# Influence of pH and Compression on Electrohydrodynamic Effects in Nanoporous Packed Beds

Authors: Bastian Schaefer; Hermann Nirschl
Journal: Physical Separation in Science and Engineering
 (2009)

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## Abstract

Fluid flow and charge transport in fine structures can be driven both by pressure gradients and by electric fields if electrochemical double layers are present on the surfaces. The interrelated electrohydrodynamic effects may be used to drive liquids without moving parts, for example, in dewatering or in electroosmotic chromatography, or to generate small electric currents. While the electrohydrodynamic transport is well understood for simple geometries, models for porous structures are complex. Furthermore, the interconnected porous structure of a packed bed itself strongly depends on the electrochemical double layers. In this study, the electrohydrodynamic transport in packed beds consisting of boehmite particles with an average diameter of 38 nm is investigated. We describe a new approach to the electrokinetic effects by treating the packed beds as theoretical sets of cylindrical capillaries. The charge transport and the electrically driven fluid flow predicted with this model agree well with experimental results. Furthermore, the hydraulic permeability was found to be a nonlinear function of the porosity, independent of whether the porosity change is caused by changing the compression or the electrochemical double layer.


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## Body

## 1. Introduction

The packed beds investigated in this study are porous structures consisting of randomly distributed, individual particles compressed between two membranes in a round tube. An increasing compression reduces the size of the pores between the particles and thus the porosity.  Generally, an electric current is driven by an electric gradient and a fluid flow is driven by a hydraulic gradient. However, in confined geometries with charged surfaces, hydraulic and electric effects are interrelated since the driving gradient shears off the diffuse part of the electrochemical double layer (EDL) on the surface. If the EDL is sheared off by a pressure-driven flow, the dragged ions constitute a streaming current. Likewise, if the ions are accelerated by an electric field, they drag the adjacent water molecules along and the electroosmotic flow results. This interrelation of fluid flow and charge transport thus depends on the surface charge of the particles, the ion distribution in the EDL, and on the pore structure of the packed bed [1–3].While relations for this electrohydrodynamic transport (EHT) are well established for simple geometries, examining EHT in a packed bed is complicated because of the complex pore structure. Further, the pore structure itself also depends on the same parameters as the surface charge and the ion distribution in the EDL, that is, the pH value and the ionic strength of the liquid [1, 4]. Many researchers describe EHT with elaborate spherical cell models, for which the continuity and stokes equation are solved (see [5] for a recent review). Others solve the Poisson-Boltzmann, continuity, and Stokes equations for different pore geometries assuming negligible surface conductivity [6] or apply molecular dynamics simulations [7]. Due to their complexity, these models are not suitable for practical applications. In this study, we propose a new method for analyzing the EHT. The packed beds are considered as theoretical sets of cylindrical capillaries with an experimentally determined geometry.The EHT in packed beds is an interesting field of research, that also has commercial applications; packed beds are permeated in many processes such as filtration or solid phase extraction, but the high-flow resistance limits the hydraulic permeation if the particles are small [8, 9]. Electric fields can help to increase the flow [1, 10]. Likewise, electroosmotic micropumps with porous substrates can drive liquids without needing any moving parts, for example, on microfluidic chips [11, 12]. The results are also relevant for understanding separator membranes in fuel cells and lithium-ion batteries [13, 14].


