Soft Matter View Article Online
Published on 06 October 2014. Downloaded by University of California - San Francisco on 27/01/2015 19:53:24.
PAPER
View Journal | View Issue
Cite this: Soft Matter, 2014, 10, 8864
Influence of polyelectrolyte shape on its sedimentation behavior: effect of relaxation electric field† Pin-Hua Yeh,a Jyh-Ping Hsu*a and Shiojenn Tseng*b The sedimentation of an isolated, charged polyelectrolyte (PE) subjected to an applied field is modeled theoretically, taking into account the variation of its shape. In particular, the effects of double-layer relaxation, effective charge density, and strength of the induced relaxation electric field are examined. We show that the interaction of these effects yields complex and interesting sedimentation behaviors. For example, the behavior of the electric force acting on a loosely structured PE can be different from that on a compactly structured one; the former is dominated mainly by the convective fluid flow. For thick double layers, electric force has a local maximum as the Reynolds number varies, but tends to increase monotonically with increasing Reynolds number if the layer is thin. The drag factor is found to
Received 23rd June 2014 Accepted 8th September 2014
behave differently from literature results. The shape of a PE significantly influences its sedimentation behavior by affecting the amount of counterions attracted to its interior and the associated local electric
DOI: 10.1039/c4sm01351d
field. Interestingly, a more stretched PE has a higher effective charge density but experiences a weaker
www.rsc.org/softmatter
electric force.
1. Introduction Particles of colloidal size are capable of settling in a dispersion medium due to the presence of gravitational force and/or other applied forces. Settling has been applied successfully in many operations, including wastewater treatment, biological processes, mining, and space propellant reignition, to name a few. If a particle is charged and the dispersion medium contains ionic species, the associated phenomena can be complex and interesting. This is because the motion of a charged particle induces a local electric eld, the nearby ionic distribution and the associated Reynolds number are inuenced, and so are the hydrodynamic and electric forces acting on the particle. Several attempts have been made for modeling the settling of a charged colloidal particle in an electrolyte medium. Lee et al.,1 for example, analyzed the sedimentation potential of a concentrated spherical colloidal suspension. They showed that if the surface potential is low (high), the ratio (scaled sedimentation potential/scaled sedimentation velocity) has a local minimum (maximum) as the double-layer thickness varies. Chih et al.2 found that if the surface potential is low, the scaled sedimentation velocity has a local maximum for a medium-
a
Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617. E-mail: [email protected]; Fax: +886-2-23623040; Tel: +886-2-23637448
b
Department of Mathematics, Tamkang University, Tamsui, Taipei, Taiwan 25137. E-mail: [email protected] † Electronic supplementary 10.1039/c4sm01351d
information
8864 | Soft Matter, 2014, 10, 8864–8874
(ESI)
available.
See
DOI:
thick double layer. Adopting a perturbation approach, Keh and Ding3 analyzed the gravitational settling of a charged, rigid sphere of arbitrary double-layer thickness in an unbounded electrolyte solution. They derived an analytical expression correlating the particle velocity with the particle zeta potential, the double-layer thickness, the uid viscosity, and the ionic diffusion coefficients. Keller et al.4 considered the settling of a rigid sphere of radius 3 107 m in an unbounded electrolyte solution subject to a gravitational or centrifugal eld. They found that although the electric force acting on the particle shows a local maximum as the Reynolds number increases, the corresponding ratio of (electric force/hydrodynamic force) decreases. Recently, Gopmandal and Bhattacharyya conducted a series of studies on the settling of a charged particle in an electrolyte medium with or without an external applied electric eld.5–9 Taking account of the effects of counterion condensation (CC) and double-layer polarization, the electric and hydrodynamic forces acting on the particle were examined by considering a charge neutralization factor and the distribution of counterions. That factor measures the degree of charge neutralization inside a porous particle, and the further the counterions are removed from a rigid particle, the more intense the induced electric eld. If an external electric eld is applied (i.e., electrophoresis), this induced electric eld reduces the electric force acting on a particle, but the situation is reversed in settling, where the longer the tail of the counterion distribution, the greater the electric force acting on a particle. Note that although the counterion distribution is capable of explaining
This journal is © The Royal Society of Chemistry 2014
View Article Online
Published on 06 October 2014. Downloaded by University of California - San Francisco on 27/01/2015 19:53:24.
Paper
Soft Matter
qualitatively how intense the induced electric eld is, a quantitative measure is desirable and necessary. In this study, we consider the settling of a charged, isolated, deformable polyelectrolyte (PE) in a salt solution subject to a gravitational eld or an applied centrifugal eld. Parameters including the xed charge density of the PE, its soness parameter and aspect ratio, the Reynolds number, and the double-layer thickness are examined for their inuences on the settling behavior of the PE. To take the inuence of the Reynolds number into account, the inertial term in the Navier– Stokes equation describing the ow eld surrounding the PE needs to be considered.8 The strength of the local electric eld arising from the movement of a PE as well as its effective charge density are evaluated so that its behavior can be explained both qualitatively and quantitatively.10
2.
Theory
As shown in Fig. 1, we consider the settling of an isolated, charged polyelectrolyte (PE) having characteristic lengths a and d in an aqueous salt solution subject to an applied eld Cg, where C and g are a proportional constant and the gravitational acceleration, respectively. For example, in the case where an ultracentrifuge eld is applied, C can be of the order of 106.11 ¯ ¼ (a3 + 3a2d/2)1/3 being The PE volume is xed at 4p¯ a3/3 with a the equivalent sphere radius. The variation in the ratio of (d/a) yields various PE shapes. r, q, and z are the cylindrical coordinates adopted with the origin at the particle center, g is in the z direction. We suppose that the PE has uniformly distributed xed charges of density rx and the liquid phase is an incompressible Newtonian uid containing z1 : z2 electrolytes with z1
and z2 being the valences of the cations and anions, respectively. Note that the present problem is q-symmetric. We assume that the system under consideration is in a steady state. Therefore, the equations governing the present problem can be summarized as: r þ h r; z rfix V2 f ¼ (1) 3 zj e V$fj ¼ V$ Dj Vnj þ nj Vf þ nj u ¼ 0 kB T
(2)
V$u ¼ 0
(3)
Vp + hV2u rVf h(r,z)gu ¼ r(u$V)u
(4)
f, p, and u are the electric potential, the hydrodynamic pressure, and the uid velocity relative to PE, respectively. nj, fj, Dj, and zj are the number concentration, the ux, the diffusivity, and the valence of ionic species j, respectively. e, kB, T, h, 3, and g are the elementary charge, Boltzmann constant, the absolute temperature, the viscosity and the permittivity of the liquid phase, and the hydrodynamic frictional coefficient of the PE per 2 X unit volume, respectively. V2, V, and r ¼ zj enj are the Lapj¼1
lace operator, the gradient operator, and the space charge density of mobile ions, respectively. h(r,z) is a region index: h(r,z) ¼ 0 for the PE exterior, and h(r,z) ¼ 1 for its interior. The terms rVf and gu in eqn (4) denote the electric body force and the frictional force acting on the liquid. l1 ¼ (g/h)1/2 is the Brinkman screening length measuring the soness of the PE material, or the shielding length measuring how deep the liquid ow can penetrate into a PE. Typically, l1 ranges from 0.1 to 10 nm for biocolloids (e.g., microorganisms) and polymer gels.12,13 For convenience, we dene the reciprocal Debye #1=2 " 2 X 2 nj0 ðezj Þ =3kB T , and the scaled screening length, k ¼ j¼1
¯2/3fref. In our case, the xed charge density of a PE, Qx ¼ rxa inertial force on the right-hand side of eqn (4) can be signicant.9 For convenience, the Reynolds number is dened as Ref ¼ 2aruref/h, where uref is the terminal velocity of a rigid sphere in the gravitational eld. Applying the result of Keh and Ding,3 the gravitational/centrifugal force in our case can be expressed as 4 Fg ¼ pa3 rp r Cgez 3 ez is the unit vector in the z direction. In our case, 2a2 rp r Cg uref ¼ 9h Fig. 1 Settling of a charged PE having characteristic lengths a and d in a dispersion medium subject to an applied field Cg, where C and g are a proportional constant and the gravitational acceleration, respectively. The PE volume is fixed with a¯ ¼ (a3 + 3a2d/2)1/3 being the equivalent sphere radius; r, q, and z are the cylindrical coordinates chosen with the origin at the particle center, g is in the z direction.
This journal is © The Royal Society of Chemistry 2014
(5)
(6)
This expression is obtained by a force balance considering the hydrodynamic force acting on a rigid sphere, the Stokes' drag FStokes ¼ 6phauref, and the gravitational/centrifugal force, Fg. To specify the boundary conditions associated with eqn (1)– (4), we assume the following: (i) the electric potential, electric
Soft Matter, 2014, 10, 8864–8874 | 8865
View Article Online
Published on 06 October 2014. Downloaded by University of California - San Francisco on 27/01/2015 19:53:24.
Soft Matter
Paper
eld, ionic concentrations, ionic uxes, uid velocity, normal and tangential stresses, and hydrodynamic stress are all continuous on the PE–liquid interface. (ii) The liquid permittivity and viscosity inside the PE are the same as those outside it. (iii) The electric potential, ionic concentrations, and liquid velocity inside the PE are nite. (iv) The electric, concentration, and ow elds far from the PE are uninuenced by its presence. These assumptions yield the boundary conditions below: f¼0
(7)
p¼0
(8)
cj ¼ Cj0
(9)
u ¼ uref ez
(10)
Cj0 is the bulk concentration of the jth ionic species. At steady state, the electric force Fe and the hydrodynamic force Fh acting on a PE in the z direction can be evaluated by1417 " 2 2 ! # ð vf vf 1 vf vf nr Fe ¼ 3 þ (11) nz dUPE vr vz 2 vr vz UPE vur vuz vuz h þ nr þ nz p þ 2h dUPE Fh ¼ vz vr vz UPE ð
(12)
UPE is the PE surface, nr and nz are the r and z components of the unit normal vector n; ur and uz are the r and z components of u, respectively. The governing equations and the associated boundary conditions are solved numerically by COMSOL Multiphysics (version 4.3a, http://www.comsol.com) operated in a highperformance cluster. The procedure to calculate the PE mobility can be found elsewhere.18,19
3.
