RESEARCH PAPER
a
b
PCCP
F. J. Arroyo,*a F. Carrique,b S. Ahuallia and A. V. Delgadoa
www.rsc.org/pccp
Dynamic mobility of concentrated suspensions. Comparison between different calculationsy
Departamento de Fı´sica Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain. E-mail:
[email protected]; Fax: þ34 958 243214; Tel: þ34 958 246103 Departamento de Fı´sica Aplicada I, Facultad de Ciencias, Universidad de Ma´laga, 29071 Ma´laga, Spain. E-mail:
[email protected]; Fax: þ34 952 132382; Tel: þ34 952 131923
Received 14th October 2003, Accepted 25th February 2004 F|rst published as an Advance Article on the web 10th March 2004
One of the fields where electroacoustic techniques (electrokinetic sonic amplitude, ESA, and colloid vibration current, CVI, are presently available) are expected to be most useful is concentrated suspensions. Here, other electrokinetic techniques, linked to the observation of individual particles, show great limitations. In spite of this, the problem of electroacoustics in concentrated suspensions is far from being fully resolved. In this work, the method considered is ESA, where the quantity of interest is the dynamic or ac mobility of the particles, ud*. Our aim is to compare in a systematic way two procedures for the calculation of ud* in concentrated suspensions: One is based upon the use of a cell model, often used in electrokinetics of concentrated systems, and the other is an analytical formula elaborated by O’Brien and coworkers taking explicitly into account particle–particle interactions. On the average both methods seem to describe properly the frequency dependence of ud*, at least up to volume fractions of solids of the order of 40%. Maxima and minima are sometimes found that can be explained through consideration of electrical double layer relaxations (alpha and, mainly, Maxwell–Wagner–O’Konski) affecting the strength of the dipole moment induced by the external field. As to the effect of ka (a: particle radius; k: reciprocal double layer thickness) on ud*, it is found that the analytical formula, in spite of being a ‘‘ large ka ’’ model can be used with confidence down to ka 10. The two methods can also predict the reduction in |ud*| upon increasing volume fraction, but only the cell model suggests that the phase angle of |ud*| goes to zero when the particle-to-particle distance is very much reduced (highest volume fractions). It appears that the phase angle can be very sensitive to the different ways in which the methods used account for the interactions between neighbouring particles, as no significant differences are found when the effect of the volume fraction (or the zeta potential) on |ud*| are estimated using any of the models.
DOI: 10.1039/b312839c
1. Introduction
1446
Electroacoustics is a branch of electrokinetics that has gained great interest in the last 20 years, although electroacoustic phenomena were first studied by Debye as early as 1933.1,2 Such interest can be justified because new electrokinetic techniques have become available that are based on such phenomena, and that have been a sort of driving force in the elaboration of new theoretical approaches. There are basically two such techniques. The first consists of the generation of a pressure wave when an alternating electric field is applied to the suspension. It is called the electrokinetic sonic amplitude (ESA effect). The second one is the reciprocal of the ESA phenomenon: a pressure wave of a suitable frequency produces an ac field in the suspension. It is called colloid vibration potential (CVP) or current (CVI). Although there had been a number of previous works, particularly those by Booth and Enderby,3,4 it is widely admitted that O’Brien5,6 set out the foundations of a theory of electroacoustic phenomena. According to his derivation, a (complex) quantity called the dynamic electrophoretic mobility, ud*, plays a fundamental role in electroacoustic phenomena, as both ESA and CVI can be related to it. This is a consequence of the linear theory of non-equilibrium thermodynamics, that establishes a reciprocal relationship between ESA and CVI. y Presented at the 17th Conference of the European Colloid & Interface Science Society, Firenze, Italy, September 21–26, 2003. Phys. Chem. Chem. Phys., 2004, 6, 1446–1452
