Sedimentation Velocity and Potential in Dilute Suspensions of Charge

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Sedimentation Velocity and Potential in Dilute Suspensions of Charge-Regulating Porous Spheres Chien Y. Lin, and Huan Jang Keh J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.9b00277 • Publication Date (Web): 05 Mar 2019 Downloaded from http://pubs.acs.org on March 14, 2019

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The Journal of Physical Chemistry

Sedimentation Velocity and Potential in Dilute Suspensions of ChargeRegulating Porous Spheres Chien Y. Lin and Huan J. Keh* Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan, ROC

ABSTRACT:

The sedimentation of a charge-regulating porous sphere surrounded by an arbitrary

electric double layer, that usually models a permeable polyelectrolyte coil or aggregate of nanoparticles, is analyzed for the first time. The hydrodynamic frictional segments and ionogenic functional groups uniformly distribute in the porous sphere, and a regulation mechanism for the dissociation and association reactions occurring at these functional groups linearly relates the local electric potential to fixed charge density. The linearized electrokinetic equations governing the ionic concentration (or electrochemical potential energy), electric potential, and fluid velocity fields are solved for the case of a small basic fixed charge density by the regular perturbation method. Analytical formulae for the sedimentation velocity of a porous sphere and sedimentation potential of a dilute suspension of porous spheres are then obtained. The charge regulation tends to reduce the electrokinetic retardation to sedimentation velocity and the sedimentation potential (can be as much as 50 and 25 per cents, respectively) compared to the case that the fixed charge density is a constant. Both the electrokinetic retardation to sedimentation velocity and the sedimentation potential vanish at the isoelectric point of the particles. The increase in the bulk concentration of the potentialdetermining ions crossing the isoelectric point changes signs of the fixed charges and thus causes a reversal in the direction of the sedimentation potential. The effects of charge regulation on the sedimentation of porous particles differ substantially from those of hard particles.

* Corresponding author. Tel.: 886-2-33663048.

Fax: 886-2-23623040.

E-mail: [email protected].

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1.

INTRODUCTION The migration of charged particles in an ionic fluid under a force field (e.g., gravitational,

centrifugal, and electromagnetic) is a common phenomenon in various areas of colloidal science as well as chemical, biomedical, and environmental engineering.1-3 This phenomenon is more complex than that of uncharged particles since the ambient fluid flow distorts the electric double layers surrounding the charged particles (known as the relaxation effect) and induces electric fields.4,5 These induced electric fields superimpose to give rise to a migration potential gradient in a suspension of charged particles. At the same time, the migration potential adjusts the velocity distribution of the ionic fluid through electrostatic interactions and diminishes the particle velocity by an electrophoretic effect. Using a regular perturbation method, Booth6 first derived closed-form formulae for the sedimentation velocity and potential in a suspension of hard (impermeable) spherical particles with arbitrary electric double layers as power expansions in their low surface potential. Later, a numerical solution relaxing the assumption of small surface potential in this analysis was obtained by Stigter7. On the other hand, Ohshima et al.8 obtained analytical formulae and numerical solutions of the sedimentation velocity and potential for a wide range of double layer thickness and surface potential. In particular, the perturbation analysis of Booth on the sedimentation velocity and potential was extended to suspensions of interacting hard spheres,9-12 porous (permeable to the solvent and small ions) spheres,13-15 soft (hard in core but porous in surface layer) spheres,16,17 and porous spherical shells (microcapsules or vesicles)18 with low fixed charge densities. The previous analytical investigations for the sedimentation phenomena of charged particles often assumed that either zeta potential or fixed charge density is constant. However, the surface potential or fixed charge density of numerous realistic particles, such as polyelectrolytes and biocolloids, are frequently determined by the dissociation and association reactions as well as the density of the ionogenic functional groups.19-24 When such charge-regulating particles are subjected to a force field (that induces electric fields), their local surface potential and fixed charge density are functions of the concentration of the potential- or charge-determining ions and they undergo migration accordingly. The sedimentation in a suspension of charge-regulating hard spherical particles with arbitrary electric double layers in an electrolyte solution was analytically studied,25 while experimental evidences for charge regulation in the sedimentation of colloidal silica in ethanol26 and of iron tailings slurry or flocs27 were provided. Also, electrokinetic phenomena of porous particles, such as flocs or polyelectrolyte coils, have been found useful in interpreting the charge regulation phenomenon of spherical protein particles from the experimental results over a wide range of the pH value.28 Even ACS Paragon Plus Environment 2

