Electrophoresis of fd-Virus Particles: Experiments and an Analysis of

Article pubs.acs.org/Langmuir

Electrophoresis of fd-Virus Particles: Experiments and an Analysis of the Effect of Finite Rod Lengths Johan Buitenhuis* Soft Matter Group, ICS-3, Forschungszentrum Jülich, 52425 Jülich, Germany ABSTRACT: The electrophoretic mobility of rodlike fd viruses is measured and compared to theory, with the theoretical calculations performed according to Stigter (Stigter, D. Charged Colloidal Cylinder with a Gouy Double-Layer. J. Colloid Interface Sci. 1975, 53, 296−306. Stigter, D. Electrophoresis of Highly Charged Colloidal Cylinders in Univalent Salt- Solutions. 1. Mobility in Transverse Field. J. Phys. Chem. 1978, 82, 1417−1423. Stigter, D. Electrophoresis of Highly Charged Colloidal Cylinders in Univalent Salt Solutions. 2. Random Orientation in External Field and Application to Polyelectrolytes. J. Phys. Chem. 1978, 82, 1424−1429. Stigter, D. Theory of Conductance of Colloidal Electrolytes in Univalent Salt Solutions. J. Phys. Chem. 1979, 83, 1663−1670), who describes the electrophoretic mobility of infinite cylinders including relaxation effects. Using the dissociation constants of the ionizable groups on the surfaces of the fd viruses, we can calculate the mobility without any adjustable parameter (apart from the possible Stern layer thickness). In addition, the approximation in the theoretical description of Stigter (and others) of using a model of infinitely long cylinders, which consequently is independent of the aspect ratio, is examined by performing more elaborate numerical calculations for finite cylinders. It is shown that, although the electrophoretic mobility of cylindrical particles in the limit of low ionic strength depends on the aspect ratio much more than “end effects”, at moderate and high ionic strengths the finite and infinite cylinder models differ only to a degree that can be attributed to end effects. Furthermore, the range of validity of the Stokes regime is systematically calculated.

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INTRODUCTION The electrical charge on proteins, DNA, and other synthetic or natural polyelectrolytes, emulsions, and solid colloidal particles and micellar structures plays an important role in every day life and in many industrial processes. An important method of characterizing the charge on these materials is the measurement of the electrophoretic mobility, which gives the velocity of the particle per unit of electrical field strength. Mostly, the electrophoretic mobility is converted to the zeta potential, which is used as a measure of the charge on the particle, which can also be used to calculate the electrostatic interaction between particles. Here the rodlike fd virus, a bacteriophage with a length of 880 nm and a diameter of 6.6 nm, is of special interest because it is often used as a well-defined model system5−7 for which the interaction between them and other particles and structures is important to know. However, when I wanted to characterize chemically modified fd viruses by electrophoretic measurements,6 I found that it was far from trivial to obtain reasonable agreement between the measured and calculated electrophoretic mobilities of the bare fd virus, and consulting the literature confirmed that the measurement and interpretation of the absolute electrophoretic mobility is often not a trivial task.8−10 This problem is the subject of this article. Specifically, the charge and electrophoretic mobility of the fd virus has been investigated. However, I also present a more general argument relating to the modeling of the charge and electrophoretic mobility of long, rodlike particles, especially the effect of finite size, and © 2012 American Chemical Society

demonstrate problems with some common experimental methods (i.e., no correct series of measurements has been published so far). Especially for rodlike particles, often only a qualitative or semiquantitative interpretation is given or the experimental results are scaled to the theoretical description such that only the ionic strength (or pH) dependence is compared. Furthermore, in many measurements either the ionic strength or the pH is not controlled if the other one is varied so that those results cannot be compared to a theoretical description and are focused on the determination of the point of zero charge. If an effort is made to provide a full quantitative interpretation, then significant deviations between theory and experiment, even on the order of a factor of 1.5 to 2, may be found.11 In many cases, an important reason is that some parameters of the experimental system are not known well enough. But even if the system is (almost) monodisperse and the surface structure and ionization are well defined, it turns out to be difficult to make a comparison between electrophoresis theory and experimental data that agrees in absolute numbers, (i.e., no scaling of the electrophoretic mobility to the data). In addition, the measurement of absolute electrophoretic mobilities is also not always as simple as may seem at first glance. Electroosmosis, aggregates and bubbles Received: December 22, 2011 Revised: July 26, 2012 Published: September 7, 2012 13354

dx.doi.org/10.1021/la302245x | Langmuir 2012, 28, 13354−13363

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dielectric constant of 73 was used that was taken from Åkerlöf,16 and the viscosity at 25 °C of 1.39 mPa s was taken from the Landolt-Börnstein series.17 Calculations are performed in Octave on a personal computer. Further details of the calculations are described in the sections below.

formed at the electrodes, impurities in dilute samples, and heating effects at higher ionic strengths make absolute measurements a critical task. Here we present a case study on the absolute electrophoretic mobility of the rodlike fd virus and compare the experimental results to theoretical calculations, without any adjustable parameter apart from the thickness of a possible Stern layer. Problems with some common experimental methods are demonstrated. In addition, the approximation in the theoretical description of Stigter of using a model of infinitely long cylinders, which consequently is independent of the aspect ratio, is examined because the electrophoretic mobility of cylindrical particles in the limit of low ionic strength (the Stokes limit) depends on the aspect ratio much more than just end effects so that even for long rods neglecting the aspect ratio dependence might result in incorrect results. Therefore, the results from Stigter's model for infinitely long cylinders are compared to more elaborate numerical calculations for long but finite cylinders at moderate and high ionic strengths. Furthermore, the range of validity of the Stokes regime is systematically calculated, aiming to be useful for experimentalists.