Results and discussion
3.1. Model verication To verify the applicability of the present model and the soware adopted, the settling of an isolated rigid sphere subject to either a gravitational eld or a centrifugal eld solved numerically by Keller et al.4 is reanalyzed. Our model is further veried by considering the sedimentation of a porous sphere solved analytically by Keh and Chen.20 The results of the verication are presented in the Section S1 of ESI.† For convenience, uref ¼ 2a2(rp r)Cg/9h is adopted in subsequent discussions. 3.2. Numerical simulation To examine the PE behavior under various conditions, a thorough numerical simulation is conducted. This is done by varying the xed charge density of a PE, Qx, its soness parameter, l¯ a, its aspect ratio, (d/a), and the scaled double-layer thickness, k¯ a. The electrostatic interactions between the negatively charged phosphate backbone on DNA and the protonated nitrogen atoms of linear poly(ethylene imine) yield PE complexes having a wide range of DNA/PEI molar ratios. Aer synthesis, analytical
8866 | Soft Matter, 2014, 10, 8864–8874
ultracentrifugation (AUC) is usually applied to determine their size and physical properties which are signicant for subsequent usage. The centrifugal force in AUC can be in the order of 5 105 to 106g.11,21,22 Due to aggregation, the size of the complexes can increase to 1000 nm at a N/P ratio of ca. 2.23 Another type of PE complex is formed by synthesizing cationic poly(diallyldimethylammonium chloride) and anionic poly(styrene sulfonic acid), the PDADMAC/PSS complex; its charge and size depend on the mixing ratio. The radii of secondary particles can exceed 100 nm due to aggregation of primary particles.24 Referring to these two examples, we assume a ¼ 500 nm in our simulation so that if a PE is stretched to d/a ¼ 10, a ¼ 198 nm and (2d + 2a) ¼ 4500 nm. Note that since the hydration radii of regular ionic species such as K+ and Cl are of the order of 0.5 nm, the inuence of their sizes on the PE behavior is unimportant. Based on the capability of modern centrifuges, the Reynolds number assumed in the numerical simulation ranges from 1.16 103 to 0.5.4,21,22 The partial specic volume of DNA is 0.54 0.02 cm3 g1.25 Typically, the density of biological materials ranges from 1040 to 2000 kg m3.26 Therefore, we assume that rp ¼ 1700 kg m3. In the calculation of the forces acting on a porous sphere, the Fe coming from the gravitational/centrifugal force is neglected, for convenience. Therefore, the Ref thus obtained does not correspond to its real velocity. The reference forces Fe,ref ¼ 3fref2 and Fh,ref ¼ FStokes are chosen with FStokes ¼ 6phauref; both are widely used in previous studies. Because the effects of doublelayer relaxation (DLR), CC,27 and electroosmotic ow (EOF)15,16 are all taken into account, the present analysis is more general than the available results in the literature. 3.2.1 Inuence of double-layer thickness k¯ a. Let us consider a positively charged PE moving in the z direction, implying that the Fe calculated by eqn (11) is negative. For convenience, we focus on its absolute value. Fig. 2 summarizes the variations in the magnitude of the scaled electric force |F*e|, the ratio of the forces acting on the PE (Fe/Fh), and the drag factor Uf with the scaled double-layer thickness k¯ a for three values of Qx at d/a ¼ 0 (i.e., spherical PE). Here, Uf is dened by28 Uf ¼
Fh Fh ¼ FStokes 6phauref
(13)
It is the ratio of (hydrodynamic drag exerted on a porous aggregate/corresponding Stokes' drag on a rigid sphere settling at the same speed). Note that because the uref dened in eqn (6) is adopted, FStokes is not necessarily a constant. Since the equivalent PE radius ¯ a is xed, varying k¯ a implies changing the bulk salt concentration. As seen in Fig. 2(a), for all the values of Qx examined, |F*e| has a local maximum occurring at k¯ a y 1. This arises from the effect of DLR due to an asymmetric ionic distribution surrounding a charged particle. This effect induces a local electric eld in the z direction, thereby retarding the PE movement. DLR is most signicant when the double-layer thickness is comparable to the size of a particle (k¯ a y 1 in our case) and its charge density high.15
This journal is © The Royal Society of Chemistry 2014
View Article Online
Published on 06 October 2014. Downloaded by University of California - San Francisco on 27/01/2015 19:53:24.
Paper
Soft Matter
Variations in the magnitude of the scaled electric force |F*e| (a), the ratio of the forces acting on the particle (Fe/Fh) (b), and the drag factor Uf (c), with ka¯ for three values of Qfix at d/a ¼ 0, la¯ ¼ 4, and Ref ¼ 0.01. Fig. 2
Fig. 2(b) shows the variation in the ratio of the forces acting on a PE, (Fe/Fh), with k¯ a at various values of Qx. This ratio measures the relative importance of the electric force and the hydrodynamic force. It is seen that (Fe/Fh) has a local maximum at k¯ a y 1 (double-layer thickness comparable to PE size). This behavior is similar to that in the electrophoresis of a rigid particle having a constant zeta potential, where double-layer polarization yields a local minimum in the particle mobility.29 As seen in Fig. 2(b), the larger the Qx the larger the ratio (Fe/Fh), which is expected because a larger Qx implies a greater electric force acting on a PE. As seen in Fig. 2(c), Uf increases with increasing Qx. This is because if Qx is sufficiently large, the larger the Qx the more signicant the effect of DLR. This effect enhances the counterion-rich liquid ow inside the PE, making the behavior of Uf similar to that of |F*e|.10 To explain the behavior of |F*e| seen in Fig. 2(a), we dene ð dþa ð 2p ðð Ez* r z dqdz Ez* r z dA* * * A ¼ ðð ¼ da ; (14) Ez;avg ð dþa 0ð 2p rðzÞdA* rðzÞdqdz * da
0
ð dþa Q*eff;avg
¼
Qeff p½rðzÞ2 dz
da ð dþa
da
p½rðzÞ2 dz
¼
∭V * Qeff dV * ∭V * dV *
;
(15)
where ) ( aj d # z # d qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (15a) rðzÞ ¼ a2 ðz dÞ2 d # z # d þ a; d a # z # d E*z,avg and Q*eff,avg are the averaged scaled strength of the local electric eld in the z direction and the averaged scaled net effective charge density of a PE, respectively. E*z ¼ Ez/Eref is the scaled strength of the local electric eld in the z direction, and Eref ¼ fref/¯ a. Note that the general behavior of E*z,avg in Fig. 3(a) is similar to that of |F*e| in Fig. 2(a). As seen in Fig. 3(b), Q*eff,avg decreases monotonically with increasing k¯ a, reecting the attraction of counterions into the PE (i.e., CC).27 Note that both E*z,avg and Q*eff,avg are capable of inuencing |F*e|. As pointed out by Gopmandal and Bhattacharyya,30 if k¯ a is sufficiently large, |F*e| is dominated by Q*eff,avg.
A
This journal is © The Royal Society of Chemistry 2014
Soft Matter, 2014, 10, 8864–8874 | 8867
View Article Online
Published on 06 October 2014. Downloaded by University of California - San Francisco on 27/01/2015 19:53:24.
Soft Matter
Paper
Fig. 3 Variations of the averaged scaled strength of the local electric field in the z direction E*z,avg, (a), and the averaged scaled net effective charge density Q*eff,avg(b), with ka¯ for three values of Qfix at d/a ¼ 0, la¯ ¼ 4, and Ref ¼ 0.01.
In our case, although |F*e| seems to be governed by E*z,avg, Q*eff,avg also contributes by reducing the magnitude of |F*e|. Fig. S3 of ESI† shows the contours of the net ionic concentration (C1 C2) (mol m3) for the case of Qx ¼ 50 in Fig. 2(a) on the half plane q ¼ 0 at three values of k¯ a. In our case, the electric force acting on a PE is obtained by integrating the Maxwell stress tensor over its surface. Therefore, to have a large value of |F*e|, both the potential gradient and the space which contains a steep potential gradient need to be large.3 As k¯ a increases from 0.8 to 2, the counterions near the rear of the PE are removed, producing a greater gradient on the integration surface. Because the space occupied by counterions also increases, |E*z,avg(k¯ a ¼ 2)| > |E*z,avg(k¯ a ¼ 0.8)|. However, if k¯ a exceeds ca. 2, the double layer becomes too thin so that the effect of CC needs to be considered. In this case, the space for a steep potential gradient is limited, and with the decrease in effective charge density, both E*z,avg and |F*e| decrease. Fig. S3 of ESI† also shows that the direction of the induced electric eld is opposite to that of the PE movement. 3.2.2 Inuence of soness parameter l¯ a. Fig. 4 summarizes the variations of the magnitude of the scaled electric force |F*e|, the averaged scaled strength of the local electric eld in the z direction E*z,avg, and the averaged scaled net effective charge density Q*eff,avg with k¯ a for three values of l¯ a. As seen in Fig. 4(a), if k¯ a is small (thick double layer), the curves of |F*e| for different values of l¯ a intersect with each other. For the values of l¯ a examined, |F*e| has a local maximum at k¯ a y 1. In general, the smaller the l¯ a, the easier it is for uid to penetrate through a PE, yielding a larger |F*e|. The behavior of |E*z,avg| seen in Fig. 4(b) is similar to that of |F*e|. The intersection of the curves corresponding to various values of l¯ a at low k¯ a results mainly from the induced electric eld and the equilibrium electric eld established by the PE. For the range of k¯ a considered, |E*z,avg(l¯ a ¼ 4)| > |E*z,avg(l¯ a ¼ 10)|. However, the curve of |E*z,avg(l¯ a ¼ 1)| intersects with the other two curves. As shown in Fig. 4(a), this is because, for low values of k¯ a, |F*e(l¯ a ¼ 1)| increases rapidly with increasing k¯ a.