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so, the effect of charge regulation on the electrophoretic motion of porous particles was found to appreciably differ from that of hard particles.29 In this article, we analyze the sedimentation phenomena in a suspension of charge-regulating porous particles in an electrolyte solution for the first time. The linearized electrokinetic equations governing the ionic concentration (or electrochemical potential energy), electric potential, and fluid velocity fields in the system are solved by the perturbation method for the case of small basic fixed charge density of the porous spherical particles. Analytical expressions for the sedimentation velocity and potential are obtained.

2.

BASIC ANALYSIS As shown in Figure 1, we consider the sedimentation of a charge-regulating spherical porous

particle of radius a in a general liquid solution containing M ionic species with the velocity Ue z (to be determined) under the action of the gravitational acceleration ge z , where e z is the unit direction. The spherical coordinate system (r ,  ,  ) originating from and

vector in the z

translating with the center of the particle is employed. The possible swelling of the porous particle under the fluid convection during its motion is assumed to be negligible so that its size and spherical shape will not change. 2.1. Equations of Charge Regulation. A general model for a porous entity to develop fixed space charges comprises the following dissociation and association reactions occurring at the amphoteric functional groups AB distributed uniformly throughout the entity:

AB2Z   AB  BZ  ,

AB  A where B

Z

Z

B

B ,

(2)

is the charge- or potential-determining cation with the valence Z

hydrogen ion H Z

(1)

Z



(usually, the

with Z  1 ). Using the Boltzmann equation for the equilibrium concentration of

at low electrical potential distribution  (r , ) (the Debye-Hückel approximation), one can

express the volumetric fixed charge density distribution Q(r , ) inside the charge-regulating porous entity as a linear function of  ,23

ZeN {[AB2Z  ]  [A Z  ]} Q  A1  A2 , [AB2Z  ]  [AB]  [A Z  ]

(3)

where A1 and A2 are constants given by 2

A1  ZeN

n1  K  K  2

n1  K  n1  K  K 

,

(4)



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2

A2  ( Ze) 2 N

K  n1 (n1  4 K  n1  K  K  ) 2

kT (n1  K  n1  K  K  ) 2

,

(5)

N is the number density of the ionogenic functional groups, K  and K  are the equilibrium

constants of the reactions in eqs 1 and 2, respectively, at the absolute temperature T , k is the  Boltzmann constant, n1 is the constant bulk concentration (number density) of B Z  , and e is the

charge of a proton. Equations 4 and 5 show that the basic fixed charge density A1 can be either positive or negative and the charge regulation coefficient A2  0 . In the limit A2  0 , the porous entity has a uniform fixed charge density Q  A1 (the charge regulation effect disappears) which   equals the saturation value ZeN (as n1   or K   0 ) or  ZeN (as n1  0 or K    ).

Equations 3 and 4 indicate that the equilibrium isoelectric point of the porous entity (with

Q

( eq )

 A1   ( eq )  0 ) for given values of

K

and

K

occurs at

n1  ( K  K  )1 / 2 . If

n1  ( K  K  )1 / 2 , A1 is negative (so are  ( eq ) and Q ( eq ) ) and its magnitude increases with a   1/ 2 ( eq ) ( eq ) decrease in n1 until ZeN is reached; if n1  ( K  K  ) , A1 is positive (so are  and Q )  and increases with an increase in n1 to the limit ZeN . The porous entity is more negatively charged ( eq ) or less positively charged with a greater value of  , and vice versa.

