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The effect of many-body interactions on the electrostatic force in an array of spherical particles
Journal of Colloid and Interface Science 346 (2010) 232–235
Contents lists available at ScienceDirect
Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
The effect of many-body interactions on the electrostatic force in an array of spherical particles Ghassan F. Hassan, Adel O. Sharif *, Ugur Tuzun, Andy Tate University of Surrey, Faculty of Engineering and Physical Sciences, Department of Chemical and Process Engineering, Centre for Osmosis Research and Applications, Guildford, Surrey GU2 7XH, UK
a r t i c l e
i n f o
Article history: Received 9 October 2009 Accepted 16 January 2010 Available online 1 March 2010 Keywords: Colloidal particle Electrostatic force Multi-body interactions Non-linear Poisson–Boltzmann equation Finite element analysis
a b s t r a c t The effect of many-body interactions on the electrostatic force between spheres in an array of charged spherical particles has been quantified by solving the non-linear Poisson–Boltzmann equation (PBE) using a finite element method (FEM) model. The equation is solved for conditions of constant surface potential. The effect of the dimensionless Debye length scaled radius of the spheres on the electrostatic force between them has been determined and a significant reduction of the force is observed as the dimensionless Debye radius is decreased. Calculations based on the dimensionless separation distance between a pair of charged spheres in isolation confirm an exponential decay of the repulsive electrostatic force as the surface to surface separation distance is increased. In contrast, calculations based on an array of interacting spheres in certain critical packing conditions reveal considerable reduction in the magnitude of the repulsive electrostatic force at separation distances less than half the sphere radius. The latter results are explained by considering the directional cancellation effects of the electrical double layer between adjacent spheres placed in certain orientations within the array. This is believed to be a surprising many-body effect which has been overlooked in previous studies and whose validity can be used to explain the stability and strength of charged sphere arrays under certain critical geometric configurations. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction It has been recognized that electrostatic interaction and hydrophobic/hydrophilic interaction between membranes and fouling materials have a significant bearing on membrane fouling. This is particularly true to more difficult fouling problems caused by adsorption of natural organic matters and biopolymers on the membrane. The balance between the forces of electrostatic repulsion and hydrophobic adhesion determines the outcomes of membrane fouling, as well as the efficiency of chemical cleaning [1]. Controlling these processes is not merely related to the membrane separate material size alone. Electrostatic interactions between the membrane and the solutes to be separated can be extremely important for these processes. Furthermore, it’s possible to control the importance of such interactions through careful selection of processing conditions, in particular pH, ionic strength and operation pressure [2]. This paper investigates the effect of multi-body interactions on the electrostatic force between any two spheres in an array of charged spheres with constant surface potential conditions, and the effect of the separation distance reduction on the behaviour
* Corresponding author. Fax: +44 (0) 1483 686581. E-mail address: [email protected] (A.O. Sharif). 0021-9797/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2010.01.053
of these systems. The non-linear Poisson–Boltzmann equation here is solved using Finite Element Methods (FEM); the resulting electrical potential distribution at the charged surfaces is then used to calculate the free energy of interaction of approaching electrical double layers and the resulting electrostatic force. Similar analyses have been used in models with hybrid Discrete Element (DEM) and Finite Element (FE) force interaction algorithms to explain the effects of capillary forces [3,4] between partially-wetted solid particles and cemented contacts in soils [5]. A common feature of the ‘‘superimposition” of the ‘‘attractive” and ‘‘repulsive” force interactions acting jointly on discrete surfaces is that the magnitude of the aggregate force potentials experienced by an assembly of particles is heavily dependent on the variations of particle surface separations within the assembly; i.e. the packing geometry. The main objective of the present study is to quantify the effect of many-body interactions on the electrostatic force between particles, in an array of charged particles, in a 1:1 symmetrical electrolyte system. Taking advantage of symmetry allows a simple geometry to be used. 2. Model and problem description The Poisson–Boltzmann equation is a second-order elliptic partial differential equation which describes the electrostatic