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THEORY OF STIGTER FOR INFINITELY LONG CYLINDERS The electrophoretic mobility of fd viruses is measured and compared to theory, with the theoretical calculations performed according to Stigter,1−4 which describes the electrophoretic mobility of infinite cylinders including relaxation effects. Stigter gives the results as equations for low surface potentials with systematically tabulated correction factors for higher potentials, which makes his results accessible for comparison to experimental results. However, this comparison between theory and experiment turned out to be less straightforward than it seemed, which resulted in the present article. As an input to the theory of Stigter, the surface charge of the virus is calculated from the dissociation constants of the carboxyl groups and amino groups on the surface of the fd virus. The vast majority of the charges come from the major coat protein, bearing five carboxyl groups from two glutamate and three aspartate residues and two amino groups from one lysine residue and one terminal amino group. From the surface pH (pHsurf) and the expected pKa values in a protein for each ionizable group (4.5 for glutamate and aspartate, 7.9 for the terminal amino group, and 10.1 for lysine), the net number of elementary charges per coat protein (Zp8) is obtained18

EXPERIMENTAL SECTION

The growth and purification of fd viruses was performed following standard biochemical protocols12 with the XL 1 blue strain of E. coli as the host bacteria. Typically, about 15 mg of the virus was produced per liter of infected bacterial suspension. After purification, the virus was redispersed in a 1 mM NaCl aqueous solution with 15% (v/v) ethanol to prevent microbiological growth, preparing a stock dispersion with an fd concentration of 10 mg mL−1. The concentration was obtained photometrically using an extinction coefficient of 3.84 mg−1 cm2 at 269 nm13. The absorption ratio at 269 and 244 nm (A269/A244) was used as a purity check, where a ratio of 1.41 indicates impurities below 1%. Samples for electrophoresis were diluted from the above-mentioned 10 mg mL−1 stock solution to obtain a concentration of 0.1 mg mL−1 in aqueous buffer solutions with 15% (v/v) ethanol with or without added NaCl. At this fd concentration and the ionic strengths used in this study, interactions between viruses can be neglected for the electrophoretic mobility.14 For the pH-dependent series, the following buffer solutions were prepared: pH 4.0, 5.0, and 5.6 from acetic acid/NaOH; pH 6.3 and 7.0 from HCl/imidazole; pH 7.7 and 8.8 from HCl/TRIS; pH 10 from HCl/NH3; and pH 11 from pure NaOH. The compositions were chosen such that the ionic strength of each buffer solution was equal to 1 mM. At pH 4.0, no measurement could be performed because of the flocculation of the fd virus. Higher ionic strengths at pH 7.0 were prepared by adding the appropriate amount of NaCl. All chemicals used for the preparation of the solutions were of high purity as ordered from Sigma-Aldrich. Measurements of the electrophoretic mobility were performed on a Malvern Zetasizer 2000 either in the M3 capillary cell or in the aqueous dip cell. A few control measurements have also been performed on a newer Malvern Zetasizer Nano ZS. For all three measurement methods, the applied electrical field strength is about 30 V/cm, which is well below the field strengths necessary for a significant field-induced orientation of the viruses at the 0.1 mg mL−1 concentration.15 Prior to the measurement, all samples were filtered through a 5 μm nylon filter. All cells were rinsed with filtered solution and checked for air bubbles prior to the measurements. Each measurement point consists of at least 10 separate runs. The measurements were checked for repeatability; if not, often bubbles or dust could be observed. Care was taken to allow for enough waiting time between the runs, which was especially important at high salt concentrations because of the heating of the solutions. Furthermore, measurements with the dip cell were found to be especially sensitive to the regular cleaning of the electrodes. All measurements were performed at 25 °C. To bring the calculations in line with the measurements, all calculations are also performed for 15% ethanol. For this, a relative

Zp8 = (1 + 10(pHsurf − 7.9))−1 + (1 + 10(pHsurf − 10.1))−1 − 2(1 + 10−(pHsurf − 4.5))−1 − 3(1 + 10−(pHsurf − 4.5))−1 (1)

and multiplication by 2700 coat proteins per virus gives the charge on the virus as Q = Zp8(2700e), with e being the elementary charge. More details about how to make this calculation if ions other than acids/bases determine the charge can be found elsewhere.8 However, to perform this equilibrium dissociation calculation, the pH (= −log[H+]) at the surface of the virus has to be known, which is related to the bulk pH by [H+]surface = [H+]bulk e−eψ0/ kT

(2)

with e being the elementary charge, k being the Boltzmann constant, T being the absolute temperature, and ψ0 being the electrostatic surface potential.8 The surface charge of fd calculated in this way is the same as the charge calculated by Zimmerman et al.,18 who showed that these calculated charges compared well with the titrated charge on the surface of fd. However, it should be noted that the charge at the end point of each charge titration is compensated for, so at the end of the titration, the surface pH and bulk pH will be (almost) equal. In other words, the charge obtained from charge titrations at a certain pH is (close to) the charge that the fd virus (or any other particle) has at that surface pH. The theory from Stigter1 gives

with Q = f([H+]surface) being the surface charge as obtained from the surface dissociation, K0 and K1 being the zeroth- and firstorder modified Bessel functions of the second kind, respectively, ε0 being the vacuum permittivity, εr being the relative dielectric 13355


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with η being the solvent viscosity and g(ζ, κa) being a correction factor that was numerically calculated by Stigter including relaxation effects. Again, I fit an empirical equation to the tabulated results from Stigter for orientationally averaged cylinders, giving

constant of the solvent, L being the length and a being the radius of the cylinder, κ being the inverse of the Debye−Hückel screening length, and β being the correction from Stigter for larger ψ0. Note that the theory is derived from an infinite cylinder and that the charge and counter charge at the end of the cylinder are neglected. The Debye−Hückel parameter κ for a 1−1 electrolyte is given by κ=

2000F 2I ε0εrRT

⎧ ⎛⎛ ⎞ ⎪ 1 ⎜ ⎜ ⎟x g (ζ , κa) = ⎨1.25 + 0.25 tanh⎜ 0.312 + 1 2̃ ⎟ 1 ⎜ ⎪ 1 + ζ ⎠ ⎝⎝ 25 ⎩ ⎞⎫ ⎟⎪ − 1.025⎟⎬ × ⎠⎪ ⎭