## 2. Theoretical Background

Fluid flow and charge transport in nanoporous packed beds interact with each other giving reason to the term EHT. The interrelation is caused by the excess of mobile counter ions in the EDL, a diffuse cloud of ions attracted by the particles’ surface charge. The surface charge depends on the pH value of the suspension because it originates from the dissociation reactions of the particles surface groups. It is positive at low pH values and negative at high pH values. The pH value where the surface charge is zero is called isoelectric point (IEP). A detailed description of the origin and the structure of the EDL can be found in [1]. The particle charge is often characterized by the zeta potential  ζ, which is the potential difference at the shear plane when the EDL is partly sheared off by a relative motion between the particle and the surrounding liquid. The zeta potential is proportional to the cross coefficients (C12  and C21   in (1) and (2)), which correlate the volumetric flow V˙   and the electric current I   to the driving gradients of pressure p  and electric potential  Ψ. The exponential decay length of the ion concentration in the diffuse part of the EDL, the Debye length κ−1, is inversely proportional to  the square root of the ionic strength. In this study, with a Debye length κ−1   of about 3 nm (at an ionic strength of 0.01 mol/L and a temperature of 288 K) and an average particle diameter a   of about 38 nm, Smoluchowski’s assumption of a thin EDL (κa≫1) is valid. Accordingly, the potential gradients may be considered as one-dimensional, so that the EHT can be expressed as(1)V˙=AL⋅(−C11⋅Δp−C12⋅ΔΨ),(2)I=AL⋅(−C21⋅Δp−C22⋅ΔΨ), with the thickness of the packed
bed L and the cross-section area of the packed bed A [2, 15].The coefficientsC11 to C22 depend on the particle charge and on the pore
structure of the packed bed, which is influenced by the formation of
agglomerates in the suspension from which the packed bed was formed. The
stability of suspensions against agglomeration is described by the DLVO theory [1, 4, 8]. A packed bed formed from a stable suspension has a dense structure with
a homogeneous pore size distribution and a low permeability. In contrast, an
agglomerated suspension results in a loosely textured packed bed with an
inhomogeneous pore size distribution. The large pores between the agglomerates
are accountable for the high permeability [1, 16].The pore size also depends on the
compression of the packed bed. Increasing the compression mainly reduces the
size of the large pores between the agglomerates [16, 17]. Additionally, when
the packed bed is permeated, a drag force acts on every particle; the contact
force between the particles increases along the direction of the flow since
each particle rests on top of another. Consequently, the contact force at the
downstream side equals the contact force at the upstream side plus the pressure
force that drives the flow [17], but this effect is neglected in this study as
the total compression is much higher than the pressure difference which drives
the liquid. Assuming a homogeneous compression, the porosity Φ is determined from the mass mS and the density ρS of the particles: (3)Φ=(L⋅A−mS/ρS)L⋅A. The porosity has a strong influence
on the EHT. The following paragraph discusses the meaning and the physical
properties of the four coefficients C11 to C22.(i) The flow rateV˙ of a liquid driven by the pressure
difference Δp depends on the cross section area A and the thickness L of
the packed bed [18] (4)V˙=−C11⋅AL⋅Δp. The coefficient C11 is
the hydraulic permeability as originally defined by Darcy [18]. It has
been largely investigated for macroporous packed beds with a homogeneous pore
size; briefly, a smaller pore size, as resulting from a smaller particle size
or from a lower porosity, causes a lower permeability [19, 20]. Today, the
permeability of a packed bed KDarcy is usually defined as the product of C11 and the viscosity of the liquid η [21], but this seems to be problematic when
dealing with charged particles, as is shown in the following.(ii) Electroosmotic
flow is driven by an externally applied electric potential differenceΔΨ if the counter-ions in the EDL of the
particles are accelerated by the electric field and drag the nearby water
molecules along [2, 3]: (5)V˙=−C12⋅AL⋅ΔΨ. Since electroosmosis is well understood
for cylindrical capillaries, we propose to consider the pores of the packed bed
as a theoretical set of cylindrical capillaries with the length LCap,
and the total cross-section area of all capillaries ACap [2]: (6)V˙=−C12,Cap⋅ACapLCap⋅ΔΨ. The electroosmosis coefficient C12,Cap includes the relative permittivity of the
liquid εL and the vacuum permittivity ε0: (7)C12,Cap=−εLε0ζη. The relation between ACap/LCap and A/L is addressed in the paragraph on the electric