G.F. Hassan et al. / Journal of Colloid and Interface Science 346 (2010) 232–235
233
Nomenclature local concentration of ions of species i in bulk solution (kmol m3) relative concentration of ion species iðni =IÞ electronic charge (1.602 1019 C) dimensionless electrostatic repulsive force dimensional repulsive force (N) total free energy around a colloidal particle (J) dimensionless free energy ionic strength of the electrolyte (kmol m3) Boltzmann constant (1.381 1023 J K1) Avogadro’s number (6.022 1023 mol1) axial coordinates (m)
ci Ci e F* F G G* I k NA x, y, z
potential around a fixed charge distribution in an ionic solution. There are three components of the solvated system that we must consider in order to accurately model the electrostatic potential: the solute molecule, the solvent, and the solvated ions. The solute molecule is modelled as a dielectric continuum of low polarizability embedded in a dielectric medium of high polarizability which represents the solvent. The solvated ions surrounding the molecule are also modelled as a continuum; distributed according to the Boltzmann distribution. Combining Poisson’s equation, used to describe the electrostatic behaviour of point charges in the dielectric continuum, with the Boltzmann charge distribution for the solvated ions gives the non-linear Poisson–Boltzmann equation (NPBE). The normalized non-linear Poisson–Boltzmann Eq. (1) for a three-dimensional electrostatic double layer in Cartesian coordinates is solved to calculate the potential distribution:
@2W @X
2
þ
@2W @Y
2
þ
@2W @Z 2
¼ r
ð1Þ
In this equation; W is the dimensionless space potential, r is the dimensionless space charge density, X, Y and Z are the dimensionless Cartesian coordinates, which are defined as following:
X ¼ jx; Y ¼ jy; Z ¼ jz; 1X r¼ zi C i expðzi WÞ 2
W ¼ eW=kT
and
where x, y and z are the dimensional coordinates, e is the elementary electric charge, w is space potential, k is Boltzmann constant, T is the absolute temperature, Ci = ci/I is the relative concentration P 2 (I ¼ 12 zi ci is the ionic strength of the electrolyte) and j is the Debye parameter which has the following form:
j¼
2 P 2 1=2 2 1=2 e zi c i 2e NA I ¼ ekT ekT
X, Y, Z zi
j e w W Ws Wtube
r
dimensionless coordinates (X ¼ jx), (Y ¼ jy), (Z ¼ jz) charge number of ion species i debye parameter (m1) permittivity of the electrolyte (=e0 er F m1), where e0 is the vacuum permittivity (8.854 1012 F m1), and er is the relative permittivity electric potential (V) reduced potential (W ¼ ew=kT) reduced potential at sphere surface reduced potential at tube surface dimensionless space charge density
Fig. 1. Array of spheres (primitive cubic).
This array system can be imagined as a big number of smaller groupings (clusters) of seven spheres (one in the middle surrounded by six spheres). The spheres clusters are of a symmetrical arrangement as can be seen in Fig. 2 which shows that one cluster of particles can be reasonably assumed to represent the whole system. The cube in the previous figure resembles the symmetry between spheres inside a cluster. Each cube surface represents a mid-plane between two adjacent spheres. This leads to Fig. 3 which show the basic solution domain of the present problem. In order to solve Eq. (1), it is necessary to define the potential W and the potential gradient in the outward normal direction @ W/@N at the boundary C in the case of constant surface potential. N = jn is the outward normal direction from the boundary C. Fig. 3 defines the main geometry of interest in the present paper. ABCD,
ð2Þ
where NA is Avogadro’s number, and e is the permittivity of the electrolyte. In the present study, the problem under consideration is that of an array of spherical charged particles, for which the equation is solved for constant reduced potential W = Ws on the spheres surfaces. Normal cubic packing system was considered here as it is already possible to implement with the meshing code, where each sphere is surrounded by six spheres with equal separation distance between each pair. Other packing systems would be implemented in future studies. Fig. 1 shows a system of array of spheres with such arrangement, with the spheres close to each other.
Fig. 2. Symmetrical nature of sphere’s cluster arrangement.