(4)

where F is the Faraday constant, R is the molar gas constant, and I is the ionic strength in mol/L given by I = 1/2Σici with ci being the concentration of each ion i in mol/L. The function β(ψ0, κa) is a numerical correction factor between the low potential and larger potential results, which is systematically calculated and tabulated by Stigter for a range of κa and y0 = ψ0e/kT values. To calculate Q and ψ0 as a function of I and pHbulk, the equations above have to be simultaneously solved numerically. Therefore, extending the suggestion by Stigter, the following semiempirical equation for β was obtained here by fitting to the tabulated values,

⎫ ⎧ ⎞ −⎛ x1− 0.72 ⎞2 ⎪ ⎛ 1 ⎝ 2.2 ⎠ ⎬ ⎨ e 1 − 1 − ⎟ ⎜ ⎪ ⎪ (1 + |ζ |̃ 1.7 /43.8) ⎠ ⎝ ⎭ ⎩ ⎪

(9)

{

1 2

}

(5)

which reproduces the values in the table from Stigter1 within 2% and within 2. This equation shows a ln(L/D) dependence, which means that the results from Stigter (and Henry21 for low potentials) cannot be right in the Stokes limit, where the electrophoretic mobility is μE = Q/fcyl, with Q being the charge on the cylinder. To determine whether the theory for infinitely long cylinders can also be applied to finite cylinders in the relevant experimental parameter range, calculations for long, finite cylinders at low surface potential were performed using the numerical method as follows. A number of calculations for finite cylinders have been published, as performed by the bead model and by the boundary element method.22−25 The bead model divides the particle in a

which reproduces the tabulated values within 2.5 and 75 mV at I = 100 mM) the effect of finite ion sizes will start to play a role,19 and the predictions from Stigter will become less accurate. However, within the experimental conditions described in this study, finite ion size effects seem to be negligible.

⎫ ⎧ ⎛ ⎞ ⎪ 2 ⎪ ⎛y ⎞ ⎜⎜ ⎟⎟sinh⎜ 0 ⎟ − 1⎬ 1 tanh(1.027x1 + 0.195 β=1+⎨ ⎪ ⎪ ⎝2⎠ ⎭ 2 ⎩⎝ y0 ⎠ + 0.0217y0 ) +

⎜

(8) 13356


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values of u(⃗ r)⃗ close to the surface of a sphere can be obtained directly by numerical integration and give the correct result for the electrophoretic mobility μE in the Hückel limit (i.e., equal to 2ε0εrζ/3η). Here, ε0 is the vacuum permittivity, εr is the relative dielectric constant, ζ is the zeta potential, and η is the viscosity of the solvent. However, to obtain the correct value of μE for a sphere at low potential but for arbitrary κa as obtained by Henry,21 two → ⎯ modifications have to be made. First, ∇′ ϕ (r′) can no longer be taken equal to be E0⃗, but the result from Henry for ϕ (r ⃗ ) has to be used,

number of more or less small beads, and the boundary element methods describe the surface of a particle with a number of small, often triangular elements. Therefore, these methods can be used to describe particles of arbitrary shape; however, demanding calculations are necessary to obtain the electrophoretic mobilities. Nevertheless, these methods have been used for calculations on finite cylinders where the results were compared to experimental results on DNA molecules, which can be considered to be rods or cylinders if they are not too long. However, for the L/D values of the fd virus, DNA molecules are flexible (i.e., they cannot be considered to be rods or cylinders anymore26), and these results cannot be used for a direct comparison to the results in this article. Here, a slightly different method is used to calculate the electrophoretic mobilities. The basic physics of this method is similar to that of the alternative and more general bead model and boundary element method, but for the present needs, the method following Burgers27 and Dhont28 is sufficient. The electrophoretic velocity of a cylinder or other particle can be calculated by the integration of the Oseen tensor over the hydrodynamic surface forces and the body forces in the electrical double layer, which in principle follows the method described by Burgers, generalized following Dhont and with some small extensions as described below. The flow field around a particle u(⃗ r)⃗ can be calculated from → ⎯ → ⎯ ⎯ → u ⃗ ( r ⃗) = dS′ T( r ⃗ − r′) · Fh(r′)

⎛ 1 a3 ⎞ ϕ(r , θ ) = −E0⎜r + ⎟cos θ 2 r2 ⎠ ⎝

with E0⃗ in the θ = 0 direction and a being the radius of the sphere. This result for ϕ(r ⃗ ) originates from the fact that the applied electrical field in the solution depends on the electrical conductivity, and the conductivity of the particle will usually be effectively zero so that the electrical field lines will bend around ⎯ ⎯→ → the sphere. If this result is used in the calculation, then Fh (r′) can no longer be taken as constant, as can be concluded from the result that u⃗(r)⃗ is no longer constant close to the surface of the ⎯ ⎯→ → sphere if Fh (r′) is taken constant. Considering the symmetry of the problem, the function

∮∂particle

−

∫r>a

→ ⎯ → ⎯ → ⎯ dV ′ T( r ⃗ − r′) ·ρ(r′) ·∇′ϕ(r′)

⎛ a 2 sin 2θ cos φ ⎞ ⎟ ⎯⎯⎯⎯→ ⎜ ⎯ | Ftotal| ⎜ a 2 sin 2θ sin φ ⎟ ⎯→ → Fh(r′) = ⎟ A total ⎜ ⎜1 + a1 + a1 cos 2θ ⎟ ⎝ ⎠ 3

(12)

→ ⎯ with the Oseen tensor T(r ⃗ − r′) describing the fluid flow at the point r ⃗ resulting from a point force applied to the liquid at point → ⎯ → ⎯ → ⎯ → ⎯ r′, ρ⃗(r′), giving the charge density at a point r′, ϕ (r′) being the → ⎯ electrical potential at a point r′ resulting from the applied ⎯ ⎯→ → electrical field, and Fh (r′) being the hydrodynamic surface force per unit surface area, which needs to be normalized such that ⎯ ⎯→ → ⎯⎯⎯⎯→ dS′ Fh(r′) = Ftotal

∮∂particle

(14)