8868 | Soft Matter, 2014, 10, 8864–8874
For thick double layers, Table S1 of ESI† shows that Q*eff,avg(k¯ a ¼ 0.5) y Q*eff,avg(k¯ a ¼ 0.8), implying that the relative magnitude of the electric force acting on a PE in these two cases depends mainly on the local electric eld. Because |E*z,avg(k¯ a¼ 0.5)| < |E*z,avg(k¯ a ¼ 0.8)|, |F*e(k¯ a ¼ 0.5)| < |F*e(k¯ a ¼ 0.8)|. This can also be inferred from Fig. S4 of ESI,† where the (C1 C2) at l¯ a¼ 4 and that at l¯ a ¼ 10 are almost uniform inside the PE, |F*e| increases monotonically with increasing double-layer thickness. However, because the extent to which counterions are removed from the PE at l¯ a ¼ 1 is greater than that in the other two cases and the space containing a steep electric potential gradient enclosed by the PE surface (i.e., the integration surface) is large, the curve of |F*e(l¯ a ¼ 1)| intersects with that of |F*e(l¯ a ¼ 4)| and that of |F*e(l¯ a ¼ 10)|. At l¯ a ¼ 1 and k¯ a ¼ 0.5, although counterions are removed appreciably from the PE, the space of a steep gradient of concentration enclosed by the PE surface is limited, so that |E*z,avg(l¯ a ¼ 1)| < |E*z,avg(l¯ a ¼ 4)|. The growth of convective ow makes it harder for counterions to be trapped inside the PE to have the xed charge neutralized. Because the smaller the l¯ a the more intense the convective ow, Q*eff,avg(l¯ a ¼ 1) > Q*eff,avg(l¯ a ¼ 4) > Q*eff,avg(l¯ a ¼ 10). In the case of l¯ a ¼ 1, as k¯ a increases from 0.5 to 0.8, the local electric eld increases appreciably. This arises from the fact that both the amount of counterions removed from the PE and the space of a steep gradient of concentration enclosed by the PE surface are appreciable. Therefore, for a medium-large value of k¯ a, the larger the l¯ a the smaller the E*z,avg. For the cases of l¯ a¼4 and l¯ a ¼ 10, because the space of a steep gradient of concentration enclosed by the PE surface is about the same, the relative magnitude of E*z,avg depends upon the amount of counterions removed from a PE. Because the smaller the l¯ a the larger that amount, |E*z,avg(l¯ a ¼ 4)| > |E*z,avg(l¯ a ¼ 10)|. As mentioned by Gopmandal and Bhattacharyya,5–9 the behavior of a porous particle is dominated by the convective uid ow when its l¯ a is small. As seen in Fig. 5(a), the drag factor Uf is also inuenced signicantly by l¯ a. Uf shows a local maximum as k¯ a varies, and the smaller the l¯ a the more
This journal is © The Royal Society of Chemistry 2014
View Article Online
Published on 06 October 2014. Downloaded by University of California - San Francisco on 27/01/2015 19:53:24.
Paper
Soft Matter
Fig. 4 Variations of the magnitude of the scaled electric force |F*e| (a), the averaged scaled strength of the local electric field in the z direction E*z,avg (b), and the averaged scaled net effective charge density Q*eff,avg (c), with ka¯ for three values of la¯ at Qfix ¼ 50, d/a ¼ 0 and Ref ¼ 0.01.
appreciable the presence of that maximum. This is because the smaller the l¯ a of a PE, the easier it is for uid to penetrate it. However, since counterions are also more easily removed by the convective ow, the associated EOF becomes less important, yielding a smaller hydrodynamic force exerted on the PE. This
Fig. 5
explains that Uf increases with increasing l¯ a. Fig. 5(b) shows the variation in the ratio of (electric force acting on a PE/corresponding hydrodynamic force), (Fe/Fh), with k¯ a for various values of l¯ a. This gure reveals that the smaller the l¯ a of a PE the greater the electric force acting on it. The (Fe/Fh) at l¯ a ¼ 1 is
Variations of the drag factor Uf (a), and the ratio (Fe/Fh) (b), with ka¯ for three values of la¯ at d/a ¼ 0, Qfix ¼ 50, and Ref ¼ 0.01.
This journal is © The Royal Society of Chemistry 2014
Soft Matter, 2014, 10, 8864–8874 | 8869
View Article Online
Published on 06 October 2014. Downloaded by University of California - San Francisco on 27/01/2015 19:53:24.
Soft Matter
much larger than that at other values of l¯ a. This is because both Q*eff,avg and E*z,avg are higher and EOF is less signicant in the former. Note that at l¯ a ¼ 1, the electric force contributes more than 50% of the total force exerted on the PE. 3.2.3 Inuence of Reynolds number Ref. Fig. 6 shows the variations of the magnitude of the scaled electric force acting on a PE, |F*e|, the averaged scaled net effective charge density, Q*eff,avg, and the averaged scaled strength of the local electric eld in the z direction, E*z,avg, with Ref for various values of k¯ a. Recall that due to its denition Ref does not correspond to the real velocity of a porous sphere. As seen in Fig. 6(a), the behavior of |F*e| for a thin double layer (k¯ a¼10) is different from that for a thick double layer. In particular, |F*e| shows a local maximum for the cases of k¯ a ¼ 0.1 and 1. As k¯ a increases from 0.1 to 1, the Ref at which the local maximum of |F*e| occurs shis to a larger value. This will be discussed later. A larger Ref implies a faster settling velocity and, therefore, a more signicant convection effect. For the range of k¯ a considered, Q*eff,avg increases monotonically with increasing Ref. As mentioned in the discussion of Fig. 4, if the soness parameter l1 of a PE is large, especially at l¯ a ¼ 1, because it is hard to drive the counterions inside the
Paper
double layer into the PE interior, they are removed from the PE through convective uid ow. A similar idea also applies to a faster settling velocity and, therefore, the effect of CC is less signicant, giving a higher effective charge density. For the values of Ref considered, the smaller the k¯ a the larger the Q*eff,avg. As can be seen in Fig. 6(b), the behavior of Q*eff,avg as Ref varies depends upon the level of k¯ a. In general, the larger the k¯ a (thinner double layer) the more signicant the CC effect. In this case, a larger Ref is needed to have counterions removed from a PE, giving a larger Q*eff,avg. Note that for the cases of k¯ a ¼ 0.1 and 1, the value of Ref at which Q*eff,avg begins to increase rapidly is about the same as that where the local maximum of |F*e| occurs. Although the contribution of Q*eff,avg is less signicant in these cases, it still inuences the values of Ref at which the local maximum of |F*e| and |E*z,avg| occurs, so that the former is slightly larger than the latter. For the case of k¯ a ¼ 1 in Fig. 6(c), |E*z,avg| has a local maximum at Ref y 0.1. This will be discussed later. Fig. 7 and 8 show the contours of the net ionic concentration (C1 C2) (mol m3) on the half plane q ¼ 0 at two levels of k¯ a. As
Fig. 6 Variations of the magnitude of the scaled electric force |F*e| (a), the averaged scaled net effective charge density Q*eff,avg (b), and the averaged scaled strength of the local electric field in the z direction E*z,avg (c), with Ref for three values of ka¯ at d/a ¼ 0, la¯ ¼ 4, and Qfix ¼ 50.
8870 | Soft Matter, 2014, 10, 8864–8874
This journal is © The Royal Society of Chemistry 2014
View Article Online
Published on 06 October 2014. Downloaded by University of California - San Francisco on 27/01/2015 19:53:24.
Paper
Soft Matter
Fig. 7 Contours of the net ionic concentration (C1 C2) (mol m3) on the half plane q ¼ 0 at Qfix ¼ 50, d/a ¼ 0, ka¯ ¼ 1, and la¯ ¼ 4 for three values of Ref, 1.16 103 (a), 0.1 (b), and 0.5 (c). White curves denote the particle surface on which the Maxwell stress tensor is integrated to obtain F*e.
can be seen in Fig. 7(a), if k¯ a ¼ 1, the ionic distribution around the PE is nearly spherical when Ref is small, giving a weak local electric eld. As Ref increases, both the PE surface on which the Maxwell stress tensor is integrated to obtain Fe and the amount of counterions removed are important. Because the space containing a steep potential gradient enclosed by the PE surface decreases appreciably at Ref ¼ 0.5, E*z,avg shows a local maximum at Ref y 0.1, as illustrated in Fig. 6(c). However, in the case of Fig. 8 where k¯ a ¼ 10, because (C1 C2) is almost uniform inside the PE for the values of Ref examined, E*z,avg depends mainly on the extent to which counterions are removed from the PE. We conclude that if the double layer is thin, E*z,avg increases monotonically with increasing Ref. Fig. S6 of ESI† shows the variation of the percentage deviation d ¼ [(FStokes F^ Stokes)/FStokes] 100% in the reference hydrodynamic drag FStokes ¼ 6phauref due to neglecting the convective term r(u$Vu) in eqn (4), where F^ Stokes is the value of FStokes without considering r(u$Vu). Here, d measures how signicant the inertial force acting on a PE is. Fig. S6 of ESI† indicates that d increases rapidly with Ref when Ref exceeds ca. 0.1. However, for the range of Ref considered, the maximum value of d is only ca. 3%. That is, the inertial force is unimportant although the particle velocity is fast (uref y 4.455 101 m s1). As illustrated in Fig. 9(a), the drag factor Uf has a local maximum as Ref varies, and the value of Ref at which the local maximum occurs increases with increasing k¯ a. For small values of Ref, the faster the settling velocity, the more thoroughly the counterions are removed from a PE such that Uf increases with increasing Ref. However, if Ref exceeds a
This journal is © The Royal Society of Chemistry 2014
certain level, the inertial force retarding the EOF is signicant. In this case, the effective charge density of a PE is high, lowering the EOF of counterions inside the PE, so that Uf decreases.8 The behavior of Uf as Ref varies depends upon the level of k¯ a. If k¯ a is small, a small variation in Ref produces an appreciable change in Uf. If k¯ a is sufficiently large, Uf behaves differently from the scaled force F*e seen as shown in Fig. 6(a). This is because the effect of EOF for thin double layers is less signicant than that of CC and, therefore, it is harder to drive counterions out of a PE. Because a larger value of Ref is needed to drive counterions out, both Q*eff,avg and the amount of counterions removed from the PE increase. A less signicant EOF also gives a smaller Uf. Fig. 9(b) shows the variation of the ratio of (electric force/ hydrodynamic force), (Fe/Fh), with Ref for three values of k¯ a. Again, for the range of Ref considered, the effect of DLR is most signicant when the Debye length is comparable to the PE size (i.e., k¯ a y 1). The behavior of (Fe/Fh) at k¯ a ¼ 10 implies that the scaled electric force F*e increases monotonically with increasing Ref. However, because it is too small, F*e is unimportant, and so is the corresponding EOF. 3.2.4 Inuence of PE shape (d/a). If d/a ¼ 0, a PE is spherical, and the larger the (d/a) the more it is stretched. Fig. 10(a) reveals that |F*e| decreases with increasing (d/a), implying that a more stretched PE experiences a smaller electric force in the z direction. Fig. 10(b) suggests that the larger the aspect ratio (d/a) (i.e., longer PE), the higher its effective charge density. If a PE is stretched to be sufficiently long (d/a exceeds ca. 8), Q*eff,avg y 1 when k¯ a is small. In addition, the shape of a
Soft Matter, 2014, 10, 8864–8874 | 8871
View Article Online
Published on 06 October 2014. Downloaded by University of California - San Francisco on 27/01/2015 19:53:24.