G.F. Hassan et al. / Journal of Colloid and Interface Science 346 (2010) 232–235
F
Dimensionless Electrostatic Force
234
B
E
A I
G
H
C
e
2
F
ð3Þ
where the dimensionless force, F* for a monovalent electrolyte system between two identical spheres has been estimated on the midplane over a mid-plane using the following form [6]:
F ¼
Z 0
1
Z
1
3.5
S.A.M.W.
3.0 2.5 2.0 1.5 1.0 0.5 0.0 0
½2ðcosh W 1Þ þ jDWj2 dX dY
1
2
3
4
5
ð4Þ
0
where X, Y are the axial coordinates spanning the mid-plane, Z is the coordinate normal to the mid-plane, all are scaled by Debye length j1, and the integrand is evaluated at the mid-plane Z = L/2, where L is the centre-line distance between the sphere surfaces. 3. Results and discussion Eq. (1) is solved numerically to obtain the potential distribution. A finite element method combined with error estimation and the Newton sequence technique is used for the non-linear terms [2]. The accuracy of the FE solution has been tested against previous numerical results [7] for constant surface potential as can be seen in Fig. 4. The process is driven by a global error estimate. Thus, the process terminates when a specified target error has been reached [8]. In the present work, a global target error of 1% has been chosen. Computed results for electrostatic effects have been analyzed in terms of electrostatic repulsive force on charged spheres at various distances from each other for constant surface potential conditions. Results have then been obtained for different reduced sphere potentials 1.0 and 2.0 for the same set of geometries. With Ws = 1.0, Fig. 5 compares the results of the array of spheres with
Dimensionless Electrostatic Force
Fig. 4. Results comparison with those of Sharif et al. [7] in terms of dimensionless electrostatic force between two isolated spheres (reduced surface potential = 2, reduced sphere radius = 1).
1.2
Two Isolated Spheres
1.0
Array of Spheres Two Spheres in Tube
0.8 0.6 0.4 0.2 0.0 0
1
2
3
4
5
6
Dimensionless Separation Distance Fig. 5. Comparison of results with different sphere array systems (reduced surface potential = 1, reduced sphere radius = 1).
the results obtained for systems of two isolated spheres, and two spheres in a charged tube (Wtube = 1.0) with tube confinement set to equate the separation distance at evaluation points to emulate array arrangement. The integration surface was chosen to be the mid-plane between the adjacent spheres in those cases. Fig. 6 shows similar comparison with Ws = 2.0 and Wtube = 2.0. It can be noted that at relatively larger distances, the electrostatic force is higher for the case of the array than that of two spheres, which means that the dispersed system would be more stable than what the two isolated sphere results can predict. The results of the arrays of spheres systems demonstrate a different behaviour from those of the two adjacent spheres in isolation: When the separation distance between particles is sufficiently reduced (i.e. <0.5 sphere radius), then the repulsive electrostatic force is forced to fall below the ‘peak’ electrostatic force value. This behaviour can be attributed to a ‘cancelling’ effect
Dimensionless Electrostatic Force
F¼
2
Present Results
D
BCGF, EFGH, ADHE, ABFE and CDHG are mid-planes of symmetry between the central sphere and each of the surrounding six spheres. Constant electric potentials for the spheres are considered with W = Ws on the boundary I (which refers to the sphere surface). The non-dimensional parameters of relevance are the reduced potential W, the dimensionless particle radius ja, and the dimensionless separation distance between spheres L. Eq. (1) is a non-linear partial differential equation with two spatial variables solved numerically using the FEM combined with error estimation [2]. Once the potential distribution is obtained electrostatic force can be calculated. The electrostatic force acting on a sphere can be calculated by integrating the Maxwell stress tensor over a suitable surface. The electrostatic interaction force F, between each two identical spheres, acting along their line-of-centres, is obtained by integrating the stress tensor over the mid-plane [2] to end up with:
4.0
Dimensionless Surface-Surface Distance
Fig. 3. Basic solution domain.
e kT
4.5
5.0
Two Isolated Spheres
4.5
Array of Spheres
4.0
Two Spheres in Tube
3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0
1
2
3
4
5
6
Dimensionless Separation Distance Fig. 6. Comparison of results with different sphere array systems (reduced surface potential = 2, reduced sphere radius = 1).