2.2. Electrokinetic Equations. The velocity of the porous particle Ue z in Figure 1 is assumed to be small so that the concentration distribution nm (r ,  ) of the ionic species m and the electrostatic potential field  (r , ) can be expressed as nm  nm( eq )  nm ,

   ( eq )   ,

(6) (7)

where, nm( eq ) (r ) and  ( eq ) (r ) are the equilibrium distributions of the ionic concentration and electrostatic potential, respectively, which relate mutually via the Boltzmann equation, and

nm (r ,  ) and  (r ,  ) are their perturbed quantities. The small perturbations nm ,  , and fluid velocity field u(r , ) are governed by the following linearized modified Stokes/Brinkman equation, continuity equations of the ionic species, and Poisson-Boltzmann equation:14

 

[ 2  h(r )2 ]  u     [ 2 ( eq )   2 ( eq ) ] ,

 2 m 

zme kT  ( eq )  ( m  u) , kT Dm

 2 

A2



m = 1, 2, …, M,

z m enm z e ( eq ) exp[ m ]( m  z m e ) . kT m 1 kT

(8) (9)

M

h(r )  

(10)

In the previous equations,  m (r ,  ) is the electrochemical potential energy distribution of the ionic species m , ACS Paragon Plus Environment 4

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 m 

kT nm  z m e ; nm( eq )

(11)

nm , z m , and Dm are the bulk concentration, valence, and diffusion coefficient, respectively, of this species (we let the species 1 be the charge-determining ion B Z  and thus z1  Z ); h(r ) is a step function equal to unity if 0  r  a and zero otherwise;  and 

are the viscosity and

1/ 2 permittivity, respectively, of the fluid;   ( f /  ) , whose reciprocal is the flow penetration length

or the square root of the fluid permeability in the porous particle (where f is the friction coefficient per unit volume). The boundary conditions to solve eqs 8-10 at the surface of the porous particle are

r  a:

u r , u ,  rr ,  r , m ,

  m ,  , and are continuous, r r

(12)

where ( u r , u ) and (  rr ,  r ) are components of the fluid velocity and viscous stress, respectively, in spherical coordinates. The continuity of the perturbed fields  / r and  m / r results from the assumptions that the dielectric permittivity of the electrolyte solution in the interior of the particle is the same as the bulk value (although the former is lower in general) and the relevant ion partitioning effect arising from interactions between the mobile ions and the fixed charges is neglected. The conditions far away from the particle can be expressed as r :

u   Ue z ,

m  0 ,

  0 ,

(13)

taking a reference frame traveling with the particle. 2.3. Equilibrium Electric Potential. For a charge-regulating porous sphere in an unbounded electrolyte solution, the equilibrium potential distribution  ( eq ) appearing in eqs 8-10 and satisfying the Poisson-Boltzmann equation is 2

 ( eq )   eq1 (r )Q  O(Q ) ,

(14)

where kT a [1  (1  a ) sinh(r )] e r kTa  eq1   (a)exp[ (r  a)] er

 eq1 

(if 0  r  a ),

(15)

(if r  a ),

(16)

 ( x)  x cosh x  sinh x ,

(17)

  a sinh(a )  a cosh(a ) ,

(18)

2 1/ 2   (m1 zm2 nm e 2 / kT )1 / 2 is the Debye screening parameter,    (1  A2 /  ) is this parameter M

2 modified by the charge regulation effect with the intensity A2 /  , and Q  A1e /  2 kT is a

dimensionless characteristic value of the fixed charge density distribution Q , whose magnitude is assumed to be small. Note that  eq1 is always positive. ACS Paragon Plus Environment 5


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2.4. Solution of the Perturbed Quantities. The small perturbed quantities ur , u , m , and

 satisfying eqs 8-13 and the continuity equation   u  0 can be solved in terms of the regular perturbation expression for the sedimentation velocity of the porous sphere, 2

3

U  U 0  U1 Q  U 2 Q  O(Q ) ,

(19)

and the results are 2

3

u r  {U 0 F0 (r )  U 1 F0 (r )Q  [U 0 F2 (r )  U 2 F0 (r )]Q } cos   O(Q ) , tan   2 u   (r ur ) , 2r r 2

 m  U 0 Fm (r ) cos  Q  O(Q ) ,

(20) (21) (22)

2

  U 0 F (r ) cos  Q  O(Q ) .

(23)

In these equations, the functions F0 (r ) , F2 (r ) , Fm (r ) , and F (r ) are listed in eqs A1, A2, A3, and A8, respectively, in the Appendix. Note that the leading relaxation effect on the electric double layer around the porous sphere is incorporated in the solutions for  m and  up to the first order of Q or A1 . 3.

