with experimental data for the thermal diffusion coefficient of fd-viruses.

(13) |

An effective volume fraction ϕ can be defined, which characterizes the importance of interactions between rods. The effective volume fraction depends on the fd number concentration and the Debye length. Due to inter-particle charge–charge interactions, the apparent radius of the core of the fd-virus particles increases by an amount that is approximately given by the Debye length. The effective volume fraction is defined as the volume fraction of rods with a core radius of a_{c} + κ^{−1}. Alternatively the apparent radius can be calculated as suggested by Onsager,^{20} based on equality of the second virial coefficient of charged rods and the equivalent hard rods with the corresponding apparent core radius. We shall be satisfied with the former simple approximation, as it does not affect our conclusions.

There are thus three parameters of interest: the number concentration of fd viruses, the Debye length, and the effective volume fraction. We therefore performed the following set of experiments:

(a) The Debye length is varied with a constant number concentration of fd-virus particles (of 1 mg ml^{−1}).

(b) The number concentration of fd-virus particles is varied with a fixed Debye length (where the buffer concentration is chosen equal to 6.11 mM).

(c) The effective volume fraction is fixed to 0.0037 with appropriate simultaneous variation of both the fd-virus concentration and the Debye length.

These experimental paths are illustrated in Fig. 3a–c, respectively. Although in our suspensions the state of the system is always isotropic, the rods in these figures are sketched with the same orientation for clarity.

Fig. 3 Illustration of the chosen experimental paths. The white rods present the core of fd-viruses and the blue parts are used to indicate the extent of the electric double layers. For clarity the rods have been drawn with the same orientation. In the experiments, however, the systems are isotropic. (a) The number concentration of fd-viruses is constant while the Debye length is increased by decreasing the buffer concentration. (b) The number concentration of fd viruses is increased at a constant Debye length. (c) Both the number concentration of fd viruses and the Debye length are changed in such a way that the effective volume fraction remains constant. |

The aim of this work is to probe the thermal diffusion behaviour of single fd-virus particles as a function of the Debye length, without the intervening effects of inter-particle interactions. In order to probe whether inter-particle interactions affect the measured mass diffusion and thermal diffusion coefficients, we performed a series of experiments as a function of the effective volume fraction. The open symbols in Fig. 4 are experimental data for the Soret coefficient and the mass- and thermal diffusion coefficient at a constant buffer concentration, and thus a fixed Debye length. The effective volume fraction for this series of experiments is changed by only changing the fd-virus concentration (path (b)). As can be seen from the lower panel in Fig. 4 (the open symbols), the thermal diffusion coefficient is insensitive to interactions, and is essentially constant up to an effective volume fraction of about 0.0065. The mass diffusion coefficient, however, significantly increases with increasing effective volume fraction, as can be seen from the middle panel (again the open symbols). The blue triangle at low concentration is taken from the literature.^{26,27} On the other hand, when the effective volume fraction is increased at a constant fd-virus number concentration (1 mg ml^{−1}) by increasing the Debye length (path (a)), significant changes are observed (see the filled symbols in Fig. 4), also for the thermal diffusion coefficient. Since inter-particle interactions are not significant as far as the thermal diffusion coefficient is concerned, its variation with the buffer concentration must be due to the variation of the double-layer thickness. In order to obtain experimental values for the single-particle Soret coefficient S^{0}_{T}, we will use the Debye-length dependent thermal diffusion coefficient (filled symbols in the bottom panel in Fig. 5) and the mass diffusion coefficient at infinite dilution D_{0} = 2.3 × 10^{−8} cm^{2} s^{−1} from the literature.^{26,27} The upper panel in Fig. 5 shows Soret coefficients obtained from the diffraction signal (cf.eqn (1) and (2)) corresponding to the ratio of the experimental values for the two diffusion coefficients.

Fig. 4
S
_{T}, D and D_{T} as a function of the effective volume fraction. Solid squares present measurements with fixed c_{fd} = 1 mg ml^{−1}. Open squares illustrate measurements with constant c_{buffer} = 6.11 mM. The open triangle presents the data of the bulk diffusion coefficient from the literature.^{26,27} The red line is a guide to the eye. |

Fig. 5
S
_{T}, D and D_{T} as a function of the Debye length. Solid squares present measurements with fixed c_{fd} = 1 mg ml^{−1}. Empty with center cross-symbols are measurements with constant ϕ = 0.0037. |

To further establish the Debye-length dependence of the mass- and thermal diffusion coefficients, we performed experiments at a fixed effective volume fraction of 0.0037 (path (c)), by appropriate simultaneous variation of both the fd number concentration and the Debye length. The Soret coefficient and the mass- and thermal diffusion coefficients along this path are plotted in Fig. 5 by the open (with center-cross) symbols as a function of the Debye length. The filled symbols in Fig. 5 refer to data obtained along path (a), where the fd number concentration is fixed to 1 mg ml^{−1}, and the Debye length is varied by changing the buffer concentration. The two sets of data points in the lower panel for the thermal diffusion coefficient coincide to within experimental error, which again confirms the insignificant inter-particle interactions.