Dimensionless Electrostatic Force
G.F. Hassan et al. / Journal of Colloid and Interface Science 346 (2010) 232–235
0.8
the repulsive electrostatic force potential is observed in Figs. 7 and 8 to move to smaller surface separations as the sphere radius is allowed to increase. This can be attributed to the fact that thickness of the electrical double-layer will shrink closer to the particle as normalized sphere radius increases when electrolyte concentration increases [9].
a=0.1
0.7
a=0.5
0.6
a=1.0
0.5 0.4 0.3
4. Conclusions
0.2 0.1 0.0 0
0.5
1
1.5
2
2.5
3
3.5
4
Dimensionless Separation Distance
Fig. 7. Electrostatic Force results between array of spheres for dimensionless radius values 0.1, 0.5, and 1.0 (reduced surface potential = 1).
Dimensionless Electrostatic Force
235
3.5
a=0.1
3.0
a=0.5
2.5
a=1.0
2.0 1.5 1.0 0.5 0.0 0
0.5
1
1.5
2
2.5
3
3.5
4
Dimensionless Separation Distance
Fig. 8. Electrostatic Force results between array of spheres for dimensionless radius values 0.1, 0.5, and 1.0 (reduced surface potential = 2).
that results from getting the spheres at two axes to minimal separation distances, which affects the electrical double layer of the spheres involved, and give rise to effective force components in different directions; resulting in reduction of the electrostatic force between adjacent spheres. This is further confirmed with the case of two charged spheres in a charged tube with varying confinement, which shows a similar behaviour albeit with smaller force values due to more effective ‘charge confinement’ for them. This behaviour of the array of spheres system is further demonstrated in Figs. 7 and 8, with a reduced sphere potential values Ws = 1 and 2, respectively, with dimensionless sphere radius values ja = 0.1, 0.5, and 1. In those figures, we can notice a maximum value for the electrostatic force on the curve, with falling values next to it. The peak of
The effect of the many-body electrostatic interactions has been quantified for a concentrated system of array of spheres with constant surface potential and constant surface charge density conditions. Results are expressed in terms of dimensionless electrostatic force between each pair of charged spheres. For constant surface potential, the results demonstrate a reduction in the repulsive force in arrays of interacting spheres in certain critical packing conditions when the separation distance between them becomes small enough that ‘canceling’ effect can take place among the spheres in different axes. This effect can be explained by considering the directional ‘cancellation’ effects of the electrical double layer between adjacent spheres placed in certain orientations, which results in reduction of the electrostatic force between adjacent spheres. Results shows that a two spheres model is not an accurate representative of multi-body systems; as it over-predicts electrostatic force at short separation distances, and under-predicts it at long distances. In such a system of charged spheres, a reduction in the repulsive force has been observed at shorter separation distances, which contradicts the two sphere results for similar surface conditions. The multi-body nature of a system is an important part that has crucial effects that should not be ignored. References [1] Charles Liu, Scott Caothien, Jennifer Hayes, Tom Caothuy, Takehiko Otoyo, Takashi Ogawa, Membrane Chemical Cleaning: From Art to Science, 1999. [2] W.R. Bowen, A.O. Sharif, Journal of Colloid and Interface Science 187 (1997) 363–374. [3] T. Groger, U. Tuzun, D.M. Heyes, Powder Technology 133 (2003) 203–218. [4] N. Israelachvili Jacob, Intermolecular and Surface Forces, second ed., Academic Press, 1992. [5] C.O. Sullivan, J.D. Bray, M.F. Reimer, Journal of Engineering Mechanics 128 (2002) 1182–1192. [6] F.G. Hasan, Finite Element Analysis of Multibody Interactions in Membranes and Colloidal Systems, Thesis, University of Surrey, 2009. [7] A.O. Sharif, M.H. Afshar, J. Moghadasi, T.J. Williams, Powder Technology 135– 136 (2003) 76–81. [8] A.O. Sharif, Z. Tabatabaian, W.R. Bowen, Journal of Colloid and Interface Science 255 (2002) 138–144. [9] R.J. Hunter, Foundation of Colloid Science, Oxford Science Publication, 2001.