Soft Matter
Paper
Fig. 8 Contours of the net ionic concentration (C1 C2) (mol m3) on the half plane q ¼ 0 at Qfix ¼ 50, d/a ¼ 0, ka¯ ¼ 10, and la¯ ¼ 4 for three values of Ref, 1.16 103 (a), 0.1 (b), and 0.5 (c). White curves denote the particle surface on which the Maxwell stress tensor is integrated to obtain F*e.
PE inuences signicantly its capability to be neutralized by the counterions in the liquid phase: the longer the PE, the harder it is for counterions to be attracted into its interior to lower its effective charge. The behavior of Q*eff,avg is dissimilar to that of F*e, implying the presence of other factors that might inuence the electric force acting on the PE. As seen in Fig. 10(c), the magnitude of the averaged scaled strength of the local electric eld in the z direction, |E*z,avg|,
Fig. 9
decreases with increasing (d/a), except for the case of d/a ¼ 0 (i.e., spherical PE). For all values of (d/a), E*z,avg has a local maximum as k¯ a varies. Fig. 10(c) reveals that if k¯ a is smaller than ca. unity, |E*z,avg(d/a ¼ 0)| < |E*z,avg(d/a ¼ 2)|, but that order is reversed if k¯ a exceeds ca. unity. This is because if k¯ a is sufficiently large, the space containing a high net ionic concentration, and therefore a steep gradient of electric potential, enclosed by the PE surface is large.
Variations of the drag factor Uf (a), and (Fe/Fh) (b), with Ref for three values of ka¯ at d/a ¼ 0, la¯ ¼ 4, and Qfix ¼ 50.
8872 | Soft Matter, 2014, 10, 8864–8874
This journal is © The Royal Society of Chemistry 2014
View Article Online
Published on 06 October 2014. Downloaded by University of California - San Francisco on 27/01/2015 19:53:24.
Paper
Soft Matter
Fig. 10 Variations of the magnitude of the scaled electric force |F*e|, (a), the averaged scaled effective charge density Q*eff,avg (b), and the averaged
scaled strength of the local electric field in the z direction E*z,avg (c), with ka¯ for various values of (d/a) at Qfix ¼ 50, la¯ ¼ 4, and Ref ¼ 0.01.
Fig. S7 (a and b) of ESI† show that if the double layer is thick (k¯ a ¼ 0.5), (C1 C2) and, therefore, the electric potential gradient is uniform near the PE surface, implying that E*z,avg depends mainly upon the amount of counterions removed from the PE. Since the amount of counterions removed at d/a ¼ 2 is greater than that at d/a ¼ 0, |E*z,avg(d/a ¼ 2)| > |E*z,avg(d/a ¼ 0)|. However, as seen in Fig. S7(b) of ESI,† if k¯ a exceeds ca. 1, |E*z,avg(d/a ¼ 0)| > |E*z,avg(d/a ¼ 2)|. As illustrated in Fig. S7 of ESI,† this is because if k¯ a is sufficiently large, the space containing a steep electric potential gradient enclosed by a spherical PE surface is larger than that with a non-spherical (d/a ¼ 2) surface. We conclude that the shape of a PE is capable of inuencing the electric force acting on it through E*z,avg. The inuence of Q*eff,avg is important only if a PE is highly stretched and the double layer is thick. Fig. 11 shows the variation of the scaled electric potential 4* ¼ 4/4ref with 4ref ¼ kBT/z1e on the half plane q ¼ 0 along the PE surface, measured by the arc length from its south pole for various values of (d/a). The corresponding contours of 4* are shown in Fig. S8 of ESI.† This gure reveals that the larger the (d/a) (more stretched PE), the less symmetric the electric potential distribution. Note that 4* is negative near the south pole of the PE, which arises from the penetration of counterions. This phenomenon has not been reported previously.
This journal is © The Royal Society of Chemistry 2014
Fig. 11 Variation of the scaled electric potential 4* ¼ 4/4ref on the half
plane q ¼ 0 along the PE surface, measured by the arc length from its south pole, at Qfix ¼ 50, ka¯ ¼ 2, la¯ ¼ 4, and Ref ¼ 0.1 for various values of (d/a). Short dashed lines: middle plane of PE; long dashed lines: north pole of PE.
Soft Matter, 2014, 10, 8864–8874 | 8873
View Article Online
Published on 06 October 2014. Downloaded by University of California - San Francisco on 27/01/2015 19:53:24.
Soft Matter
The behavior of the drag factor Uf seen in Fig. S9(a) of ESI† is similar to that of the corresponding |Fe*| presented in Fig. 10(a). As mentioned in the discussion of Fig. 2(b), this is due to the effect of DLR, which enhances the counterion-rich liquid ow inside a PE. Due to the variation in the degree of EOF effect with the PE shape, the curves of Uf(d/a ¼ 0) and Uf(d/a ¼ 2) intersect. As seen in Fig. S9(b) of ESI,† although |Fe*| has its largest value at d/a ¼ 0, that value is unimportant compared with the corresponding hydrodynamic force. We conclude that the behavior of Uf seen in Fig. S9(b)† results from competition of the effects of DLR, CC, and EOF. As mentioned in the discussion of Fig. 10(a), a stretched PE experiences a smaller electric force than a spherical PE.
4. Conclusions The sedimentation of an isolated, charged PE in an aqueous salt solution subject to an applied eld is modeled, taking account of the variation of its shape. We show that the interaction of the effects of double-layer relaxation, effective charge density, and strength of induced relaxation electric eld produces the following complex and interesting sedimentation behaviors: (i) the higher the xed charge density of a PE the more signicant the effect of double-layer relaxation, thereby enhancing the counterion-rich liquid ow inside it, making the behavior of the drag factor similar to that of the magnitude of the electric force acting on the PE. (ii) For thick double layers, because the electric force acting on a loosely structured PE is dominated by the convective uid ow, its effective charge density is higher than that of a compactly structured PE. Due to the effect of electroosmotic ow, the drag acting on a loosely structured PE is smaller than that on a more compactly structured PE. (iii) The behavior of the averaged effective charge density of a PE as the Reynolds number varies depends upon the level of double-layer thickness. The thinner the double layer, the larger the Reynolds number needed to drive counterions out of a PE. (iv) For thick double layers, the magnitude of the electric force acting on a PE has a local maximum as the Reynolds number varies, but increases monotonically with increasing Reynolds number for thin double layers. For the range of the salt concentration considered, the drag factor always shows a local maximum as the settling velocity varies. This behavior is different from those observed in the literature. (v) The longer the PE the harder it is for counterions to be attracted into its interior to have its xed charge neutralized, producing a greater electric force. However, the effective charge density of the PE is not necessarily higher because the induced relaxation electric eld also plays a role.
References 1 E. Lee, J.-W. Chu and J.-P. Hsu, J. Chem. Phys., 1999, 110, 11643–11651. 2 M.-H. Chih, E. Lee, J.-W. Chu and J.-P. Hsu, Langmuir, 2000, 16, 1650–1654.
8874 | Soft Matter, 2014, 10, 8864–8874
Paper
3 H. J. Keh and J. M. Ding, J. Colloid Interface Sci., 2000, 227, 540–552. 4 F. Keller, M. Feist, H. Nirschl and W. D¨ orer, J. Colloid Interface Sci., 2010, 344, 228–236. 5 S. Bhattacharyya and P. P. Gopmandal, 3rd Micro and Nano Flows Conference, Thessaloniki, Greece, 22–24 August 2011. 6 S. Bhattacharyya and P. P. Gopmandal, Colloids Surf., A, 2011, 390, 86–94. 7 S. Bhattacharyya and P. P. Gopmandal, Proc. World Acad. Sci. Eng. Tech., 2012, 6, 1088–1094. 8 S. Bhattacharyya and P. P. Gopmandal, So Matter, 2013, 9, 1871–1884. 9 P. P. Gopmandal and S. Bhattacharyya, Colloid Polym. Sci., 2013, 1–10. 10 K.-L. Liu, J.-P. Hsu, W.-L. Hsu, L.-H. Yeh and S. Tseng, Electrophoresis, 2012, 33, 1068–1078. 11 J. P. Clamme, J. Azoulay and Y. M´ ely, Biophys. J., 2003, 84, 1960–1968. 12 J. F. L. Duval, K. J. Wilkinson, H. P. van Leeuwen and J. Buffle, Environ. Sci. Technol., 2005, 39, 6435–6445. 13 T. Hoare and R. Pelton, Polymer, 2005, 46, 1139–1150. 14 J.-P. Hsu, L.-H. Yeh and M.-H. Ku, J. Colloid Interface Sci., 2007, 305, 324–329. 15 L.-H. Yeh and J.-P. Hsu, So Matter, 2011, 7, 396–411. 16 J.-P. Hsu, C.-Y. Lin, L.-H. Yeh and S.-H. Lin, So Matter, 2012, 8, 9469–9479. 17 M. Zhang, L.-H. Yeh, S. Qian, J.-P. Hsu and S. W. Joo, J. Phys. Chem. C, 2012, 116, 4793–4801. 18 L.-H. Yeh, M. Zhang, S. Qian and J.-P. Hsu, Nanoscale, 2012, 4, 2685–2693. 19 Y.-Y. Chen, J.-P. Hsu and S. Tseng, J. Colloid Interface Sci., 2014, 421, 154–159. 20 H. J. Keh and W. C. Chen, J. Colloid Interface Sci., 2006, 296, 710–720. 21 W. M¨ achtle, in Encyclopedia of Analytical Chemistry, John Wiley & Sons, Ltd, 2006. 22 W. Maechtle and L. B¨ orger, Analytical Ultracentrifugation of Polymers and Nanoparticles, Springer, 2006. 23 V. Doln´ık, Electrophoresis, 2008, 29, 143–156. 24 V. Starchenko, M. M¨ uller and N. Lebovka, J. Phys. Chem. B, 2012, 116, 14961–14967. 25 H. Eisenberg, Biological macromolecules and polyelectrolytes in solution, Clarendon Press, 1976. 26 P. Y. Chen, A. Y. M. Lin, Y. S. Lin, Y. Seki, A. G. Stokes, J. Peyras, E. A. Olevsky, M. A. Meyers and J. McKittrick, J. Mech. Behav. Biomed. Mater., 2008, 1, 208–226. 27 L.-H. Yeh, J.-P. Hsu, S. Qian and S. Tseng, Electrochem. Commun., 2012, 19, 97–100. 28 G. Neale, N. Epstein and W. Nader, Chem. Eng. Sci., 1973, 28, 1865–1874. 29 K. Makino and H. Ohshima, Langmuir, 2010, 26, 18016– 18019. 30 P. P. Gopmandal and S. Bhattacharyya, Colloids Surf., A, 2013, 433, 64–76.