SEDIMENTATION VELOCITY The total force exerted on the settling charged porous sphere is the total of the gravitational force,

electric force, as well as hydrodynamic drag force on the particle. The gravitational and buoyant force is Fg 

4 3 πa (1   p )(  p   ) ge z , 3

(24)

which is independent of the characteristic fixed charge density Q , where  is the fluid density;

 p and  p are the porosity and density of the porous sphere, respectively. The electrostatic force acting on the charge-regulating porous sphere is

Fe  2 π 

π

0

a

 {[A  A  0

1

2

( eq )

]  A2 ( eq ) }r 2sin drd .

(25)

Substituting eqs 14 and 23 into eq 25, we obtain 2 3 4 kT Fe   π a 2U 0 F (a )[ 2  A2 eq1 (a )]Q  O(Q ) , 3 e

(26)

2

where the leading term is of the second order Q . The hydrodynamic force exerted on the porous particle is Fh  2 πf 

π

0



a

0

u r 2sin drd .

(27)

Substitution of eqs 20 and 21 into eq 27 leads to 2 3 4 Fh  π2 a 3{F0 (a )(U 0  U1 Q)  [U 2 F0 (a )  U 0 F2 (a )]Q  O(Q )}e z . 3 ACS Paragon Plus Environment 6

(28)

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The total force acting on the settling porous sphere is zero. Applying this constraint to the sum of eqs 24, 26, and 28, we obtain the particle velocity in eq 19 with 3 (a)  2 cosh(a)(a)3 2 U0  a (1  ε p )(  p   ) g , 9 (a)( λa ) 2 U1  0 , F (a) U0 U2   {F2 (a )  [ 2 kT  A2e eq1 (a )]  2 } ,  ae F0 (a )

(29) (30) (31)

where U 0 is the sedimentation velocity of an uncharged porous sphere. As expected, the effect of the fixed charge density on the sedimentation velocity U of the particle starts from the second order

Q

2

because of the interaction force between the sedimentation potential (having the order Q as

shown in the next section) and the fixed charge density as given by eq 26. The velocity coefficient

U 2 , which should be negative, depends on the shielding parameter a , dimensionless ionic 2 2 2 diffusivities e Dm / k T , and charge regulation characteristics of the particle and electrolyte 2 solution. Noting that the quantity (a ) 2 Q (  a A1e / kT ) is independent of  for a characteristic 2

charge density A1 , we shall present the results of both  U 2 /(a) 4U 0 and  U 2 Q /U 0 (which are positive and dimensionless) computed from eqs 29 and 31 for the electrokinetic retardation to the sedimentation velocity of the porous sphere in Section 5.

4.

SEDIMENTATION POTENTIAL For a dilute suspension of charged spherical particles undergoing sedimentation, the

sedimentation potential is the average electric field subject to the constraint of vanishing electric current and can be expressed as14

ESED 

eU 0e z 3

a kTΛ



M

z m1

 m m

n Dm (r 3

dFm dr

2

 r 2 Fm ) r  Q  O(Q ) ,

(32)

where Λ  m1 zm2 nm Dm e 2 / kT is the electric conductivity of the ionic fluid and  is the M

apparent volume fraction of the particles. The substitution of eq 29 for U 0 and eq A3 for Fm (r ) (with eqs 15 and 16 for  eq1 ) into eq 32 results in  ESED  (1   p )(  p   ) g   Ee z , Λ

(33)

where

E 

A1 {sinh(a)[acosh(a) L1  (2   2 )sinh(a) L3 ] 2 2 3  (   )a  (a) 2 4

2

 acosh(a)[sinh(a) L2  a(2   2 )cosh(a) L4 ]} ,

(34)

L1  3 22 (1  a)   2 [3 2 (1  a)  2 (3  3a   2 a 2  2 3a 3  22 a 3 )]   4 (3  3a  3 2 a 2  22 a 3 ) ,


(35)


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L2  3 2 2 (1  a)   2 [3 2 (1  a)  2 (3  3a  2 2 a 2   3a 3  22 a 2 )]   4 (3  3a  3 3a 3  2a 22 ) , L3  3 2 (1  a )   2 (3  3a  3 3 a 3  22 a 3 ) ,