Contents lists available at ScienceDirect
Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
The effect of many-body interactions on the electrostatic force in an array of spherical particles Ghassan F. Hassan, Adel O. Sharif *, Ugur Tuzun, Andy Tate University of Surrey, Faculty of Engineering and Physical Sciences, Department of Chemical and Process Engineering, Centre for Osmosis Research and Applications, Guildford, Surrey GU2 7XH, UK
a r t i c l e
i n f o
Article history: Received 9 October 2009 Accepted 16 January 2010 Available online 1 March 2010 Keywords: Colloidal particle Electrostatic force Multi-body interactions Non-linear Poisson–Boltzmann equation Finite element analysis
a b s t r a c t The effect of many-body interactions on the electrostatic force between spheres in an array of charged spherical particles has been quantified by solving the non-linear Poisson–Boltzmann equation (PBE) using a finite element method (FEM) model. The equation is solved for conditions of constant surface potential. The effect of the dimensionless Debye length scaled radius of the spheres on the electrostatic force between them has been determined and a significant reduction of the force is observed as the dimensionless Debye radius is decreased. Calculations based on the dimensionless separation distance between a pair of charged spheres in isolation confirm an exponential decay of the repulsive electrostatic force as the surface to surface separation distance is increased. In contrast, calculations based on an array of interacting spheres in certain critical packing conditions reveal considerable reduction in the magnitude of the repulsive electrostatic force at separation distances less than half the sphere radius. The latter results are explained by considering the directional cancellation effects of the electrical double layer between adjacent spheres placed in certain orientations within the array. This is believed to be a surprising many-body effect which has been overlooked in previous studies and whose validity can be used to explain the stability and strength of charged sphere arrays under certain critical geometric configurations. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction It has been recognized that electrostatic interaction and hydrophobic/hydrophilic interaction between membranes and fouling materials have a significant bearing on membrane fouling. This is particularly true to more difficult fouling problems caused by adsorption of natural organic matters and biopolymers on the membrane. The balance between the forces of electrostatic repulsion and hydrophobic adhesion determines the outcomes of membrane fouling, as well as the efficiency of chemical cleaning [1]. Controlling these processes is not merely related to the membrane separate material size alone. Electrostatic interactions between the membrane and the solutes to be separated can be extremely important for these processes. Furthermore, it’s possible to control the importance of such interactions through careful selection of processing conditions, in particular pH, ionic strength and operation pressure [2]. This paper investigates the effect of multi-body interactions on the electrostatic force between any two spheres in an array of charged spheres with constant surface potential conditions, and the effect of the separation distance reduction on the behaviour
* Corresponding author. Fax: +44 (0) 1483 686581. E-mail address: [email protected] (A.O. Sharif). 0021-9797/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2010.01.053
of these systems. The non-linear Poisson–Boltzmann equation here is solved using Finite Element Methods (FEM); the resulting electrical potential distribution at the charged surfaces is then used to calculate the free energy of interaction of approaching electrical double layers and the resulting electrostatic force. Similar analyses have been used in models with hybrid Discrete Element (DEM) and Finite Element (FE) force interaction algorithms to explain the effects of capillary forces [3,4] between partially-wetted solid particles and cemented contacts in soils [5]. A common feature of the ‘‘superimposition” of the ‘‘attractive” and ‘‘repulsive” force interactions acting jointly on discrete surfaces is that the magnitude of the aggregate force potentials experienced by an assembly of particles is heavily dependent on the variations of particle surface separations within the assembly; i.e. the packing geometry. The main objective of the present study is to quantify the effect of many-body interactions on the electrostatic force between particles, in an array of charged particles, in a 1:1 symmetrical electrolyte system. Taking advantage of symmetry allows a simple geometry to be used. 2. Model and problem description The Poisson–Boltzmann equation is a second-order elliptic partial differential equation which describes the electrostatic