This journal is © The Royal Society of Chemistry 2014
Published on 06 October 2014. Downloaded by University of California - San Francisco on 27/01/2015 19:53:24.
PAPER
View Journal | View Issue
Cite this: Soft Matter, 2014, 10, 8864
Influence of polyelectrolyte shape on its sedimentation behavior: effect of relaxation electric field† Pin-Hua Yeh,a Jyh-Ping Hsu*a and Shiojenn Tseng*b The sedimentation of an isolated, charged polyelectrolyte (PE) subjected to an applied field is modeled theoretically, taking into account the variation of its shape. In particular, the effects of double-layer relaxation, effective charge density, and strength of the induced relaxation electric field are examined. We show that the interaction of these effects yields complex and interesting sedimentation behaviors. For example, the behavior of the electric force acting on a loosely structured PE can be different from that on a compactly structured one; the former is dominated mainly by the convective fluid flow. For thick double layers, electric force has a local maximum as the Reynolds number varies, but tends to increase monotonically with increasing Reynolds number if the layer is thin. The drag factor is found to
Received 23rd June 2014 Accepted 8th September 2014
behave differently from literature results. The shape of a PE significantly influences its sedimentation behavior by affecting the amount of counterions attracted to its interior and the associated local electric
DOI: 10.1039/c4sm01351d
field. Interestingly, a more stretched PE has a higher effective charge density but experiences a weaker
www.rsc.org/softmatter
electric force.
1. Introduction Particles of colloidal size are capable of settling in a dispersion medium due to the presence of gravitational force and/or other applied forces. Settling has been applied successfully in many operations, including wastewater treatment, biological processes, mining, and space propellant reignition, to name a few. If a particle is charged and the dispersion medium contains ionic species, the associated phenomena can be complex and interesting. This is because the motion of a charged particle induces a local electric eld, the nearby ionic distribution and the associated Reynolds number are inuenced, and so are the hydrodynamic and electric forces acting on the particle. Several attempts have been made for modeling the settling of a charged colloidal particle in an electrolyte medium. Lee et al.,1 for example, analyzed the sedimentation potential of a concentrated spherical colloidal suspension. They showed that if the surface potential is low (high), the ratio (scaled sedimentation potential/scaled sedimentation velocity) has a local minimum (maximum) as the double-layer thickness varies. Chih et al.2 found that if the surface potential is low, the scaled sedimentation velocity has a local maximum for a medium-
a
Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617. E-mail: [email protected]; Fax: +886-2-23623040; Tel: +886-2-23637448
b
Department of Mathematics, Tamkang University, Tamsui, Taipei, Taiwan 25137. E-mail: [email protected] † Electronic supplementary 10.1039/c4sm01351d
information
8864 | Soft Matter, 2014, 10, 8864–8874
(ESI)
available.
See
DOI:
thick double layer. Adopting a perturbation approach, Keh and Ding3 analyzed the gravitational settling of a charged, rigid sphere of arbitrary double-layer thickness in an unbounded electrolyte solution. They derived an analytical expression correlating the particle velocity with the particle zeta potential, the double-layer thickness, the uid viscosity, and the ionic diffusion coefficients. Keller et al.4 considered the settling of a rigid sphere of radius 3 107 m in an unbounded electrolyte solution subject to a gravitational or centrifugal eld. They found that although the electric force acting on the particle shows a local maximum as the Reynolds number increases, the corresponding ratio of (electric force/hydrodynamic force) decreases. Recently, Gopmandal and Bhattacharyya conducted a series of studies on the settling of a charged particle in an electrolyte medium with or without an external applied electric eld.5–9 Taking account of the effects of counterion condensation (CC) and double-layer polarization, the electric and hydrodynamic forces acting on the particle were examined by considering a charge neutralization factor and the distribution of counterions. That factor measures the degree of charge neutralization inside a porous particle, and the further the counterions are removed from a rigid particle, the more intense the induced electric eld. If an external electric eld is applied (i.e., electrophoresis), this induced electric eld reduces the electric force acting on a particle, but the situation is reversed in settling, where the longer the tail of the counterion distribution, the greater the electric force acting on a particle. Note that although the counterion distribution is capable of explaining
This journal is © The Royal Society of Chemistry 2014
View Article Online
Published on 06 October 2014. Downloaded by University of California - San Francisco on 27/01/2015 19:53:24.
Paper
Soft Matter
qualitatively how intense the induced electric eld is, a quantitative measure is desirable and necessary. In this study, we consider the settling of a charged, isolated, deformable polyelectrolyte (PE) in a salt solution subject to a gravitational eld or an applied centrifugal eld. Parameters including the xed charge density of the PE, its soness parameter and aspect ratio, the Reynolds number, and the double-layer thickness are examined for their inuences on the settling behavior of the PE. To take the inuence of the Reynolds number into account, the inertial term in the Navier– Stokes equation describing the ow eld surrounding the PE needs to be considered.8 The strength of the local electric eld arising from the movement of a PE as well as its effective charge density are evaluated so that its behavior can be explained both qualitatively and quantitatively.10
2.
Theory
As shown in Fig. 1, we consider the settling of an isolated, charged polyelectrolyte (PE) having characteristic lengths a and d in an aqueous salt solution subject to an applied eld Cg, where C and g are a proportional constant and the gravitational acceleration, respectively. For example, in the case where an ultracentrifuge eld is applied, C can be of the order of 106.11 ¯ ¼ (a3 + 3a2d/2)1/3 being The PE volume is xed at 4p¯ a3/3 with a the equivalent sphere radius. The variation in the ratio of (d/a) yields various PE shapes. r, q, and z are the cylindrical coordinates adopted with the origin at the particle center, g is in the z direction. We suppose that the PE has uniformly distributed xed charges of density rx and the liquid phase is an incompressible Newtonian uid containing z1 : z2 electrolytes with z1
and z2 being the valences of the cations and anions, respectively. Note that the present problem is q-symmetric. We assume that the system under consideration is in a steady state. Therefore, the equations governing the present problem can be summarized as: r þ h r; z rfix V2 f ¼ (1) 3 zj e V$fj ¼ V$ Dj Vnj þ nj Vf þ nj u ¼ 0 kB T
(2)
V$u ¼ 0
(3)
Vp + hV2u rVf h(r,z)gu ¼ r(u$V)u
(4)
f, p, and u are the electric potential, the hydrodynamic pressure, and the uid velocity relative to PE, respectively. nj, fj, Dj, and zj are the number concentration, the ux, the diffusivity, and the valence of ionic species j, respectively. e, kB, T, h, 3, and g are the elementary charge, Boltzmann constant, the absolute temperature, the viscosity and the permittivity of the liquid phase, and the hydrodynamic frictional coefficient of the PE per 2 X unit volume, respectively. V2, V, and r ¼ zj enj are the Lapj¼1
lace operator, the gradient operator, and the space charge density of mobile ions, respectively. h(r,z) is a region index: h(r,z) ¼ 0 for the PE exterior, and h(r,z) ¼ 1 for its interior. The terms rVf and gu in eqn (4) denote the electric body force and the frictional force acting on the liquid. l1 ¼ (g/h)1/2 is the Brinkman screening length measuring the soness of the PE material, or the shielding length measuring how deep the liquid ow can penetrate into a PE. Typically, l1 ranges from 0.1 to 10 nm for biocolloids (e.g., microorganisms) and polymer gels.12,13 For convenience, we dene the reciprocal Debye #1=2 " 2 X 2 nj0 ðezj Þ =3kB T , and the scaled screening length, k ¼ j¼1
¯2/3fref. In our case, the xed charge density of a PE, Qx ¼ rxa inertial force on the right-hand side of eqn (4) can be signicant.9 For convenience, the Reynolds number is dened as Ref ¼ 2aruref/h, where uref is the terminal velocity of a rigid sphere in the gravitational eld. Applying the result of Keh and Ding,3 the gravitational/centrifugal force in our case can be expressed as 4 Fg ¼ pa3 rp r Cgez 3 ez is the unit vector in the z direction. In our case, 2a2 rp r Cg uref ¼ 9h Fig. 1 Settling of a charged PE having characteristic lengths a and d in a dispersion medium subject to an applied field Cg, where C and g are a proportional constant and the gravitational acceleration, respectively. The PE volume is fixed with a¯ ¼ (a3 + 3a2d/2)1/3 being the equivalent sphere radius; r, q, and z are the cylindrical coordinates chosen with the origin at the particle center, g is in the z direction.
This journal is © The Royal Society of Chemistry 2014
(5)
(6)
This expression is obtained by a force balance considering the hydrodynamic force acting on a rigid sphere, the Stokes' drag FStokes ¼ 6phauref, and the gravitational/centrifugal force, Fg. To specify the boundary conditions associated with eqn (1)– (4), we assume the following: (i) the electric potential, electric
Soft Matter, 2014, 10, 8864–8874 | 8865
View Article Online
Published on 06 October 2014. Downloaded by University of California - San Francisco on 27/01/2015 19:53:24.
Soft Matter
Paper
eld, ionic concentrations, ionic uxes, uid velocity, normal and tangential stresses, and hydrodynamic stress are all continuous on the PE–liquid interface. (ii) The liquid permittivity and viscosity inside the PE are the same as those outside it. (iii) The electric potential, ionic concentrations, and liquid velocity inside the PE are nite. (iv) The electric, concentration, and ow elds far from the PE are uninuenced by its presence. These assumptions yield the boundary conditions below: f¼0
(7)
p¼0
(8)
cj ¼ Cj0
(9)
u ¼ uref ez
(10)
Cj0 is the bulk concentration of the jth ionic species. At steady state, the electric force Fe and the hydrodynamic force Fh acting on a PE in the z direction can be evaluated by1417 " 2 2 ! # ð vf vf 1 vf vf nr Fe ¼ 3 þ (11) nz dUPE vr vz 2 vr vz UPE vur vuz vuz h þ nr þ nz p þ 2h dUPE Fh ¼ vz vr vz UPE ð
(12)
UPE is the PE surface, nr and nz are the r and z components of the unit normal vector n; ur and uz are the r and z components of u, respectively. The governing equations and the associated boundary conditions are solved numerically by COMSOL Multiphysics (version 4.3a, http://www.comsol.com) operated in a highperformance cluster. The procedure to calculate the PE mobility can be found elsewhere.18,19
3.