(15)

was found to give good results, where θ and φ describe spherical ⎯⎯⎯⎯→ coordinates with Ftotal in the z (θ = 0) direction. The optimum values for a1 and a2 are obtained by calculating the velocity u⃗(r)⃗ at several points along the surface of the sphere and minimizing the standard deviation of the velocity. This results, for instance, in a1 = −0.25 and a2 = −0.26 at κa = 3.41, whereas a1 and a2 are both equal to −0.48 at κa = 10.78. Because a1 and a2 have almost the same value, they can also be taken to be equal to obtain an easier optimization. The limitation of the method seems to be given by the numerical accuracy, especially for larger κa values, because the larger the κa, the more similar the surface and the volume integral become, whereas the mobility is given by the difference of these two integrals. Apart from this limitation, the method was tested and shown to give the proper results for the electrophoretic mobility of a sphere at low zeta potential. Having shown that on the basis of eq 12 the correct result is obtained for the electrophoretic mobility of the spheres, we now turn to cylinders. First, it was determined whether the friction coefficient for a finite cylinder as given in eqs 10 and 11 can be reproduced from eq 12 by omitting the volume integral over the electrical double layer. Following Tirado and Garcia de la Torre, the calculations were orientationally averaged and performed for ⎯ ⎯→ → closed cylinders with flat ends. For Fh (r′), we first used a function proposed by Burgers27 ⎯⎯⎯⎯→ ⎯ F ⎯→ → Fh(r′) = total (a′0 + a′2 z 2 + a′4 z4) A total (16)

(13)

⎯ ⎯⎯⎯⎯→ ⎯→ → The simplest approximation for Fh (r′) is Ftotal /Atotal with ⎯⎯⎯⎯→ Ftotal = Q·E0⃗, where Q is the total charge of the particle, E0⃗ is the applied electrical field, and Atotal is the total surface area of the particle. If sticky boundary conditions are assumed at the surface of the particle, then velocity u⃗(r)⃗ close to the surface of the particle must be equal to the velocity of the particle u⃗ at all points ⎯ ⎯→ → ⎯⎯⎯⎯→ close to the surface. However, taking Fh (r′) = Ftotal /Atotal and evaluating eq 12 will, apart from some special cases, generally not result in constant values for u⃗(r)⃗ . This means that eq 12 with an ⎯ ⎯→ → inaccurate estimate for Fh (r′) does not give a good approximation of the flow field, and the average value of u⃗(r)⃗ will generally not be a good approximation of the electrophoretic velocity. ⎯ ⎯⎯⎯⎯→ ⎯→ → An exception where Fh (r′) = Ftotal /A total is a good approximation is found if we first consider the calculation for a → ⎯ sphere and take ∇′ ϕ (r′) equal to the applied electrical field E0⃗ → ⎯ independent of r′ and take ρ(r)⃗ from linearized Poison− Boltzman theory for low surface potentials. In that case, the

where the cylinder is assumed to be oriented in the z direction with the origin of the coordinate system in the center of the 13357


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cylinder and with a′0 being adjusted such that eq 13 holds. In this way, the results from Tirado and Garcia de la Torre can be reproduced with an accuracy of about 1% for an aspect ratio of L/ D = 5 or 10. However, for larger aspect ratios, especially those used in the electrophoretic calculations described below, the polynomial expansion from eq 16 is less efficient, and the modified form ⎯⎯⎯⎯→ ⎛ ⎯ F ⎯→ → z Fh(r′) = total ⎜⎜a ′′0 + a ′′2 A total ⎝ 1/2L

Q = ψ0 2πε0εrLκa

(17)

was used, with a″2 being optimized and with a″0 being adjusted such that eq 13 holds. After these preliminary test calculations, we turn to the actual calculation of the electrophoretic mobility of a cylinder. In the calculations, the cylinder is always oriented in the z direction. Therefore, if the electrical field is in the z direction then the cylinder is oriented parallel to the field and the corresponding electrophoretic mobility μE∥ is obtained. The electrophoretic mobility of a cylinder oriented perpendicular to the electrical field, μE⊥, is always calculated with the field in the x direction. The orientationally averaged electrophoretic mobility μE is obtained from μE =

⎛ a2 ⎞ ϕ(r , φ) = −E0⎜r + ⎟cos φ r ⎠ ⎝

⎯⎯⎯⎯→ ⎛ ⎯ | Ftotal| ⎜ ⎯→ → z Fh(r′) = c0 + c 2 A total ⎜⎝ (1/2L)

K 0(κr ) K 0(κa)

with c0 fixed by eq 13. In preliminary optimizations, c1 and c′1 were found to be almost equal so that they were taken to be equal in definitive calculations. The cylinder ends are not charged (note that end effects are neglected in this respect), and the hydrodynamic force is taken as

(18)

⎯⎯⎯⎯→ ⎛ ⎯ F ⎯→ → z Fh(r′) = total ⎜⎜c0 + c 2 A total ⎝ 1/2L

(L / D) ⎞

⎯⎯⎯⎯→ ⎟ = Ftotal (c0 + c 2) ⎟ A total ⎠ (24)

Final calculations of the electrophoretic mobility μE were performed by defining a number of points along and around but always close to the surface of the cylinder and calculating u⃗(r)⃗ from eq 12 using eqs 17−24 and under the condition of eq 13. Parameters c1 = c′1, c1, and a″ are optimized by minimizing the standard deviation of u⃗(r)⃗ . After this optimization procedure, the length of the average of u⃗(r)⃗ gives electrophoretic velocities u∥ and u⊥, which gives the corresponding mobilities after dividing by E0. The number of points required to calculate u⃗(r)⃗ and their distance to the surface of the particle were varied to find a good compromise between the calculation time and accuracy. For calculations on cylinders with dimensions of the fd virus, for instance, u⃗(r)⃗ was calculated from 48 points. The condition from eq 13 was calculated analytically and checked numerically by the same integration routine used to integrate eq 12, and in the same way, the charge in the electrical double layer was integrated numerically and found that the value deviated less than 1/2000 from the charge on the particle. The numerical integrations were implemented by a repeated 3/8 Simpson rule, and the dependence on the integration step size was carefully checked and adjusted. If the integration steps that are taken are small enough, then the results of the calculation should converge as can be shown analyically and as expected, which is in agreement with the numerical results. As discussed above, the friction coefficient of a cylinder could be calculated and was found to agree well with the results from Tirado and Garcia de la Torre.29 Several