conductivity C22.(iii) The streaming currentIStr is the charge transport created when
counter-ions are dragged from the EDL by a pressure-driven liquid flow.
The streaming current coefficient C21 equals the electroosmosis coefficient C12 because of Onsager’s relation [2, 3]. As
above, we propose to treat the porous structure as a theoretical set of
cylindrical capillaries to make the EHT easier to understand:   (8)IStr=−C21⋅AL⋅Δp,(9)IStr=−C21,Cap⋅ACapLCap⋅Δp,with the streaming
current coefficientC21,Cap.
The streaming current could be measured if the inlet area of the structure was
connected to the outlet area with a low-resistance ampere meter (short-circuit
conditions) [2]. However, it is not investigated in this study since collecting
the ions on the large cross-section area of the packed bed with point
electrodes is inaccurate.If the faces are
not connected or connected with a high-resistance voltage meter, the streaming
current is balanced by a charge transport against the pressure-driven flow, the
so-called counter current. An electric potential, the streaming potentialΔΨStr,
builds up to drive the counter current against the electric resistance of the
packed bed. The streaming potential may be calculated from (2) if the electric
current I is set to zero [2]. The streaming potential can also be expressed in terms of the capillary
geometry with the streaming current coefficient for a capillary and the
capillary conductivity  C22,Cap   as introduced in the following:   (10)ΔΨStr=−C21C22⋅Δp,(11)ΔΨStr=−C21,CapC22,Cap⋅Δp.  Measuring the
streaming potential between
an upstream and a downstream electrode allows the determination of the zeta
potential ζ. It is important to make a correction for surface
conduction (see what
follows), since it reduces the potential needed to induce the counter
current [1, 2].The streaming
potential causes an electroosmotic backflow against the pressure gradient, as
can be seen whenΔΨStr is inserted into (1): (12)V˙reduced=−(C11−C12⋅C21C22)⋅ALΔp.This so-called
electroviscous effect appears as an increased apparent viscosityη* of the liquid. Imagine a
packed bed that is hydraulically permeated while the faces are short circuited,
so that no streaming potential and no electroosmotic backflow evolve. If the
short circuiting is now cut, the flow rate is reduced by the electroviscosity
ratio (13)η*η=C11C11−(C12⋅C21/C22).(iv) The electric currentI driven by the electric potential
difference ΔΨ depends on the electric conductivity C22 of the porous structure, which is
inversely proportional to the ohmic resistance  ROhm.
Considering the packed bed as a theoretical set of cylindrical capillaries
removes the influence of the pore structure: (14)I=−C22⋅AL⋅ΔΨ=−ΔΨROhm,(15)I=−C22,Cap⋅ACapLCap⋅ΔΨ. If the capillary
conductivity C22,Cap is known, it is possible to calculate the
geometry factor which relates the capillary geometry to the geometry of the
packed bed as   (16)ACap⋅LA⋅LCap=C22C22,Cap. If surface
conduction was negligible, this approach would be analogous to Archie’s law. He
found out that the conductivity of a porous system is a function of the liquid conductivity KL and the pore structure [22], but in the
presence of an EDL, the capillary conductivity C22,Cap also comprises surface conductivity Kσ (by movement of ions in the immobile part of
the EDL). Kσ is related to the particle diameter a to account for the specific surface area [23, 24]   (17)C22,Cap=KL+4⋅Kσ/a. The dimensionless
Dukhin number  Du
is the ratio of surface conduction to liquid
conduction [2, 23]   (18)Du=4⋅Kσa⋅KL. Surface conduction
must not be neglected for a correct interpretation of electrohydrodynamics
since it causes a “short-circuiting” behind the shear plane of the EDL and thus
does not contribute to the electroosmotic flow [2]. For a rigid structure, the
geometry factor may be determined by filling the pores with a high electrolyte
solution and measuring the total conductivity C22;
if liquid conduction is much higher than surface conduction, C22,Cap equals approximately KL.
However, the technique is not applicable to packed beds since the increase of
ionic strength influences the packed bed’s structure substantially. Another
approach is to determine the capillary conductivity C22,Cap from the electroosmotic drag V˙/I,
that relates the electroosmotic flow to the electric current  [15]. The electroosmotic drag may be
expressed as the flow rate per coulomb or as the number of water molecules that
are dragged per number of transported protons (19)C22,Cap=−εL⋅ε0⋅ζη⋅V˙/I.As fluid flow and charge transport
in porous structures are closely interrelated, it is difficult to examine them
separately. Especially, the charge transport mechanism has a strong influence
on the electrohydrodynamic transport. The separation between surface and liquid
conduction permits to access the electroosmotic flow and the streaming current
with a capillary model.