In order to asses the importance of the finite extent of the electric double layer, we include a comparison to eqn (12) in the asymptotic limit of very thin double layers, where κa_{c} → ∞. For such thin double layers we have from eqn (12),

(14) |

The corresponding expression for spherical particles (without the factor L/2a_{c}) has been derived independently by Fayolle et al.^{12} and Duhr and Braun.^{13} There is so far no extension of Ruckenstein's theory^{9} to rod-like colloids. It is not obvious that we can simply take Ruckenstein's expression for the Soret coefficient for spheres and multiply that with the number of beads to obtain the corresponding expression for rods. We will therefore refrain from a comparison with any possible extension of Ruckenstein's theory to rods.

The result in eqn (12) and the above expression account only for the contribution of the electric double layer to the Soret coefficient. There is an additional “ideal gas” contribution equal to 1/T, and contributions due to thermal properties of, for example, the solvation layer and the core of the colloids. These contributions are insensitive to salt concentration, so that they determine the “offset” in plots of the Soret coefficient as a function of the Debye length. The offset and the surface charge density σ are used as fitting parameters in a comparison of experiments with theory. Fig. 6 shows the Soret coefficient S^{0}_{T} = D_{T}/D_{0} determined from the measured thermal diffusion coefficient D_{T} and the diffusion at infinite dilution D_{0} as a function of the Debye length. The fits are plotted in Fig. 6, while the offset and surface charge density for the best fits to various theories are given in Table 1. The solid line (blue) is a fit to limiting expression (eqn (14)) for very thin double layers. The fit is slightly improved by accounting for the finite extent of the electric double layer, as discussed in Section 3 (the dashed green line). Similar results have been found for charged spherical particles (Ludox silica particles),^{2,13} of which the experimental data are presented in Fig. 6 as open circles, which fitted with the model described in Section 3 for spherical colloids.

Fig. 6 The Soret coefficient S_{T} as a function of the Debye length. In contrast to Fig. 4 and 5 the open squares present the calculated S^{0}_{T} = D_{T}/D_{0} for the fd-virus with an effective volume fraction of ϕ = 0.0037 and the open circles show S_{T} of Ludox silica particles. The data of the fd-virus are fitted by the two models discussed in the main text. The data of Ludox particle are fitted by Dhont's model for spheres.^{2} |

Model |
σ/e nm^{−2} |
Offset |
---|---|---|

Eqn (14) (based on ref. 12 and 13) | 0.023 ± 0.002 | −0.74 |

Eqn (12) (this work) | 0.050 ± 0.003 | −1.39 |

Calculated free surface charge | 0.066 |

The question now is whether the surface charge density of σ = (0.050 ± 0.003) e nm^{−2} that is found from the fit to our theory is a reasonable value. Fd-virus consists of a DNA strand that is covered by 2700 proteins, which carry a bare charge of 9.5 ± 0.5 negative elementary charges per nm. According to a calculation by Buitenhuis,^{28} about 90% of these groups at pH = 8.2 are dissociated, so that the total bare charge of an fd particle is equal to N = 7500 ± 400 e. A large fraction of this surface charge is neutralized by ion-condensation. According to Manning's ion-condensation theory,^{29,30} the ratio b/l_{B} < 1 of the typical distance b = L/N between the charges on a cylindrical rod and the Bjrerrum length l_{B} is equal to the fraction of bare charges that is not neutralized by condensed ions (for monovalent ions). Since l_{B} = 0.71 nm for water at room temperature, it is found that b/l_{B} is equal to 0.16 ± 0.01. The total number of charges close to the surface of an fd particle that determine the structure of the diffuse electric double layer is thus equal to 1239 ± 66, which corresponds to surface charge densities of 0.066 ± 0.004 e nm^{−2} (where the indicated error is certainly much smaller than the actual error that are implicit in the approximations made in the theory on which this estimate is based). This value of the surface charge density is in reasonable agreement with the experimentally found surface charge density of 0.050 ± 0.003. The surface charge density obtained from a fit to the same theory for very thin double layers gives a significantly lower surface charge density (see Table 1).

Within Manning's ion-condensation theory, there is no dependence of the number of condensed ions on the overall ionic strength, which suggests that the contribution of the condensed ions to the thermal diffusion coefficient is independent of the Debye length. The contribution of the condensed ions to D_{T} is thus incorporated in the offset as introduced earlier, while the Debye-length dependence of D_{T} is determined by the diffuse electric double layer.

Future developments related to charged colloids could include (i) the extension of the theory beyond the Debye–Hückel approximation and (ii) the thermal diffusion of other types of non-spherical colloids, like disks, and of charged rods with varying flexibility.

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