(19)

with ψ0 being the surface potential and K0 being the zeroth-order modified Bessel function of the second kind. Because we consider only 1−1 electrolytes, the charge density is obtained from ⎛ e ⎞ ρ(r ) = −FI 2000⎜ψ (r ) ⎟ ⎝ kT ⎠

⎛1 + c1 cos 2φ ⎞ ⎟⎜⎜ c′ sin 2φ ⎟⎟ ⎟⎜ 1 ⎟ ⎠⎝ ⎠ 0

(L / D) ⎞

(23)

⊥

The calculation of μE∥ is relatively easy because, following Henry and Stigter and others, we assume that the applied electrical field is constant in direction and magnitude all along the cylinder, and ⎯→ ⃗ eq 17 is used for Fh (r′). For the charge density in the electrical double layer, an approximation is made, and the result for an infinitely long cylinder is used. This means that the calculations are performed for a finite cylinder, but without a charge or double layer at the ends. For cylinders with larger aspect ratios, this will give only a relatively small error. For the hydrodynamic interactions between surface elements of the rod, no infinite rod result can be used because they are long-ranged and lead to the ln(L/D) term in eq 10, and determining whether this dependence will also be found for the electrophoretic mobility is the reason for the calculations. The electrical double layer potential ψ(r) as can be obtained from the linearized Poisson−Boltzmann equation is given by ψ (r ) = ψ0

(22)

with φ = 0 in the x direction. This nonhomogeneous electrical field around the cylinder also results in a more complicated ⎯ ⎯→ → equation for Fh (r′), where the following equation was used,

μ E + 2μ E 3

(21)

where ε0 is the vacuum permittivity, εr is the relative dielectric constant of the solvent, and K1 is the first-order modified Bessel function of the second kind. All equations needed to calculate μE∥ are now given above, but for the calculation of μE⊥, account has to be taken for the nonhomogeneous electrical field around the cylinder. For an electrical nonconducting cylinder the electrical potential of the applied field perpendicular to the cylinder in cylindrical coordinates is21

L/D ⎞

⎟⎟ ⎠

K1(κa) K 0(κa)

(20)

with e being the elementary charge, k being the Boltzmann constant, T being the absolute temperature, F being the Faraday constant, and I being the ionic strength of the solution in mol/L. ⎯⎯⎯⎯→ Furthermore, for Ftotal = Q·E0⃗ the total charge Q on the particle is needed, which is given by 13358


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⎯ ⎯→ → variations of the functions describing Fh (r′) were tried and selected. One aspect that deserves some additional attention is the position and especially the distance to the particle surface of the points at which u⃗(r)⃗ is calculated, which we will refer to as speed points hereafter. First, it is important that a speed point does not coincide with a point used to calculate the numerical integration → ⎯ because of the 1/(r − r′) term. Therefore, it can be taken in between two integration points. Taking speed points at 1.01a gives good results, but also 1.001a worked well if the integration steps were small enough. A large increase in calculation speed can be obtained if the speed points are taken inside the particle instead of outside. This may sound strange, but it turns out that if the speed at the particle surface is constant, then the same speed is found if a speed point is taken inside the particle. This can be understood as follows. Consider that we could produce by some means a constant fluid velocity on a closed surface inside a liquid with the shape of the particle but without the particle being there. In that case, one would have the same constant speed everywhere within that closed surface, and outside one would have the same flow profile as with the particle. Therefore, speed points were first taken inside the particle at 0.8a, 0.9a, or 0.95a, which has the advantage that larger integration steps can be used. Then the parameters are optimized at 48 speed points, and after this optimization, the result can be checked by analyzing one or a few speed points outside the particle. If the integration step size is carefully adjusted, then these results inside and outside always agreed well with each other. In this way, the results from the calculations given below can be performed on a normal personal computer in about a month.

determined by the difference between these two integrals, as illustrated in Figure 1. Clearly, small errors in (one of) the integrals can make a large error in the electrophoretic velocity.

Figure 1. Plot of u/usurface against κa, with u and usurface being the magnitudes of the corresponding vector quantities u⃗ and u⃗surface close to the particle, with usurface being the magnitude of the average of the surface integral from eq 12 close to the particle. Note that the retardation is given by 1 − (u/usurface), which reaches 98% at κa = 11.2; in other words, at κa = 11.2 the result from eq 12 is only a 2% difference between the two integrals. Note that this also means that the velocity of the particle u is only 2% of the velocity in the Stokes limit for the same surface charge.

Following Hunter,8 the results are expressed in terms of g(κa), which is defined as μE g (κa ) = 2ε0εrζ /3η (25)

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RESULTS AND DISCUSSION Calculations for Finite Cylinders. As discussed above, the theory from Stigter is made for infinitely long cylinders so that the theoretical results do not depend on the L/D ratio of the cylinder, whereas the rotationally averaged friction coefficient fcyl of an uncharged finite cylinder in a viscous liquid does depend on L/D as given in eq 10, which means that the results from Stigter cannot be right in the Stokes limit, where the electrophoretic mobility is μE = Q/fcyl. To check whether the discrepancies between experimental and theoretical results can be explained by the use of infinite instead of finite rods, calculations for finite cylinders were performed using the method described above. Table 1 gives results for finite cylinders with dimensions of the fd virus (i.e., L = 880 nm and D = 6.6 nm). The standard

where the zeta potential ζ is set equal to the surface potential ψ0. As can be seen, the numerical results agree well with the results according to Stigter as derived for infinite cylinders. To determine whether a ln(L/D) dependence as found for the friction coefficient of cylindrical rods is also found for the electrophoretic mobility, calculations for cylinders with an aspect ratio of 1/5 of that of fd viruses are also performed, the results of which are shown in Table 2. Here too the numerical results agree Table 2. Same as in Table 1, but Now for L = Lfd/5 (i.e., L = 176 nm)

Table 1. Simulated Electrophoretic Mobility of a Cylinder with the Dimensions of an fd Virus I (mM)

κa

g(κa) perpendicular

g(κa) parallel

g(κa) average

g(κa) average Stigter

0.1 1 10 100 1000

0.1125 0.356 1.125 3.558 11.25

0.741 0.765 0.841 1.009 1.228

1.434 1.453 1.466 1.481 1.508

0.972 0.994 1.049 1.166 1.321

1.005 1.019 1.064 1.177 1.335

I (mM)