## 3. Experiments

The electro-compression-permeability-cell
(ECP-Cell) (see Figure1) is designed for the compression and permeation of
packed beds. These consist of ceramic particles compressed between two membranes
by applying a force to the plunger. The two-sided filtration on a cross-section
area of 0.005 m2 provides an even
compression and thus homogeneous structure of the packed bed. After reaching a
constant thickness of approximately 3 mm to 8 mm (as measured with a position
encoder at the piston), the packed bed is cooled to 288 K by circulating
electrolyte solution through the heat exchangers. The solution has the same
ionic strength and pH value as the suspension from which the packed bed is
formed.Figure 1
Electro-compression-permeability-cell.Applying pressures of 20 and 40 kPa
to the liquid in the fluid reservoir, the flow rate and the streaming potential
are measured with the scale and the measuring electrodes. These point
electrodes, made from platinum, are directly in contact with the membranes.
They are suitable to measure a voltage, but not an electric current. The
voltmeter has an electric resistance of 10 MΩ.Subsequently, an electric voltage
is applied to the large-area working electrodes to drive electric currents of
10, 20, 50, and 100 mA. The electrodes consist of titanium that is coated with
iridium oxide to prevent oxidation. Reversing the electric field every 30 seconds
prevents the formation of electrolytic gas; the electrodes take up hydrogen
during the cathode cycle and release it during the anode cycle. Electrolytic
gas would be seen on the scale as an apparent mass flow, since the gas would
displace the water from the cooling circuit. The driving electric field is
measured with the measuring electrodes because the electric resistance of the
liquid in the cooling circuit reduces the effective voltage. The electric
current is measured at the working electrodes. All parts of the ECP-Cell except
for the electrodes are nonconducting.For each pH value a new packed bed
is formed. The suspensions are prepared by dispersing 25 g of boehmite
particles (Disperal from Sasol, Hamburg, Germany) in 225 g potassium nitrate
solution with an ionic strength of 0.01 M and pH 4. At a low pH value the
particles are easily dispersed by stirring with an agitator (F 520 75 from
Krups, Offenbach, Germany) at 800 rpm for 30 minutes. During this time, the pH
value is constantly readjusted by adding nitric acid. Subsequently, the final
pH value is adjusted by adding 1 M nitric acid or caustic potash while stirring
with a horseshoe mixer. This mixer reaches the bottom as well as the wall of
the beaker and has a helix form to evoke axial transport. A homogenous mixing
and shear rate during the flocculation are critical because the flocculated
suspensions are pseudoplastic. Using a propeller mixer would lead to an
inhomogeneous shear stress and thus to an inhomogeneous distribution of the
agglomerate size and the viscosity. The pH value of the suspension is
constantly readjusted during 45 minutes of stirring.Disperal was chosen for the
experiments because it is easy to disperse at low pH values. The primary
particles have a narrow-size distribution (x0,10=28nm, x0,50=38nm,
and x0,90=56nm as measured with a Nanotrac from Mircotrac
Inc., Pa, USA).
Boehmite has an IEP of 9.5 and the zeta potential is +61 mV at pH 4, +44 mV at pH
7, +22 mV at pH 8.5, +3 mV at pH 9.5,  −13 mV at pH 10, and −29 mV at pH 12. Disperal
is commonly used as a filler material in synthetics and as a raw material for
the aluminium oxide industry. Boehmite is dissolvable in aqueous solutions,
especially at high and low pH values. However, at temperatures below 50°C the
dissolution is so slow that it may be neglected [25]. The nylon membranes (Ultipor
from Pall, NY, USA)
have a nominal pore size of 0.1 μm. The particles being smaller than the pores
of the membrane are retarded because the pores get clogged.