(36)

L4  3 (1  a)   (3  3a  3 a  2 a ) ,

(38)

2

2

2 2

(37)

2 2

for the sedimentation potential in a suspension of charge-regulating porous spheres to the first order of Q or A1 . According to an Onsager reciprocal relation,30  E is also the electrophoretic mobility of the charge-regulating porous spheres, which has the same sign as A1 and does not depend on the ionic diffusivities Dm to this order. Note that the direction of the sedimentation potential depends on the 2 product of  p   and  E . In the next section, we shall present the results of both  E / A1a

(which is positive, dimensionless, and independent of  ) and e E / kT (   2 Q E / A1 ) for the sedimentation potential of the suspension. 5.

RESULTS AND DISCUSSION The sedimentation velocity and potential in a dilute suspension of charge-regulating porous

spheres in a general electrolyte solution can be calculated using eqs 19, 29-31, and 33-38 obtained in the previous sections. The typical physical characteristics in the aqueous solution of a univalent electrolyte ( M  2 ,

Z  z1   z2  1 , and

n2  n1 ) at room temperature with T  298 K ,

  6.95 10 10 C 2 J 1m 1 , a  107 m (giving ea 2 / kT  5.6  10 4 m 3 / C ), and

N  10 23 m 3

will be used for the calculations in this section. The corresponding equilibrium values of the 2 normalized extent of the charge regulation effect A2 /  , basic fixed charge density (a ) 2 Q 2 ( eq ) (  ea A1 / kT ), fixed charge density distribution Q (r ) / eN , and electric potential distribution

 ( eq ) (r ) / eNa 2 around a porous sphere as functions of the electrolyte concentration n1 and reaction equilibrium constants K  and K  were graphically presented elsewhere.23 5.1. Sedimentation Velocity. The sedimentation velocity of a charge-regulating porous sphere 2

can be calculated to O(Q ) using eqs 19 and 29-31. Figure 2 is plotted for the dimensionless coefficient  U 2 /(a) 4U 0 of the charge-induced retardation to this velocity in the KCl solution  2 2 2 ( D2  D1 and k T /D1e  0.259 )31 versus the electrolyte concentration n1 for various values

of the shielding parameter a (relative hydrodynamic resistance) of the porous sphere. The solid 4 and dashed curves represent the cases that K   K   10 M (with Q as a function of the local 2 electric potential) and that Q  A1  eN (no charge regulation with A2 /   0 and    ),14

respectively. As expected, the velocity retardation coefficient  U 2 /(a) 4U 0

is positive and

 decreases monotonically with an increase in  a for a given value of n1 . In the limit  a   , the

porous particle behaves like an impermeable but conductive sphere and thus  U 2 /(a ) 4U 0  0 ACS Paragon Plus Environment 8

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(different from a dielectric hard sphere with or without charge regulation effect).25 For a fixed value  of  a , the coefficient  U 2 /(a) 4U 0 has maxima at some finite values of n1 and vanishes in the   limiting cases n1  0 ( a  0 ) and n1   ( a   ), consistent with the corresponding results

for a hard sphere. This coefficient has only one local maximum for the simple case of constant fixed charge density Q  A1 , but may have two local maxima for the case with charge regulation. In fact, the charge regulation effect can reduce the value of  U 2 /(a) 4U 0 significantly (by more than 50 percent). 2

In Figure 3, the normalized velocity retardation  U 2 Q /U 0 for the sedimentation of a charge regulating porous sphere in the KCl solution is plotted versus the electrolyte concentration n1 for

a  1 and various values of the reaction equilibrium constants K  and K  . Evidently, the value 2

 1/ 2 of  U 2 Q /U 0 vanishes and no velocity retardation occurs at the isoelectric point n1  ( K  K  ) 2