κa


g(κa) parallel

g(κa) average


1 10 100 1000

0.356 1.125 3.558 11.25

0.736 0.814 0.969 1.135

1.331 1.381 1.412 1.429

0.934 1.003 1.117 1.233

1.019 1.064 1.177 1.335

well with the results from Stigter, where the remaining differences are within 10%, which might be attributed to end effects. For comparison, the ratio of the ln(L/D) + ν term from eq 10 of a rod with an aspect ratio of fd and a rod with an aspect ratio of 1/5 of that of fd is 1.44, which is also a 44% difference. Clearly, a comparable dependence is not found for the electrophoretic mobility. This means that the possible difference between the experimental and theoretical electrophoretic mobilities is not the result of a missing ln(L/D) dependence in the electrophoresis theory.

deviation of the x, y, and z components of u⃗(r)⃗ was typically about 1% of the largest component, apart from the calculations at κa = 11.25 where the value reached about 5%. This might be a result of the surface integral and the volume integral in eq 12 becoming more and more similar for increasing κa, whereas μE is 13359


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results shown in Tables 1 and 2. Also note the large difference between g(κa) for aspect ratios of 26.7 and 133.3 in Table 3 as compared to those in Tables 1 and 2. Surprisingly, it is shown in Figure 1 that for κa = 0.11 and L/D = 133.3 the retarded result is only a bit more than a factor of 2 different from the Stokes limit, but the result for g(κa) still differs only about 3% from the infinite cylinder result from Stigter. Apparently, for the aspect ratio dependence to show up one has to be quite close to the Stokes limit. As far as I know, these results considering the validity range of the Stokes limit have not been demonstrated before by direct calculations including full hydrodynamics. Measurements on fd Viruses. Figure 3 shows the electrophoretic mobility of wt-fd as a function of the pH at a

Showing when a ln(L/D) dependence appears in the electrophoretic mobility (as it should in the Stokes limit) can be performed with the present numerical calculations only if an exact solution for the linearized Poisson−Boltzmann equation for finite cylinders is added, which is outside the scope of the present article. Although the transition from the Stokes limit to a result that can be approximated by an infinite cylinder cannot be calculated perfectly with the present numerical calculations because of the missing charge at the end of the finite cylinders, calculations close to the Stoke limit can be performed in an accurate way because the influence of the electrical double layer is small anyway and the onset of the influence of the double layer is not expected to be influenced much by omitting only the charge at the ends of the cylinder. Therefore, to determine the range of validity of the Stokes limit, the results in Figure 1 were extended to smaller κa values to reach the Stokes limit within 2%, and the same calculations were also performed for cylinders with aspect ratios that were 0.1, 0.2, and 0.5 times that of the fd virus. The κa values at which the Stokes limit is approached at 10, 5, and 2% were determined from plots such as Figure 1 by spline interpolation, and the cumulative results are given in Figure 2. The points given

Figure 3. Electrophoretic mobility against pH at a constant ionic strength of 1 mM, compared to calculations (−) according to Stigter with Stern layers of 0, 1, and 2 nm (i.e., taking the potential at these distances from the surface equal to the zeta potential). The dashed line through the dip cell measurements made with the Zetasizer 2000 is a guide for the eyes.

constant ionic strength of 1 mM, compared to theoretical calculations according to Stigter as described above. Figure 4

Figure 2. Validity of the Stokes limit within 2, 5, and 10% plotted as a function of κa and L/D. Further details are given in the text.

for L/D = 1 are calculated from the known result for spheres,8 which fit the results of the cylinders quite well, considering that these lack charge at their ends and are not spherocylinders. The results clearly show that the value of κa alone is not enough to decide on the validity of the Stokes limit. Calculations at the smallest κa value (of 3.56 × 10−4) used to determine Figure 2, where the Stokes limit is approached to within at least 2%, were used to check for an aspect ratio dependence. The results are given in Table 3, demonstrating a clear aspect ratio dependence. Note the much larger difference between g(κa) and the result from Stigter as compared to the Table 3. Simulated Electrophoretic Mobility for a Cylinder with Different Aspect Ratios at κa = 3.56 × 10−4, Which Is within 2% of the Stokes Limit for All Aspect Ratios L/D

ln (L/D)


g(κa) parallel

g(κa) average


13.3 26.7 66.7 133.3

2.59 3.28 4.20 4.89

0.313 0.379 0.466 0.530

0.450 0.573 0.746 0.879

0.359 0.444 0.559 0.646

1.000 1.000 1.000 1.000

Figure 4. Electrophoretic mobility against ionic strength at a constant pH of 7, compared to calculations (−) according to Stigter with Stern layers of 0 and 1 nm (i.e., taking the potential at these distances from the surface equal to the zeta potential). Also shown are published results from Hoss et al.14 and Wen and Tang,30 where these results are corrected for the differences in the viscosity and dielectric constant of the solvent. 13360


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of the thickness of the Stern layer is made. This implies that the calculation of the surface potential (and charge) is probably correct, and these calculations can be used in experimental studies where the fd virus is used as a model system. As an example, using eqs 2−7 we calculate the real surface potential and an effective potential ψeff 0 that can be used with a Debye−Hückel (DH) potential distribution ψ(r)eff for low potentials such that the long-distance potential curve is equal to that of the real potential, as shown in Figure 5. In this way, the effective surface