Results and discussion
3.1. Model verication To verify the applicability of the present model and the soware adopted, the settling of an isolated rigid sphere subject to either a gravitational eld or a centrifugal eld solved numerically by Keller et al.4 is reanalyzed. Our model is further veried by considering the sedimentation of a porous sphere solved analytically by Keh and Chen.20 The results of the verication are presented in the Section S1 of ESI.† For convenience, uref ¼ 2a2(rp r)Cg/9h is adopted in subsequent discussions. 3.2. Numerical simulation To examine the PE behavior under various conditions, a thorough numerical simulation is conducted. This is done by varying the xed charge density of a PE, Qx, its soness parameter, l¯ a, its aspect ratio, (d/a), and the scaled double-layer thickness, k¯ a. The electrostatic interactions between the negatively charged phosphate backbone on DNA and the protonated nitrogen atoms of linear poly(ethylene imine) yield PE complexes having a wide range of DNA/PEI molar ratios. Aer synthesis, analytical
8866 | Soft Matter, 2014, 10, 8864–8874
ultracentrifugation (AUC) is usually applied to determine their size and physical properties which are signicant for subsequent usage. The centrifugal force in AUC can be in the order of 5 105 to 106g.11,21,22 Due to aggregation, the size of the complexes can increase to 1000 nm at a N/P ratio of ca. 2.23 Another type of PE complex is formed by synthesizing cationic poly(diallyldimethylammonium chloride) and anionic poly(styrene sulfonic acid), the PDADMAC/PSS complex; its charge and size depend on the mixing ratio. The radii of secondary particles can exceed 100 nm due to aggregation of primary particles.24 Referring to these two examples, we assume a ¼ 500 nm in our simulation so that if a PE is stretched to d/a ¼ 10, a ¼ 198 nm and (2d + 2a) ¼ 4500 nm. Note that since the hydration radii of regular ionic species such as K+ and Cl are of the order of 0.5 nm, the inuence of their sizes on the PE behavior is unimportant. Based on the capability of modern centrifuges, the Reynolds number assumed in the numerical simulation ranges from 1.16 103 to 0.5.4,21,22 The partial specic volume of DNA is 0.54 0.02 cm3 g1.25 Typically, the density of biological materials ranges from 1040 to 2000 kg m3.26 Therefore, we assume that rp ¼ 1700 kg m3. In the calculation of the forces acting on a porous sphere, the Fe coming from the gravitational/centrifugal force is neglected, for convenience. Therefore, the Ref thus obtained does not correspond to its real velocity. The reference forces Fe,ref ¼ 3fref2 and Fh,ref ¼ FStokes are chosen with FStokes ¼ 6phauref; both are widely used in previous studies. Because the effects of doublelayer relaxation (DLR), CC,27 and electroosmotic ow (EOF)15,16 are all taken into account, the present analysis is more general than the available results in the literature. 3.2.1 Inuence of double-layer thickness k¯ a. Let us consider a positively charged PE moving in the z direction, implying that the Fe calculated by eqn (11) is negative. For convenience, we focus on its absolute value. Fig. 2 summarizes the variations in the magnitude of the scaled electric force |F*e|, the ratio of the forces acting on the PE (Fe/Fh), and the drag factor Uf with the scaled double-layer thickness k¯ a for three values of Qx at d/a ¼ 0 (i.e., spherical PE). Here, Uf is dened by28 Uf ¼
Fh Fh ¼ FStokes 6phauref
(13)
It is the ratio of (hydrodynamic drag exerted on a porous aggregate/corresponding Stokes' drag on a rigid sphere settling at the same speed). Note that because the uref dened in eqn (6) is adopted, FStokes is not necessarily a constant. Since the equivalent PE radius ¯ a is xed, varying k¯ a implies changing the bulk salt concentration. As seen in Fig. 2(a), for all the values of Qx examined, |F*e| has a local maximum occurring at k¯ a y 1. This arises from the effect of DLR due to an asymmetric ionic distribution surrounding a charged particle. This effect induces a local electric eld in the z direction, thereby retarding the PE movement. DLR is most signicant when the double-layer thickness is comparable to the size of a particle (k¯ a y 1 in our case) and its charge density high.15
This journal is © The Royal Society of Chemistry 2014
View Article Online
Published on 06 October 2014. Downloaded by University of California - San Francisco on 27/01/2015 19:53:24.
Paper
Soft Matter
Variations in the magnitude of the scaled electric force |F*e| (a), the ratio of the forces acting on the particle (Fe/Fh) (b), and the drag factor Uf (c), with ka¯ for three values of Qfix at d/a ¼ 0, la¯ ¼ 4, and Ref ¼ 0.01. Fig. 2
Fig. 2(b) shows the variation in the ratio of the forces acting on a PE, (Fe/Fh), with k¯ a at various values of Qx. This ratio measures the relative importance of the electric force and the hydrodynamic force. It is seen that (Fe/Fh) has a local maximum at k¯ a y 1 (double-layer thickness comparable to PE size). This behavior is similar to that in the electrophoresis of a rigid particle having a constant zeta potential, where double-layer polarization yields a local minimum in the particle mobility.29 As seen in Fig. 2(b), the larger the Qx the larger the ratio (Fe/Fh), which is expected because a larger Qx implies a greater electric force acting on a PE. As seen in Fig. 2(c), Uf increases with increasing Qx. This is because if Qx is sufficiently large, the larger the Qx the more signicant the effect of DLR. This effect enhances the counterion-rich liquid ow inside the PE, making the behavior of Uf similar to that of |F*e|.10 To explain the behavior of |F*e| seen in Fig. 2(a), we dene ð dþa ð 2p ðð Ez* r z dqdz Ez* r z dA* * * A ¼ ðð ¼ da ; (14) Ez;avg ð dþa 0ð 2p rðzÞdA* rðzÞdqdz * da
0
ð dþa Q*eff;avg
¼
Qeff p½rðzÞ2 dz
da ð dþa
da
p½rðzÞ2 dz
¼
∭V * Qeff dV * ∭V * dV *
;
(15)
where ) ( aj d # z # d qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (15a) rðzÞ ¼ a2 ðz dÞ2 d # z # d þ a; d a # z # d E*z,avg and Q*eff,avg are the averaged scaled strength of the local electric eld in the z direction and the averaged scaled net effective charge density of a PE, respectively. E*z ¼ Ez/Eref is the scaled strength of the local electric eld in the z direction, and Eref ¼ fref/¯ a. Note that the general behavior of E*z,avg in Fig. 3(a) is similar to that of |F*e| in Fig. 2(a). As seen in Fig. 3(b), Q*eff,avg decreases monotonically with increasing k¯ a, reecting the attraction of counterions into the PE (i.e., CC).27 Note that both E*z,avg and Q*eff,avg are capable of inuencing |F*e|. As pointed out by Gopmandal and Bhattacharyya,30 if k¯ a is sufficiently large, |F*e| is dominated by Q*eff,avg.
A
This journal is © The Royal Society of Chemistry 2014
Soft Matter, 2014, 10, 8864–8874 | 8867
View Article Online
Published on 06 October 2014. Downloaded by University of California - San Francisco on 27/01/2015 19:53:24.
Soft Matter
Paper
Fig. 3 Variations of the averaged scaled strength of the local electric field in the z direction E*z,avg, (a), and the averaged scaled net effective charge density Q*eff,avg(b), with ka¯ for three values of Qfix at d/a ¼ 0, la¯ ¼ 4, and Ref ¼ 0.01.
In our case, although |F*e| seems to be governed by E*z,avg, Q*eff,avg also contributes by reducing the magnitude of |F*e|. Fig. S3 of ESI† shows the contours of the net ionic concentration (C1 C2) (mol m3) for the case of Qx ¼ 50 in Fig. 2(a) on the half plane q ¼ 0 at three values of k¯ a. In our case, the electric force acting on a PE is obtained by integrating the Maxwell stress tensor over its surface. Therefore, to have a large value of |F*e|, both the potential gradient and the space which contains a steep potential gradient need to be large.3 As k¯ a increases from 0.8 to 2, the counterions near the rear of the PE are removed, producing a greater gradient on the integration surface. Because the space occupied by counterions also increases, |E*z,avg(k¯ a ¼ 2)| > |E*z,avg(k¯ a ¼ 0.8)|. However, if k¯ a exceeds ca. 2, the double layer becomes too thin so that the effect of CC needs to be considered. In this case, the space for a steep potential gradient is limited, and with the decrease in effective charge density, both E*z,avg and |F*e| decrease. Fig. S3 of ESI† also shows that the direction of the induced electric eld is opposite to that of the PE movement. 3.2.2 Inuence of soness parameter l¯ a. Fig. 4 summarizes the variations of the magnitude of the scaled electric force |F*e|, the averaged scaled strength of the local electric eld in the z direction E*z,avg, and the averaged scaled net effective charge density Q*eff,avg with k¯ a for three values of l¯ a. As seen in Fig. 4(a), if k¯ a is small (thick double layer), the curves of |F*e| for different values of l¯ a intersect with each other. For the values of l¯ a examined, |F*e| has a local maximum at k¯ a y 1. In general, the smaller the l¯ a, the easier it is for uid to penetrate through a PE, yielding a larger |F*e|. The behavior of |E*z,avg| seen in Fig. 4(b) is similar to that of |F*e|. The intersection of the curves corresponding to various values of l¯ a at low k¯ a results mainly from the induced electric eld and the equilibrium electric eld established by the PE. For the range of k¯ a considered, |E*z,avg(l¯ a ¼ 4)| > |E*z,avg(l¯ a ¼ 10)|. However, the curve of |E*z,avg(l¯ a ¼ 1)| intersects with the other two curves. As shown in Fig. 4(a), this is because, for low values of k¯ a, |F*e(l¯ a ¼ 1)| increases rapidly with increasing k¯ a.