 (where A1  0 ). For constant values of a , n1 , and K  K  , the quantity  U 2 Q /U 0 decreases

with an increase in the ratio K  / K  , as shown in Figure 3a. For fixed values of K  , K  , and a , 2

 interestingly,  U 2 Q /U 0 increases with decreases or increases in n1 from zero at the isoelectric  1/ 2  point n1  ( K  K  ) , reaches local maxima, and then decreases to zero in the limits n1  0 and

n1   . These corresponding maxima shift toward the isoelectric point and become larger as K  / K  decreases, as shown in Figure 3a, and shift toward greater values of n1 as K  K  increases, as indicated in Figure 3b, keeping the other constant. The effect of charge regulation on the sedimentation velocity of a porous particle differs substantially from that of a hard particle.25 5.2. Sedimentation Potential. The sedimentation potential in a dilute suspension of chargeregulating porous spheres can be calculated to the first order of Q or A1 using eqs 33-38. The 2 dimensionless coefficient  E / A1a (which is positive) of the sedimentation potential is plotted  versus the electrolyte concentration n1 in Figure 4 for various values of the shielding parameter

a

of the porous spheres, in which the solid and dashed curves denote the cases of

K   K   104 M and Q  A1 , respectively. Again,  E / A1a 2 decreases monotonically with an  increase in  a for a given value of n1 . For a specified value of  a , this sedimentation potential   coefficient in general decreases with an increase in n1 (or a ) from a constant at n1  0  ( a  0 ) to another at n1   ( a   ), consistent with the corresponding results for hard

spheres with or without charge regulation effect.25 The charge regulation effect can reduce the value 2  of  E / A1a significantly (by more than 25 percent), and even make it increase with n1 .

In Figure 5, numerical values of the normalized sedimentation potential e E / kT in a  1/ 2 suspension of charge-regulating porous spheres, which has the same sign as n1  ( K  K  ) (or A1 )  1/ 2 and vanishes at the isoelectric point n1  ( K  K  ) , are plotted versus the electrolyte concentration

n1 for a  1 and different values of K  and K  . For constant values of a and n1 , this ACS Paragon Plus Environment 9


normalized sedimentation potential decreases as K  K  increases, as shown in Figure 5b, and its magnitude decreases with increases in K  / K  , as indicated in Figure 5a, keeping the other fixed.  Although e E / kT in general increases with increases in n1 , its value may not be a monotonic   function of n1 and have local extrema at some finite values of n1 , for an otherwise specified

condition. Again, these effects of charge regulation on the sedimentation potential of porous particles differ substantially from those of hard particles.25 6.

CONCLUSIONS In this paper, the sedimentation phenomena in a dilute suspension of charge-regulating porous

spherical particles in an arbitrary electrolyte solution are analytically studied. The fixed charge density distribution in the porous particles is related to the local electric potential through a linear regulation model dependent on the equilibrium constants K  and K  of the dissociation and  association reactions occurring at the ionogenic functional groups and the bulk concentration n1 of

the potential-determining ions. The linearized electrokinetic equations governing the ionic concentration (or electrochemical potential energy), electric potential, and fluid velocity fields are solved for the case of a small basic fixed charge density by the regular perturbation method. The sedimentation velocity of the particles is expressed by eqs 19 and 29-31 and the sedimentation potential of the suspension is given by eqs 33-38 as functions of the charge regulation characteristics. Both the charge-induced retardation to sedimentation velocity and the sedimentation potential decrease with an increase in the shielding parameter a and vanish at the isoelectric point

n1  ( K  K  )1 / 2 of the particles. The increase in n1 crossing the isoelectric point changes signs of the fixed charges and thus causes a reversal in the direction of the sedimentation potential. The effects of charge regulation on the sedimentation of porous particles are somewhat similar to but still differ substantially from those of hard particles. The derivation of eqs 31 and 34 for the sedimentation of charge-regulating porous spheres is based on the Debye-Hückel approximation with small electric potential or fixed charge density. Similar formulas for the sedimentation of hard spherical particles with this approximation were found to be quite accurate as long as the reduced zeta potential  ( eq ) (a) e / kT is not greater than 2.8 Thus, our results may be tentatively used for cases of reasonably high electric potential or fixed charge density (say, A1  2  10 6 C/m 3 in aqueous solutions with 1   1 nm or Q  A1e /  2 kT  0.1 ). APPENDIX For conciseness some functions in Section 2 are given here. In eqs 20 and 22, ACS Paragon Plus Environment 10