shows the same comparison between theory and experiment as a function of the ionic strength at a constant pH of 7. Note that the shape of the theoretical curves agrees with the experimental results as long as only experimental results obtained by the same method are considered. Clearly, the theoretical results according to Stigter (solid curves) and generally all other theories for the electrophoretic mobility suffer from the problem that the width of the Stern layer is not well known. Still, we can try to make an estimate of a possible Stern layer thickness, which means theoretically taking the potential at the Stern layer distance from the surface as being equal to the zeta potential. It is known that the ends of the major coat protein on the fd virus are flexible over a length of about 2 nm.31−33 If these ends are fully stretched into the solution and the liquid in between is fully immobile, then the Stern layer could be 2 nm. However, this is an extreme case, and a Stern layer around 1 nm or smaller seems to be more realistic. In trying to compare theory and experiment for the electrophoretic mobility, surprisingly we found that the experimental results depend on the precise method of measurement. This holds for our own measurements as well as for the few published measurements where the ionic strength and pH are reasonably well known. In Figure 4, we see that the results from Hoss et al.14 correspond to our measurements on the Zetasizer 2000 using the dip cell, whereas our measurements on the Zetasizer 2000 using the more sophisticated M3 capillary cell are in reasonable agreement with our test measurements on a new Zetasizer NanoZS. Measurements of the electrophoretic mobility on a microscope using fluorescently labeled fd viruses by Wen and Tang30 gave results that are in between. However, considering a recent Malvern Instruments publication9 by Corbett and Jack in 2011 discussing the measurement accuracy of modern microelectrophoresis for the mobility of proteins, modern methods using the Zetasizer 2000 with the M3 capillary cell and the Zetasizer Nano ZS seem to give more reliable results. Still, we note that the Zetasizer 2000 gives similar results for a standard latex sample measured in the dip cell and in the M3 capillary, whereas surprisingly the results for fd viruses are not similar but deviate by almost a factor of 2. We tried to optimize the results, as described in the Experimental Section, but the reason for this discrepancy remains unclear. It might have to do with the small electrode distance used in the dip cell making the measurements more sensitive to aggregates and bubbles formed at the electrode as well as to changing electrode overpotentials. A detailed study concerning these effects is outside the scope of this work. If we now compare the experimental results measured on the Zetasizer 2000 with the M3 capillary and the Zetasizer Nano ZS to the theoretical calculations, reasonable agreement is obtained for a Stern layer of 1 nm. The slight deviation between these experiments and the 1 nm theory curve could be explained by a change in the Stern layer thickness with pH and ionic strength. The flexible ends of the coat proteins are expected to be more stretched for lower ionic strength and higher surface charge (i.e., higher pH), which would correspond to a thicker Stern layer and therefore agrees with the direction of change in the Stern layer needed to obtain even better agreement. Maybe the effect of more or less stretched out peptide chains and its change with ionic strength and pH can be described by extending the theory of Stigter in a similar way as was done for spheres.34 In summary, the most modern electrophoresis instruments give results for the electrophoretic mobility that are in good agreement with theoretical calculations if a reasonable estimate

Figure 5. Example comparing the real potential distribution from eq 6 to the low-potential version using the first part of eq 6 in combination with an effective surface potential ψeff 0 that is chosen such that the longdistance potential distribution of the two curves coincides. The vertical dashed line gives the radius of the fd virus.

potential with the much simpler DH potential distribution can be applied at a longer distance (i.e., the low-potential part of eq 6 can be used to calculate the interaction between two rods). A calculation of the real and effective surface potential at pH 7 as a function of the ionic strength is given in Figure 6. As expected, the difference between the effective and real surface potentials becomes smaller for lower surface potentials (i.e., for higher ionic strengths).

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CONCLUSIONS Modern electrophoresis instruments seem to allow for reliable measurements of the electrophoretic mobility of fd virus

Figure 6. Comparison of the real surface potential ψ0 with the effective surface potential ψeff 0 as a function of the ionic strength at pH 7. 13361


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(11) Semenov, I.; Papadopoulos, P.; Stober, G.; Kremer, F. Ionic Concentration- and pH-Dependent Electrophoretic Mobility As Studied by Single Colloid Electrophoresis. J. Phys.: Condens. Matter 2010, 22, 494109. (12) Sambrook, J.; Russell, D. W. Molecular Cloning: A Laboratory Manual, 3rd ed.; Cold Spring Harbor Laboratory Press: New York, 2001. (13) Berkowitz, S. A.; Day, L. A. Mass, Length, Composition and Structure of Filamentous Bacterial Virus fd. J. Mol. Biol. 1976, 102, 531− 547. (14) Hoss, U.; Batzill, S.; Deggelmann, M.; Graf, C.; Hagenbuchle, M.; Johner, C.; Kramer, H.; Martin, C.; Overbeck, E.; Weber, R. Electrokinetic Properties of Aqueous Suspensions of Rodlike fd VirusParticles in the Gas-Like and Liquid-Like Phase. Macromolecules 1994, 27, 3429−3431. (15) Kramer, H.; Graf, C.; Hagenbuchle, M.; Johner, C.; Martin, C.; Schwind, P.; Weber, R. Electrooptic Effects of Aqueous fd-Virus Suspensions at Very-Low Ionic-Strength. J. Phys. II 1994, 4, 1061−1074. (16) Åkerlöf, G. Dielectric Constants of Some Organic Solvent-Water Mixtures at Various Temperatures. J. Am. Chem. Soc. 1932, 54, 4125− 4139. (17) Wohlfarth, C. W. B. Viscosity of Pure Organic Liquids and Binary Liquid Mixtures. In Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology, New Series; Lechner, W. B., Ed.; Springer: Berlin, Vol. IV/18. (18) Zimmermann, K.; Hagedorn, H.; Heuck, C. C.; Hinrichsen, M.; Ludwig, H. The Ionic Properties of the Filamentous Bacteriophages Pf1 and Fd. J. Biol. Chem. 1986, 261, 1653−1655. (19) Lopez-Garcia, J. J.; Horno, J.; Grosse, C. Poisson-Boltzmann Description of the Electrical Double Layer Including Ion Size Effects. Langmuir 2011, 27, 13970−13974. (20) Tirado, M. M.; Martinez, C. L.; Garcia de la Torre, J. Comparison of Theories for the Translational and Rotational Diffusion-Coefficients of Rod-Like Macromolecules: Application to Short DNA Fragments. J. Chem. Phys. 1984, 81, 2047−2052. (21) Henry, D. C. The Cataphoresis of Suspended Particles Part I: The Equation of Cataphoresis. Proc. R. Soc. London, Ser. A 1931, 133, 106− 129. (22) Allison, S. A. Boundary Element Modeling of Biomolecular Transport. Biophys. Chem. 2001, 93, 197−213. (23) Xin, Y.; Mitchell, H.; Cameron, H.; Allison, S. A. Modeling the Electrophoretic Mobility and Diffusion of Weakly Charged Peptides. J. Phys. Chem. B 2006, 110, 1038−1045. (24) Allison, S. A. Numerical Solution of the Nonlinear Poisson− Boltzmann Equation for a Macroion Modeled As an Array of Non Overlapping Beads. J. Phys. Chem. B 2011, 115, 4872−4879. (25) Allison, S.; Chen, C. Y.; Stigter, D. The Length Dependence of Translational Diffusion, Free Solution Electrophoretic Mobility, and Electrophoretic Tether Force of Rigid Rod-Like Model Duplex DNA. Biophys. J. 2001, 81, 2558−2568. (26) Frank, S.; Winkler, R. G. Polyelectrolyte Electrophoresis: Field Effects and Hydrodynamic Interactions. Eur. Phys. Lett. 2008, 83, 38004. (27) Burgers, J. M. On the Motion of Small Particles of Elongated Form Suspended in a Viscous Liquid. Verhandelingen der Koninklijke Nederlandse Akademie van Wetenschappen, Afd. Natuurkunde, Sec. I 1938, 16, 113. (28) Dhont, J. K. G. An Introduction to Dynamics of Colloids; Elsevier: Amsterdam, 1996. (29) Tirado, M. M.; Garcia de la Torre, J. Translational Friction Coefficients of Rigid, Symmetric Top Macromolecules: Application to Circular-Cylinders. J. Chem. Phys. 1979, 71, 2581−2587. (30) Wen, Q.; Tang, J. X. Absence of Charge Inversion on Rodlike Polyelectrolytes with Excess Divalent Counterions. J. Chem. Phys. 2004, 121, 12666−12670. (31) Bhattacharjee, S.; Glucksman, M. J.; Makowski, L. Structural Polymorphism Correlated to Surface-Charge in Filamentous Bacteriophages. Biophys. J. 1992, 61, 725−735. (32) Zeri, A. C.; Mesleh, M. F.; Nevzorov, A. A.; Opella, S. J. Structure of the Coat Protein in fd Filamentous Bacteriophage Particles