8868 | Soft Matter, 2014, 10, 8864–8874
For thick double layers, Table S1 of ESI† shows that Q*eff,avg(k¯ a ¼ 0.5) y Q*eff,avg(k¯ a ¼ 0.8), implying that the relative magnitude of the electric force acting on a PE in these two cases depends mainly on the local electric eld. Because |E*z,avg(k¯ a¼ 0.5)| < |E*z,avg(k¯ a ¼ 0.8)|, |F*e(k¯ a ¼ 0.5)| < |F*e(k¯ a ¼ 0.8)|. This can also be inferred from Fig. S4 of ESI,† where the (C1 C2) at l¯ a¼ 4 and that at l¯ a ¼ 10 are almost uniform inside the PE, |F*e| increases monotonically with increasing double-layer thickness. However, because the extent to which counterions are removed from the PE at l¯ a ¼ 1 is greater than that in the other two cases and the space containing a steep electric potential gradient enclosed by the PE surface (i.e., the integration surface) is large, the curve of |F*e(l¯ a ¼ 1)| intersects with that of |F*e(l¯ a ¼ 4)| and that of |F*e(l¯ a ¼ 10)|. At l¯ a ¼ 1 and k¯ a ¼ 0.5, although counterions are removed appreciably from the PE, the space of a steep gradient of concentration enclosed by the PE surface is limited, so that |E*z,avg(l¯ a ¼ 1)| < |E*z,avg(l¯ a ¼ 4)|. The growth of convective ow makes it harder for counterions to be trapped inside the PE to have the xed charge neutralized. Because the smaller the l¯ a the more intense the convective ow, Q*eff,avg(l¯ a ¼ 1) > Q*eff,avg(l¯ a ¼ 4) > Q*eff,avg(l¯ a ¼ 10). In the case of l¯ a ¼ 1, as k¯ a increases from 0.5 to 0.8, the local electric eld increases appreciably. This arises from the fact that both the amount of counterions removed from the PE and the space of a steep gradient of concentration enclosed by the PE surface are appreciable. Therefore, for a medium-large value of k¯ a, the larger the l¯ a the smaller the E*z,avg. For the cases of l¯ a¼4 and l¯ a ¼ 10, because the space of a steep gradient of concentration enclosed by the PE surface is about the same, the relative magnitude of E*z,avg depends upon the amount of counterions removed from a PE. Because the smaller the l¯ a the larger that amount, |E*z,avg(l¯ a ¼ 4)| > |E*z,avg(l¯ a ¼ 10)|. As mentioned by Gopmandal and Bhattacharyya,5–9 the behavior of a porous particle is dominated by the convective uid ow when its l¯ a is small. As seen in Fig. 5(a), the drag factor Uf is also inuenced signicantly by l¯ a. Uf shows a local maximum as k¯ a varies, and the smaller the l¯ a the more
This journal is © The Royal Society of Chemistry 2014
View Article Online
Published on 06 October 2014. Downloaded by University of California - San Francisco on 27/01/2015 19:53:24.
Paper
Soft Matter
Fig. 4 Variations of the magnitude of the scaled electric force |F*e| (a), the averaged scaled strength of the local electric field in the z direction E*z,avg (b), and the averaged scaled net effective charge density Q*eff,avg (c), with ka¯ for three values of la¯ at Qfix ¼ 50, d/a ¼ 0 and Ref ¼ 0.01.
appreciable the presence of that maximum. This is because the smaller the l¯ a of a PE, the easier it is for uid to penetrate it. However, since counterions are also more easily removed by the convective ow, the associated EOF becomes less important, yielding a smaller hydrodynamic force exerted on the PE. This
Fig. 5
explains that Uf increases with increasing l¯ a. Fig. 5(b) shows the variation in the ratio of (electric force acting on a PE/corresponding hydrodynamic force), (Fe/Fh), with k¯ a for various values of l¯ a. This gure reveals that the smaller the l¯ a of a PE the greater the electric force acting on it. The (Fe/Fh) at l¯ a ¼ 1 is
Variations of the drag factor Uf (a), and the ratio (Fe/Fh) (b), with ka¯ for three values of la¯ at d/a ¼ 0, Qfix ¼ 50, and Ref ¼ 0.01.
This journal is © The Royal Society of Chemistry 2014
Soft Matter, 2014, 10, 8864–8874 | 8869
View Article Online
Published on 06 October 2014. Downloaded by University of California - San Francisco on 27/01/2015 19:53:24.
Soft Matter
much larger than that at other values of l¯ a. This is because both Q*eff,avg and E*z,avg are higher and EOF is less signicant in the former. Note that at l¯ a ¼ 1, the electric force contributes more than 50% of the total force exerted on the PE. 3.2.3 Inuence of Reynolds number Ref. Fig. 6 shows the variations of the magnitude of the scaled electric force acting on a PE, |F*e|, the averaged scaled net effective charge density, Q*eff,avg, and the averaged scaled strength of the local electric eld in the z direction, E*z,avg, with Ref for various values of k¯ a. Recall that due to its denition Ref does not correspond to the real velocity of a porous sphere. As seen in Fig. 6(a), the behavior of |F*e| for a thin double layer (k¯ a¼10) is different from that for a thick double layer. In particular, |F*e| shows a local maximum for the cases of k¯ a ¼ 0.1 and 1. As k¯ a increases from 0.1 to 1, the Ref at which the local maximum of |F*e| occurs shis to a larger value. This will be discussed later. A larger Ref implies a faster settling velocity and, therefore, a more signicant convection effect. For the range of k¯ a considered, Q*eff,avg increases monotonically with increasing Ref. As mentioned in the discussion of Fig. 4, if the soness parameter l1 of a PE is large, especially at l¯ a ¼ 1, because it is hard to drive the counterions inside the
Paper
double layer into the PE interior, they are removed from the PE through convective uid ow. A similar idea also applies to a faster settling velocity and, therefore, the effect of CC is less signicant, giving a higher effective charge density. For the values of Ref considered, the smaller the k¯ a the larger the Q*eff,avg. As can be seen in Fig. 6(b), the behavior of Q*eff,avg as Ref varies depends upon the level of k¯ a. In general, the larger the k¯ a (thinner double layer) the more signicant the CC effect. In this case, a larger Ref is needed to have counterions removed from a PE, giving a larger Q*eff,avg. Note that for the cases of k¯ a ¼ 0.1 and 1, the value of Ref at which Q*eff,avg begins to increase rapidly is about the same as that where the local maximum of |F*e| occurs. Although the contribution of Q*eff,avg is less signicant in these cases, it still inuences the values of Ref at which the local maximum of |F*e| and |E*z,avg| occurs, so that the former is slightly larger than the latter. For the case of k¯ a ¼ 1 in Fig. 6(c), |E*z,avg| has a local maximum at Ref y 0.1. This will be discussed later. Fig. 7 and 8 show the contours of the net ionic concentration (C1 C2) (mol m3) on the half plane q ¼ 0 at two levels of k¯ a. As
Fig. 6 Variations of the magnitude of the scaled electric force |F*e| (a), the averaged scaled net effective charge density Q*eff,avg (b), and the averaged scaled strength of the local electric field in the z direction E*z,avg (c), with Ref for three values of ka¯ at d/a ¼ 0, la¯ ¼ 4, and Qfix ¼ 50.
8870 | Soft Matter, 2014, 10, 8864–8874
This journal is © The Royal Society of Chemistry 2014
View Article Online
Published on 06 October 2014. Downloaded by University of California - San Francisco on 27/01/2015 19:53:24.
Paper
Soft Matter
Fig. 7 Contours of the net ionic concentration (C1 C2) (mol m3) on the half plane q ¼ 0 at Qfix ¼ 50, d/a ¼ 0, ka¯ ¼ 1, and la¯ ¼ 4 for three values of Ref, 1.16 103 (a), 0.1 (b), and 0.5 (c). White curves denote the particle surface on which the Maxwell stress tensor is integrated to obtain F*e.
can be seen in Fig. 7(a), if k¯ a ¼ 1, the ionic distribution around the PE is nearly spherical when Ref is small, giving a weak local electric eld. As Ref increases, both the PE surface on which the Maxwell stress tensor is integrated to obtain Fe and the amount of counterions removed are important. Because the space containing a steep potential gradient enclosed by the PE surface decreases appreciably at Ref ¼ 0.5, E*z,avg shows a local maximum at Ref y 0.1, as illustrated in Fig. 6(c). However, in the case of Fig. 8 where k¯ a ¼ 10, because (C1 C2) is almost uniform inside the PE for the values of Ref examined, E*z,avg depends mainly on the extent to which counterions are removed from the PE. We conclude that if the double layer is thin, E*z,avg increases monotonically with increasing Ref. Fig. S6 of ESI† shows the variation of the percentage deviation d ¼ [(FStokes F^ Stokes)/FStokes] 100% in the reference hydrodynamic drag FStokes ¼ 6phauref due to neglecting the convective term r(u$Vu) in eqn (4), where F^ Stokes is the value of FStokes without considering r(u$Vu). Here, d measures how signicant the inertial force acting on a PE is. Fig. S6 of ESI† indicates that d increases rapidly with Ref when Ref exceeds ca. 0.1. However, for the range of Ref considered, the maximum value of d is only ca. 3%. That is, the inertial force is unimportant although the particle velocity is fast (uref y 4.455 101 m s1). As illustrated in Fig. 9(a), the drag factor Uf has a local maximum as Ref varies, and the value of Ref at which the local maximum occurs increases with increasing k¯ a. For small values of Ref, the faster the settling velocity, the more thoroughly the counterions are removed from a PE such that Uf increases with increasing Ref. However, if Ref exceeds a
This journal is © The Royal Society of Chemistry 2014
certain level, the inertial force retarding the EOF is signicant. In this case, the effective charge density of a PE is high, lowering the EOF of counterions inside the PE, so that Uf decreases.8 The behavior of Uf as Ref varies depends upon the level of k¯ a. If k¯ a is small, a small variation in Ref produces an appreciable change in Uf. If k¯ a is sufficiently large, Uf behaves differently from the scaled force F*e seen as shown in Fig. 6(a). This is because the effect of EOF for thin double layers is less signicant than that of CC and, therefore, it is harder to drive counterions out of a PE. Because a larger value of Ref is needed to drive counterions out, both Q*eff,avg and the amount of counterions removed from the PE increase. A less signicant EOF also gives a smaller Uf. Fig. 9(b) shows the variation of the ratio of (electric force/ hydrodynamic force), (Fe/Fh), with Ref for three values of k¯ a. Again, for the range of Ref considered, the effect of DLR is most signicant when the Debye length is comparable to the PE size (i.e., k¯ a y 1). The behavior of (Fe/Fh) at k¯ a ¼ 10 implies that the scaled electric force F*e increases monotonically with increasing Ref. However, because it is too small, F*e is unimportant, and so is the corresponding EOF. 3.2.4 Inuence of PE shape (d/a). If d/a ¼ 0, a PE is spherical, and the larger the (d/a) the more it is stretched. Fig. 10(a) reveals that |F*e| decreases with increasing (d/a), implying that a more stretched PE experiences a smaller electric force in the z direction. Fig. 10(b) suggests that the larger the aspect ratio (d/a) (i.e., longer PE), the higher its effective charge density. If a PE is stretched to be sufficiently long (d/a exceeds ca. 8), Q*eff,avg y 1 when k¯ a is small. In addition, the shape of a
Soft Matter, 2014, 10, 8864–8874 | 8871
View Article Online
Published on 06 October 2014. Downloaded by University of California - San Francisco on 27/01/2015 19:53:24.