G.F. Hassan et al. / Journal of Colloid and Interface Science 346 (2010) 232–235
233
Nomenclature local concentration of ions of species i in bulk solution (kmol m3) relative concentration of ion species iðni =IÞ electronic charge (1.602 1019 C) dimensionless electrostatic repulsive force dimensional repulsive force (N) total free energy around a colloidal particle (J) dimensionless free energy ionic strength of the electrolyte (kmol m3) Boltzmann constant (1.381 1023 J K1) Avogadro’s number (6.022 1023 mol1) axial coordinates (m)
ci Ci e F* F G G* I k NA x, y, z
potential around a fixed charge distribution in an ionic solution. There are three components of the solvated system that we must consider in order to accurately model the electrostatic potential: the solute molecule, the solvent, and the solvated ions. The solute molecule is modelled as a dielectric continuum of low polarizability embedded in a dielectric medium of high polarizability which represents the solvent. The solvated ions surrounding the molecule are also modelled as a continuum; distributed according to the Boltzmann distribution. Combining Poisson’s equation, used to describe the electrostatic behaviour of point charges in the dielectric continuum, with the Boltzmann charge distribution for the solvated ions gives the non-linear Poisson–Boltzmann equation (NPBE). The normalized non-linear Poisson–Boltzmann Eq. (1) for a three-dimensional electrostatic double layer in Cartesian coordinates is solved to calculate the potential distribution:
@2W @X
2
þ
@2W @Y
2
þ
@2W @Z 2
¼ r
ð1Þ
In this equation; W is the dimensionless space potential, r is the dimensionless space charge density, X, Y and Z are the dimensionless Cartesian coordinates, which are defined as following:
X ¼ jx; Y ¼ jy; Z ¼ jz; 1X r¼ zi C i expðzi WÞ 2
W ¼ eW=kT
and
where x, y and z are the dimensional coordinates, e is the elementary electric charge, w is space potential, k is Boltzmann constant, T is the absolute temperature, Ci = ci/I is the relative concentration P 2 (I ¼ 12 zi ci is the ionic strength of the electrolyte) and j is the Debye parameter which has the following form:
j¼
2 P 2 1=2 2 1=2 e zi c i 2e NA I ¼ ekT ekT
X, Y, Z zi
j e w W Ws Wtube
r
dimensionless coordinates (X ¼ jx), (Y ¼ jy), (Z ¼ jz) charge number of ion species i debye parameter (m1) permittivity of the electrolyte (=e0 er F m1), where e0 is the vacuum permittivity (8.854 1012 F m1), and er is the relative permittivity electric potential (V) reduced potential (W ¼ ew=kT) reduced potential at sphere surface reduced potential at tube surface dimensionless space charge density
Fig. 1. Array of spheres (primitive cubic).
This array system can be imagined as a big number of smaller groupings (clusters) of seven spheres (one in the middle surrounded by six spheres). The spheres clusters are of a symmetrical arrangement as can be seen in Fig. 2 which shows that one cluster of particles can be reasonably assumed to represent the whole system. The cube in the previous figure resembles the symmetry between spheres inside a cluster. Each cube surface represents a mid-plane between two adjacent spheres. This leads to Fig. 3 which show the basic solution domain of the present problem. In order to solve Eq. (1), it is necessary to define the potential W and the potential gradient in the outward normal direction @ W/@N at the boundary C in the case of constant surface potential. N = jn is the outward normal direction from the boundary C. Fig. 3 defines the main geometry of interest in the present paper. ABCD,
ð2Þ
where NA is Avogadro’s number, and e is the permittivity of the electrolyte. In the present study, the problem under consideration is that of an array of spherical charged particles, for which the equation is solved for constant reduced potential W = Ws on the spheres surfaces. Normal cubic packing system was considered here as it is already possible to implement with the meshing code, where each sphere is surrounded by six spheres with equal separation distance between each pair. Other packing systems would be implemented in future studies. Fig. 1 shows a system of array of spheres with such arrangement, with the spheres close to each other.
Fig. 2. Symmetrical nature of sphere’s cluster arrangement.