solutions at low concentration (0.1 mg/mL) that compare well with theoretical predictions according to Stigter in combination with a surface ionization model, where the Stern layer thickness is the only adjustable parameter. Nevertheless, although not precisely known, even for the Stern layer a reasonable estimate can be made on the basis of independent experimental NMR results concerning the surface structure of the fd viruses. In addition, the approximation in the theory of Stigter to describe long but finite rods by an infinitely long cylinder was validated. More specifically, although the electrophoretic mobility of a cylinder in the limit of low ionic strength depends on the aspect ratio much more than end effects, no aspect ratio dependence exceeding end effects is found for long rods at moderate to high ionic strengths, as could be demonstrated by more elaborate numerical calculations for finite rods. These calculations for finite cylinders were performed at low surface potential for rods with the dimensions of the fd virus (with L/D = 133) as well as for rods with L/D = 27 and were compared to the result according to Stigter, with the result that no difference exceeding end effects (i.e., no ln(L/D) dependence) is observed. Therefore, the infinite cylinder model is found to be a good approximation of the calculation of the electrophoretic mobility of long but finite cylinders at moderate to high ionic strengths. Furthermore, the range of validity of the Stokes regime was systematically calculated by using the calculations for finite cylinders.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS I thank J. K. G. Dhont, G. Nägele, and P. R. Lang for useful as well as stimulating discussions and A. Csiszar for help with the measurements on the Malvern Nano ZS.

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REFERENCES

(1) Stigter, D. Charged Colloidal Cylinder with a Gouy Double-Layer. J. Colloid Interface Sci. 1975, 53, 296−306. (2) Stigter, D. Electrophoresis of Highly Charged Colloidal Cylinders in Univalent Salt Solutions. 1. Mobility in Transverse Field. J. Phys. Chem. 1978, 82, 1417−1423. (3) Stigter, D. Electrophoresis of Highly Charged Colloidal Cylinders in Univalent Salt-Solutions. 2. Random Orientation in External Field and Application to Polyelectrolytes. J. Phys. Chem. 1978, 82, 1424− 1429. (4) Stigter, D. Theory of Conductance of Colloidal Electrolytes in Univalent Salt Solutions. J. Phys. Chem. 1979, 83, 1663−1670. (5) Dogic, Z.; Fraden, S. Ordered Phases of Filamentous Viruses. Curr. Opin. Colloid Interface Sci. 2006, 11, 47−55. (6) Zhang, Z.; Buitenhuis, J.; Cukkemane, A.; Brocker, M.; Bott, M.; Dhont, J. K. G. Charge Reversal of the Rodlike Colloidal fd Virus through Surface Chemical Modification. Langmuir 2010, 26, 10593− 10599. (7) Kang, K.; Dhont, J. K. G. Electric-Field Induced Transitions in Suspensions of Charged Colloidal Rods. Soft Matter 2010, 6, 273−286. (8) Hunter, R. J. Zeta Potential in Colloid Science: Principles and Applications; Academic Press: London, 1981. (9) Corbett, J. C. W.; Jack, R. O. Measuring Protein Mobility Using Modern Microelectrophoresis. Colloids Surf., A 2011, 376, 31−41. (10) Delgado, A. V.; Gonzalez-Caballero, F.; Hunter, R. J.; Koopal, L. K.; Lyklema, J. Measurement and interpretation of Electrokinetic Phenomena. J. Colloid Interface Sci. 2007, 309, 194−224. 13362


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Determined by Solid-State NMR Spectroscopy. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 6458−6463. (33) Wang, Y. A.; Yu, X.; Overman, S.; Tsuboi, M.; Thomas, G. J.; Egelman, E. H. The Structure of a Filamentous Bacteriophage. J. Mol. Biol. 2006, 361, 209−215. (34) Hill, R. J.; Saville, D. A. 'Exact' Solutions of the Full Electrokinetic Model for Soft Spherical Colloids: Electrophoretic Mobility. Colloids Surf., A 2005, 267, 31−49.

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