If the suspension is subjected to an electric field E and a pressure gradient Hp simultaneously, then the velocity of the particles, v, and the average current density through the suspension, h ji will depend linearly on both E and Hp: v ¼ aHp þ ud E; h ji ¼ bHp þ K E; ð1Þ where a and b are transport properties of the suspension, and K* is its complex electrical conductivity. From Onsager’s reciprocity relationship, b and ud* will be proportional, so that the dynamic mobility is essential in the value of the colloid vibration current. In an ESA experiment, the amplitude of the pressure wave generated by application of an ac field of frequency o is proportional to both ud* and the so-called density contrast, Dr ¼ rp rm , where rp (rm) is the particle (medium) density. As before, ud* is crucial for the determination of the ESA signal. It must be mentioned that other authors use a different approach. In particular, Dukhin et al.7,8 stress that Onsager reciprocity relationships are a consequence of time reversibility in the classical equations, and as such, they are strictly valid in the stationary case (i.e., constant applied fields, o ! 0). Accordingly, they propose a theory for CVI based on a completely different approach in which ud* is absent. This is not a central issue of the present work. Hence we pursue our main objective of the calculation of ud*. The point now is that the great advantage of electroacoustic techniques over other classical electrokinetic methods is that they can, in principle, be applied to colloidal suspensions of any particle
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concentration f (f is the volume fraction of solids), since they are based on the determination of a collective average response, and no individual particles need to be tracked. However, this implies to know the f-dependence of ud* (or ESA, or CVI), in order to extract the zeta potential, surface conductivity, or any other parameters characterizing the electrical state of the interface. This is in fact our final target. Again, there are two independent approaches both of which contribute to enrich our understanding of electroacoustics. O’Brien and coworkers use a ‘‘ first-principles ’’ approach; in ref. 9, Rider and O’Brien calculated ud* for a pair of particles in mutual interaction, thus extending the range of validity of their previous calculations to volume fractions 10%. Later, O’Brien et al.10 gave a more complete account of particle– particle interactions assuming that only nearest-neighbour interactions take place; their formula gave excellent results for near-neutrally buoyant particles (Dr 0). More recently, O’Brien et al.11 found a formula for ud* that they claim to be valid for high volume fractions and arbitrary Dr. The expression is11,12 u d ¼
In this paper, we show our own calculations of ud* using a cell model and explicitly compare them to the above-mentioned formula, based on the calculation of many-particle interactions. Our estimations are valid for any zeta potential, this requiring a numerical integration of the governing equations. In a forthcoming paper, these calculations will be checked against ESA experimental data.
2. Fundamental equations Only a brief survey of the set of differential equations and boundary conditions governing the problem of calculating ud* for a concentrated suspension of spheres will be given. More details can be found in refs. 19, 20. The electric potential C(r,t) depends on the electric charge density rel(r,t) at position r and time t according to
em z ð2 2fÞ 3fðF 1Þ ð2l2 =ð3 þ 3l þ l2 ÞÞ ; Zð2 þ fÞ 1 þ ðDr=rm ÞffF þ ð2l2 =3ð3 þ 3l þ l2 ÞÞg ð2Þ
where em and Z are, respectively, the permittivity and dynamic viscosity of the medium, z is the zeta potential, and F and l are defined as: 2 1 þ ð4l2 I þ f1 þ 2lg expð2lÞÞJ 2 ; F¼ 3 2 expðlÞ ; J¼ 1 þ l þ ðl2 =3Þ Z1 I¼ ½gðrÞ 1r expð2lrÞdr; 1
rffiffiffiffiffiffiffiffiffiffi oa2 ; l ¼ ð1 þ iÞ 2n
ð3Þ
In these expressions, o is the angular frequency, a is the particle radius, and n is the kinematic viscosity. The pair distribution function, g(r), is calculated according to the Percus– Yevick formula.13 According to the authors, eqn. (2) is valid if the following conditions are met: (i) The double layer is thin, i.e., ka 1, with k the reciprocal Debye length. (ii) The zeta potential is low (