G.F. Hassan et al. / Journal of Colloid and Interface Science 346 (2010) 232–235
F
Dimensionless Electrostatic Force
234
B
E
A I
G
H
C
e
2
F
ð3Þ
where the dimensionless force, F* for a monovalent electrolyte system between two identical spheres has been estimated on the midplane over a mid-plane using the following form [6]:
F ¼
Z 0
1
Z
1
3.5
S.A.M.W.
3.0 2.5 2.0 1.5 1.0 0.5 0.0 0
½2ðcosh W 1Þ þ jDWj2 dX dY
1
2
3
4
5
ð4Þ
0
where X, Y are the axial coordinates spanning the mid-plane, Z is the coordinate normal to the mid-plane, all are scaled by Debye length j1, and the integrand is evaluated at the mid-plane Z = L/2, where L is the centre-line distance between the sphere surfaces. 3. Results and discussion Eq. (1) is solved numerically to obtain the potential distribution. A finite element method combined with error estimation and the Newton sequence technique is used for the non-linear terms [2]. The accuracy of the FE solution has been tested against previous numerical results [7] for constant surface potential as can be seen in Fig. 4. The process is driven by a global error estimate. Thus, the process terminates when a specified target error has been reached [8]. In the present work, a global target error of 1% has been chosen. Computed results for electrostatic effects have been analyzed in terms of electrostatic repulsive force on charged spheres at various distances from each other for constant surface potential conditions. Results have then been obtained for different reduced sphere potentials 1.0 and 2.0 for the same set of geometries. With Ws = 1.0, Fig. 5 compares the results of the array of spheres with
Dimensionless Electrostatic Force
Fig. 4. Results comparison with those of Sharif et al. [7] in terms of dimensionless electrostatic force between two isolated spheres (reduced surface potential = 2, reduced sphere radius = 1).
1.2
Two Isolated Spheres
1.0
Array of Spheres Two Spheres in Tube
0.8 0.6 0.4 0.2 0.0 0
1
2
3
4
5
6
Dimensionless Separation Distance Fig. 5. Comparison of results with different sphere array systems (reduced surface potential = 1, reduced sphere radius = 1).
the results obtained for systems of two isolated spheres, and two spheres in a charged tube (Wtube = 1.0) with tube confinement set to equate the separation distance at evaluation points to emulate array arrangement. The integration surface was chosen to be the mid-plane between the adjacent spheres in those cases. Fig. 6 shows similar comparison with Ws = 2.0 and Wtube = 2.0. It can be noted that at relatively larger distances, the electrostatic force is higher for the case of the array than that of two spheres, which means that the dispersed system would be more stable than what the two isolated sphere results can predict. The results of the arrays of spheres systems demonstrate a different behaviour from those of the two adjacent spheres in isolation: When the separation distance between particles is sufficiently reduced (i.e. <0.5 sphere radius), then the repulsive electrostatic force is forced to fall below the ‘peak’ electrostatic force value. This behaviour can be attributed to a ‘cancelling’ effect
Dimensionless Electrostatic Force
F¼
2
Present Results
D
BCGF, EFGH, ADHE, ABFE and CDHG are mid-planes of symmetry between the central sphere and each of the surrounding six spheres. Constant electric potentials for the spheres are considered with W = Ws on the boundary I (which refers to the sphere surface). The non-dimensional parameters of relevance are the reduced potential W, the dimensionless particle radius ja, and the dimensionless separation distance between spheres L. Eq. (1) is a non-linear partial differential equation with two spatial variables solved numerically using the FEM combined with error estimation [2]. Once the potential distribution is obtained electrostatic force can be calculated. The electrostatic force acting on a sphere can be calculated by integrating the Maxwell stress tensor over a suitable surface. The electrostatic interaction force F, between each two identical spheres, acting along their line-of-centres, is obtained by integrating the stress tensor over the mid-plane [2] to end up with:
4.0
Dimensionless Surface-Surface Distance
Fig. 3. Basic solution domain.
e kT
4.5
5.0
Two Isolated Spheres
4.5
Array of Spheres
4.0
Two Spheres in Tube
3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0
1
2
3
4
5
6
Dimensionless Separation Distance Fig. 6. Comparison of results with different sphere array systems (reduced surface potential = 2, reduced sphere radius = 1).