## 4. Results

The gradients, the fluid flow, and
the electric current are defined in the direction of the pressure-driven flow.
With the upper working electrode defined as the anode, a negative electric
potential difference means that the electric potential decreases along this
direction. Protons move toward the lower working electrode to reduce their
potential causing an electric current in the same direction.Both porosityΦ and permeability C11 (calculated with (3) and (4)) strongly depend
on the pH value of the suspension (see Figure 2). The high permeability around
the IEP (at a pH of about 9.5) is caused by the agglomeration of the particles.
Above the IEP, the porosity and the permeability do not decrease because the electrostatic
repulsion is too weak to desagglomerate the particles. A substantial change of
porosity and permeability occurs between pH 8.5 and pH 4, where the
electrostatic repulsion between the charged particles prevents
agglomeration.  An increase of the pH value
(for example from 4 to 8.5 at 1 bar) causes a small increase of the porosity (by
a factor of 1.26, but a large increase of permeability (by a factor of 7.10). This
indicates a shrinking of the large pores between the agglomerates, which are
accountable for the major part of the fluid transport [7, 20]. Since the flow
is laminar, the average velocity is higher in larger pores, so that they have a
disproportionately high contribution to the fluid flow.Figure 2
PorosityΦ   (full
lines, left axis) and permeability C11 (dashed lines, right axis) at
different levels of compression.PorosityΦ and permeability C11 also decrease upon an increase of compression.
The porosity change is about the same in both compression steps (from 1 bar to
2 bar and from 2 bar to 4 bar). The difference of the permeabilities at 1 bar
and at 2 bar is bigger than the difference between 2 bar and 4 bar. Again, this
indicates that first the largest pores between the agglomerates are narrowed,
later also the smaller pores are affected.Figure3 shows the liquid
conductivity KL (full lines) and the conductivity of the
packed beds C22 (dashed lines) as functions of the pH value.
The liquid conductivity is nearly constant over a wide range of the pH and has a
maximum at the upper end, coming from abundant oxonium ions. The conductivity
of the packed bed C22 is calculated using (14). It depends on the pH
value and on the compression. A higher compression causes a higher volume
fraction of the nonconducting particles and thus a lower conductivity. This
also explains the low conductivity at low pH values. Another reason for the packed bed conductivity deviating from the liquid conductivity is the strong
contribution of surface conductivity, which is indicated by the high Dukhin number (see Figure 5).Figure 3
Liquid conductivityKL (full lines) and conductivity of the packed beds C22 (dashed lines)
at different levels of compression. The liquid conductivity does not change
with compression.The electroosmotic dragV˙/I,
that is the quotient of the electroosmotic flow rate to the driving electric
current, is negative below the IEP and positive above (see Figure 4, left axis,
full lines). The quotients have minima at a pH of around 7, which can be
explained with the relatively high zeta potential and relatively low Dukhin
number (see Figure 5), which means that surface conduction is low.  Surface conduction does not drag water
molecules along and thus does not evoke electroosmotic flow.Figure 4
Electroosmotic dragV˙/I (full lines, left axis) and geometry factor (ACap⋅L)/(A⋅LCap) (dashed lines, right axis) at different levels
of compression.Figure 5
Capillary
conductivityC22,Cap (full lines, left axis) and Dukhin number Du (dashed
lines, right axis) at different levels of compression.The geometry factor relatesACap/LCap to A/L (see Figure 4, right axis, dashed lines). It
is calculated with (16) and (19) using the measured electroosmotic drag and the
zeta potential of the particles as measured with the AcoustoSizer II. The
geometry factor is smaller than 1 for all experiments because the path through
the packed bed is tortuous and thus makes the capillaries longer and thinner
than the packed bed. An increased compression makes the pores even more
tortuous and thus reduces the geometry factor. The geometry factor has a
maximum at the IEP, where the particles are strongly agglomerated.Figure5 shows the capillary
conductivity C22,Cap and the Dukhin number Du depending on the pH value. The capillary
conductivity has a minimum around the IEP, because of the zero particle charge and the low liquid conductivity. The Dukhin
number correlates to the capillary conductivity in the medium pH range. The
deviation at high pH values is caused by the high liquid conductivity. An
increased compression leads to higher capillary conductivity C22,Cap and Dukhin number. Surface conduction is
favored by a closer contact between the particles and thus stronger overlap of