Soft Matter
Paper
Fig. 8 Contours of the net ionic concentration (C1 C2) (mol m3) on the half plane q ¼ 0 at Qfix ¼ 50, d/a ¼ 0, ka¯ ¼ 10, and la¯ ¼ 4 for three values of Ref, 1.16 103 (a), 0.1 (b), and 0.5 (c). White curves denote the particle surface on which the Maxwell stress tensor is integrated to obtain F*e.
PE inuences signicantly its capability to be neutralized by the counterions in the liquid phase: the longer the PE, the harder it is for counterions to be attracted into its interior to lower its effective charge. The behavior of Q*eff,avg is dissimilar to that of F*e, implying the presence of other factors that might inuence the electric force acting on the PE. As seen in Fig. 10(c), the magnitude of the averaged scaled strength of the local electric eld in the z direction, |E*z,avg|,
Fig. 9
decreases with increasing (d/a), except for the case of d/a ¼ 0 (i.e., spherical PE). For all values of (d/a), E*z,avg has a local maximum as k¯ a varies. Fig. 10(c) reveals that if k¯ a is smaller than ca. unity, |E*z,avg(d/a ¼ 0)| < |E*z,avg(d/a ¼ 2)|, but that order is reversed if k¯ a exceeds ca. unity. This is because if k¯ a is sufficiently large, the space containing a high net ionic concentration, and therefore a steep gradient of electric potential, enclosed by the PE surface is large.
Variations of the drag factor Uf (a), and (Fe/Fh) (b), with Ref for three values of ka¯ at d/a ¼ 0, la¯ ¼ 4, and Qfix ¼ 50.
8872 | Soft Matter, 2014, 10, 8864–8874
This journal is © The Royal Society of Chemistry 2014
View Article Online
Published on 06 October 2014. Downloaded by University of California - San Francisco on 27/01/2015 19:53:24.
Paper
Soft Matter
Fig. 10 Variations of the magnitude of the scaled electric force |F*e|, (a), the averaged scaled effective charge density Q*eff,avg (b), and the averaged
scaled strength of the local electric field in the z direction E*z,avg (c), with ka¯ for various values of (d/a) at Qfix ¼ 50, la¯ ¼ 4, and Ref ¼ 0.01.
Fig. S7 (a and b) of ESI† show that if the double layer is thick (k¯ a ¼ 0.5), (C1 C2) and, therefore, the electric potential gradient is uniform near the PE surface, implying that E*z,avg depends mainly upon the amount of counterions removed from the PE. Since the amount of counterions removed at d/a ¼ 2 is greater than that at d/a ¼ 0, |E*z,avg(d/a ¼ 2)| > |E*z,avg(d/a ¼ 0)|. However, as seen in Fig. S7(b) of ESI,† if k¯ a exceeds ca. 1, |E*z,avg(d/a ¼ 0)| > |E*z,avg(d/a ¼ 2)|. As illustrated in Fig. S7 of ESI,† this is because if k¯ a is sufficiently large, the space containing a steep electric potential gradient enclosed by a spherical PE surface is larger than that with a non-spherical (d/a ¼ 2) surface. We conclude that the shape of a PE is capable of inuencing the electric force acting on it through E*z,avg. The inuence of Q*eff,avg is important only if a PE is highly stretched and the double layer is thick. Fig. 11 shows the variation of the scaled electric potential 4* ¼ 4/4ref with 4ref ¼ kBT/z1e on the half plane q ¼ 0 along the PE surface, measured by the arc length from its south pole for various values of (d/a). The corresponding contours of 4* are shown in Fig. S8 of ESI.† This gure reveals that the larger the (d/a) (more stretched PE), the less symmetric the electric potential distribution. Note that 4* is negative near the south pole of the PE, which arises from the penetration of counterions. This phenomenon has not been reported previously.
This journal is © The Royal Society of Chemistry 2014
Fig. 11 Variation of the scaled electric potential 4* ¼ 4/4ref on the half
plane q ¼ 0 along the PE surface, measured by the arc length from its south pole, at Qfix ¼ 50, ka¯ ¼ 2, la¯ ¼ 4, and Ref ¼ 0.1 for various values of (d/a). Short dashed lines: middle plane of PE; long dashed lines: north pole of PE.
Soft Matter, 2014, 10, 8864–8874 | 8873
View Article Online
Published on 06 October 2014. Downloaded by University of California - San Francisco on 27/01/2015 19:53:24.
Soft Matter
The behavior of the drag factor Uf seen in Fig. S9(a) of ESI† is similar to that of the corresponding |Fe*| presented in Fig. 10(a). As mentioned in the discussion of Fig. 2(b), this is due to the effect of DLR, which enhances the counterion-rich liquid ow inside a PE. Due to the variation in the degree of EOF effect with the PE shape, the curves of Uf(d/a ¼ 0) and Uf(d/a ¼ 2) intersect. As seen in Fig. S9(b) of ESI,† although |Fe*| has its largest value at d/a ¼ 0, that value is unimportant compared with the corresponding hydrodynamic force. We conclude that the behavior of Uf seen in Fig. S9(b)† results from competition of the effects of DLR, CC, and EOF. As mentioned in the discussion of Fig. 10(a), a stretched PE experiences a smaller electric force than a spherical PE.
4. Conclusions The sedimentation of an isolated, charged PE in an aqueous salt solution subject to an applied eld is modeled, taking account of the variation of its shape. We show that the interaction of the effects of double-layer relaxation, effective charge density, and strength of induced relaxation electric eld produces the following complex and interesting sedimentation behaviors: (i) the higher the xed charge density of a PE the more signicant the effect of double-layer relaxation, thereby enhancing the counterion-rich liquid ow inside it, making the behavior of the drag factor similar to that of the magnitude of the electric force acting on the PE. (ii) For thick double layers, because the electric force acting on a loosely structured PE is dominated by the convective uid ow, its effective charge density is higher than that of a compactly structured PE. Due to the effect of electroosmotic ow, the drag acting on a loosely structured PE is smaller than that on a more compactly structured PE. (iii) The behavior of the averaged effective charge density of a PE as the Reynolds number varies depends upon the level of double-layer thickness. The thinner the double layer, the larger the Reynolds number needed to drive counterions out of a PE. (iv) For thick double layers, the magnitude of the electric force acting on a PE has a local maximum as the Reynolds number varies, but increases monotonically with increasing Reynolds number for thin double layers. For the range of the salt concentration considered, the drag factor always shows a local maximum as the settling velocity varies. This behavior is different from those observed in the literature. (v) The longer the PE the harder it is for counterions to be attracted into its interior to have its xed charge neutralized, producing a greater electric force. However, the effective charge density of the PE is not necessarily higher because the induced relaxation electric eld also plays a role.
References 1 E. Lee, J.-W. Chu and J.-P. Hsu, J. Chem. Phys., 1999, 110, 11643–11651. 2 M.-H. Chih, E. Lee, J.-W. Chu and J.-P. Hsu, Langmuir, 2000, 16, 1650–1654.
8874 | Soft Matter, 2014, 10, 8864–8874
Paper
3 H. J. Keh and J. M. Ding, J. Colloid Interface Sci., 2000, 227, 540–552. 4 F. Keller, M. Feist, H. Nirschl and W. D¨ orer, J. Colloid Interface Sci., 2010, 344, 228–236. 5 S. Bhattacharyya and P. P. Gopmandal, 3rd Micro and Nano Flows Conference, Thessaloniki, Greece, 22–24 August 2011. 6 S. Bhattacharyya and P. P. Gopmandal, Colloids Surf., A, 2011, 390, 86–94. 7 S. Bhattacharyya and P. P. Gopmandal, Proc. World Acad. Sci. Eng. Tech., 2012, 6, 1088–1094. 8 S. Bhattacharyya and P. P. Gopmandal, So Matter, 2013, 9, 1871–1884. 9 P. P. Gopmandal and S. Bhattacharyya, Colloid Polym. Sci., 2013, 1–10. 10 K.-L. Liu, J.-P. Hsu, W.-L. Hsu, L.-H. Yeh and S. Tseng, Electrophoresis, 2012, 33, 1068–1078. 11 J. P. Clamme, J. Azoulay and Y. M´ ely, Biophys. J., 2003, 84, 1960–1968. 12 J. F. L. Duval, K. J. Wilkinson, H. P. van Leeuwen and J. Buffle, Environ. Sci. Technol., 2005, 39, 6435–6445. 13 T. Hoare and R. Pelton, Polymer, 2005, 46, 1139–1150. 14 J.-P. Hsu, L.-H. Yeh and M.-H. Ku, J. Colloid Interface Sci., 2007, 305, 324–329. 15 L.-H. Yeh and J.-P. Hsu, So Matter, 2011, 7, 396–411. 16 J.-P. Hsu, C.-Y. Lin, L.-H. Yeh and S.-H. Lin, So Matter, 2012, 8, 9469–9479. 17 M. Zhang, L.-H. Yeh, S. Qian, J.-P. Hsu and S. W. Joo, J. Phys. Chem. C, 2012, 116, 4793–4801. 18 L.-H. Yeh, M. Zhang, S. Qian and J.-P. Hsu, Nanoscale, 2012, 4, 2685–2693. 19 Y.-Y. Chen, J.-P. Hsu and S. Tseng, J. Colloid Interface Sci., 2014, 421, 154–159. 20 H. J. Keh and W. C. Chen, J. Colloid Interface Sci., 2006, 296, 710–720. 21 W. M¨ achtle, in Encyclopedia of Analytical Chemistry, John Wiley & Sons, Ltd, 2006. 22 W. Maechtle and L. B¨ orger, Analytical Ultracentrifugation of Polymers and Nanoparticles, Springer, 2006. 23 V. Doln´ık, Electrophoresis, 2008, 29, 143–156. 24 V. Starchenko, M. M¨ uller and N. Lebovka, J. Phys. Chem. B, 2012, 116, 14961–14967. 25 H. Eisenberg, Biological macromolecules and polyelectrolytes in solution, Clarendon Press, 1976. 26 P. Y. Chen, A. Y. M. Lin, Y. S. Lin, Y. Seki, A. G. Stokes, J. Peyras, E. A. Olevsky, M. A. Meyers and J. McKittrick, J. Mech. Behav. Biomed. Mater., 2008, 1, 208–226. 27 L.-H. Yeh, J.-P. Hsu, S. Qian and S. Tseng, Electrochem. Commun., 2012, 19, 97–100. 28 G. Neale, N. Epstein and W. Nader, Chem. Eng. Sci., 1973, 28, 1865–1874. 29 K. Makino and H. Ohshima, Langmuir, 2010, 26, 18016– 18019. 30 P. P. Gopmandal and S. Bhattacharyya, Colloids Surf., A, 2013, 433, 64–76.
This journal is © The Royal Society of Chemistry 2014