Dimensionless Electrostatic Force
G.F. Hassan et al. / Journal of Colloid and Interface Science 346 (2010) 232–235
0.8
the repulsive electrostatic force potential is observed in Figs. 7 and 8 to move to smaller surface separations as the sphere radius is allowed to increase. This can be attributed to the fact that thickness of the electrical double-layer will shrink closer to the particle as normalized sphere radius increases when electrolyte concentration increases [9].
a=0.1
0.7
a=0.5
0.6
a=1.0
0.5 0.4 0.3
4. Conclusions
0.2 0.1 0.0 0
0.5
1
1.5
2
2.5
3
3.5
4
Dimensionless Separation Distance
Fig. 7. Electrostatic Force results between array of spheres for dimensionless radius values 0.1, 0.5, and 1.0 (reduced surface potential = 1).
Dimensionless Electrostatic Force
235
3.5
a=0.1
3.0
a=0.5
2.5
a=1.0
2.0 1.5 1.0 0.5 0.0 0
0.5
1
1.5
2
2.5
3
3.5
4
Dimensionless Separation Distance
Fig. 8. Electrostatic Force results between array of spheres for dimensionless radius values 0.1, 0.5, and 1.0 (reduced surface potential = 2).
that results from getting the spheres at two axes to minimal separation distances, which affects the electrical double layer of the spheres involved, and give rise to effective force components in different directions; resulting in reduction of the electrostatic force between adjacent spheres. This is further confirmed with the case of two charged spheres in a charged tube with varying confinement, which shows a similar behaviour albeit with smaller force values due to more effective ‘charge confinement’ for them. This behaviour of the array of spheres system is further demonstrated in Figs. 7 and 8, with a reduced sphere potential values Ws = 1 and 2, respectively, with dimensionless sphere radius values ja = 0.1, 0.5, and 1. In those figures, we can notice a maximum value for the electrostatic force on the curve, with falling values next to it. The peak of
The effect of the many-body electrostatic interactions has been quantified for a concentrated system of array of spheres with constant surface potential and constant surface charge density conditions. Results are expressed in terms of dimensionless electrostatic force between each pair of charged spheres. For constant surface potential, the results demonstrate a reduction in the repulsive force in arrays of interacting spheres in certain critical packing conditions when the separation distance between them becomes small enough that ‘canceling’ effect can take place among the spheres in different axes. This effect can be explained by considering the directional ‘cancellation’ effects of the electrical double layer between adjacent spheres placed in certain orientations, which results in reduction of the electrostatic force between adjacent spheres. Results shows that a two spheres model is not an accurate representative of multi-body systems; as it over-predicts electrostatic force at short separation distances, and under-predicts it at long distances. In such a system of charged spheres, a reduction in the repulsive force has been observed at shorter separation distances, which contradicts the two sphere results for similar surface conditions. The multi-body nature of a system is an important part that has crucial effects that should not be ignored. References [1] Charles Liu, Scott Caothien, Jennifer Hayes, Tom Caothuy, Takehiko Otoyo, Takashi Ogawa, Membrane Chemical Cleaning: From Art to Science, 1999. [2] W.R. Bowen, A.O. Sharif, Journal of Colloid and Interface Science 187 (1997) 363–374. [3] T. Groger, U. Tuzun, D.M. Heyes, Powder Technology 133 (2003) 203–218. [4] N. Israelachvili Jacob, Intermolecular and Surface Forces, second ed., Academic Press, 1992. [5] C.O. Sullivan, J.D. Bray, M.F. Reimer, Journal of Engineering Mechanics 128 (2002) 1182–1192. [6] F.G. Hasan, Finite Element Analysis of Multibody Interactions in Membranes and Colloidal Systems, Thesis, University of Surrey, 2009. [7] A.O. Sharif, M.H. Afshar, J. Moghadasi, T.J. Williams, Powder Technology 135– 136 (2003) 76–81. [8] A.O. Sharif, Z. Tabatabaian, W.R. Bowen, Journal of Colloid and Interface Science 255 (2002) 138–144. [9] R.J. Hunter, Foundation of Colloid Science, Oxford Science Publication, 2001.