the EDL. This is not contradicted by the decreasing conductivity of the packed
bed C22 upon an increase of compression (see Figure 3).
While the surface conductivity and thus the capillary conductivity increase;
this effect is dominated by the decreasing geometry factor. The relatively high
Dukhin number Du is the reason for the minima of the
electroosmotic drag at pH 7 (see Figure 4, left axis, full lines). Charge
transport behind the shear plane of the EDL does not drag water molecules, so
it does not contribute to the electroosmotic flow.Figure6 (full lines) shows the
electroosmosis coefficient C12 as calculated with (5), (6), and (7) and with
the geometry factor. It correlates well with the measurements (Figure 6, dashed
lines). Below the IEP the surface charge is positive, so that the counter ions
in the diffuse layer are negative. If an electric field in the positive
direction is applied, the negative counter ions are accelerated toward the
positive electrode, which is at the upper side of the packed bed. The resulting
flow is consequently negative. Accordingly, C12 is negative below the IEP and positive above.
The electroosmotic flow is stronger at a low compression, as can be explained
with the higher geometry factor and an identical C12,Cap in (6). The minima at pH 7 can be explained by
the relatively high zeta potential and the relatively high geometry factor. Due
to Onsager’s relation, the streaming current coefficient C21 equals the electroosmosis coefficient C12,
but no experimental values exist for C21 since the streaming current cannot be measured with the point electrodes in the ECP-Cell.Figure 6
Electroosmosis
coefficientC12 as calculated (full lines) and measured (dashed
lines) at different levels of compression.Knowing the capillary conductivityC22,Cap,
the quotient of the streaming potential and the pressure gradient ΔΨ/Δp is calculated from (11) (see Figure 7, full
lines). The quotient is positive below the IEP and negative above, which is
opposite to the electroosmosis coefficient C12.
At low pH values, the counter ions sheared off by the flow (negative pressure
gradient) have a negative surface charge, so negative ions will accumulate at
the downstream side, resulting in a negative electric gradient. A stronger
compression leads to a smaller modulus of the quotient because of the
characteristic functions of C12 and C22.
This is also why the modulus of the quotient is smaller above the IEP than
below. The calculated values agree well with the measurements (Figure 7, dashed
lines). One has to consider the experimental errors of the streaming potential
measurements caused by concentration polarization of the electrodes.Figure 7
Streaming potentialΔΨ related to pressure difference Δp as calculated (full lines) and measured (dashed
lines) at different levels of compression.The electroviscosity ratioη*/η has a minimum at the IEP, where it is close to
the theoretical value of 1, meaning that the apparent viscosity equals the real
viscosity of the liquid in the pores (see Figure 8). An increase of the surface
charge leads to an increase of the electroviscosity ratio because the pressure
driven flow is retarded by the electroosmotic counter flow driven by the
streaming potential. Above the IEP, the electroviscosity ratio is smaller than below because of the
increased porosity. A higher porosity leads to larger pores and a smaller part
of the flow taking place within the EDL. Consequently, the electroviscosity
ratio increases with increasing compression.
Since the increased apparent viscosity would have to be
considered for the calculation of the permeability KDarcy we use C11 instead of KDarcy.Figure 8
Electroviscosity
ratioη*/η for different levels of compression. The error
bars represent the 95% confidence interval.


## 5. Summary

The EHT in nanoporous packed beds
depends on the surface charge of the particles as well as on the pore
structure, which itself depends on the surface charge. A capillary model is
introduced to separate the two influences, using an experimentally determined
geometry factor. The calculated charge transport and electrically driven fluid
flow agree well with experimental results. The model explains why the EHT
coefficients can decrease with increasing modulus of surface charge if the
porosity decreases substantially.The hydraulic permeability,
however, is better described as a nonlinear relation to the porosity. The
direct influence of the pH value on the hydraulic permeability is in the range
of a few percent.The present study is limited to the
range of Smoluchowski’s thin double layer approximation, which is fulfilled in many
technical applications of packed beds. Here, the focus is on the influence of
the pH value and the compression. The effect of changing the ionic strength and
the particle size will be addressed in the future.



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*Source: 817135-2009-03-24.xml*