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a if r  a , F0 (r )  3 A[ (a)  2( )3 (r )] r a a if r  a ; F0 (r )  1  3( λa ) 2 A ( λa )  AB( )3 r r a 2 r 1 r F2 (r )  C1  C2 (r )( ) 3  2  G (r )dr  3  r 3G (r )dr r 3 0 r 0 r r 2  5 3 [ (r )   (r )G (r )dr   (r )   (r )G (r )dr ] if r  a , 0 0 r a a 1 r 5 1 r 1 r F2 (r )  C3  C4 ( ) 3  r G (r )dr   r 3G (r )dr   r 2G (r )dr 3   r r 15r 3r 3  r2 r   G (r )dr if r  a ; 15   d d z e 1 r Fm (r )  m [ 2  r 3 F0 (r ) eq1 dr  r  F0 (r ) eq1 dr ] , r 3Dm r 0 dr dr where the function  (x) was defined by eq 17, e d eq1 M  G (r )   z n F (r ) , kT rdr m1 m m m a

a





(A1a) (A1b)

(A2a)

(A2b) (A3)

(A4)

C1  A{a 2 (a )  G (r )dr   (a )  r 2G (r )dr

a 2 2 a  2   (r )G (r )dr  cosh(a )  r 3G (r )dr}  2  G (r )dr , 0  0 3 3 0 a a 1 C2  2 A{ r 2G (r )dr  5 3 [3 (a )  2 sinh(a )(a ) 3 ]  (r )G (r )dr  0 a 3 a 1 a 2 a  (a 2  2 )  G (r )dr  2 3  r 3G (r )dr}  5 3   (r )G (r )dr ,   a 0 a 0 a B a 1 a C3  A{a 2  G (r )dr   (a )[(a ) 2  r 2G (r )dr   r 3G (r )dr ]   3 a 0 a 1 a 3  2a 2   (r )G (r )dr}  r G (r )dr , 0 3a  a B a 1 C4  A{  r 2G (r )dr  2 [(2 a 2  5) B  3 (a)(a) 2  2 sinh(a)(a) 4 ] G (r )dr  3  5 a a a 3 1 1  2(a 2  2 )   (r )G (r )dr   (a )  r 3G (r )dr )}  r 5 G ( r ) dr , 3  0 0 a 15a  3 1 A  [3 (a)  2 cosh(a)(a) ] ,

2

a

(A5a)

(A5b)

(A5c)

(A5d) (A6a)

B   (a)[( λa )  6]  2 sinh(a)( λa ) ,

(A6b)

 ( x)  x sinh x  cosh x ,  ( x)  x cosh x  sinh x .

(A7a) (A7b)

2

2

In eq 23,

F (r )  F (r ) 

e

kT 2r e

kT 3 r 2

{ (r )[Y1  2

2 2 S ( 0 , r ,  )]   (r ) S (0, r , )} 2  2

{ (r )[Y2  S  (r , ,  )]   (r )[Y2  S (r , ,  )]} ACS Paragon Plus Environment 11

if r  a ,

(A8a)

if r  a ,

(A8b)


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which satisfies

1 d 2 dF 2 e M  (r )  [ 2   2  h(r )( 2   2 )]F    z m nm Fm , 2 dr r dr r kT m1

(A9)

where

Y1 

 W

{[  (a ) ' (a )   (a )  ' (a )][S (a, ,  )  S (a, ,  )]}

2  { (a )[ ' (a )   ' (a )]   ' (a )[ (a )   (a )]}S (0, a, ) W 2 2  2 S  (0, a, ) ,  Y2 

(A10a)

1 {[ (a ) ' (a )   (a ) ' (a )]S  (a, ,  ) W  [ (a )  ' (a )   (a ) ' (a )]S (a, ,  )} 

3 [  (a ) ' (a )   (a )  ' (a )] S (0, a, ) , W 2

(A10b)

M

S ( x, y,  )    (r ) zm nm Fm (r )dr ,

(A11a)

S  ( x, y,  )    (r ) zm nm Fm (r )dr ,

(A11b)

W   ' (a )[ (a )   (a ) ]   (a ) [ ' (a )   ' (a )] ,

(A12)

y

x

y

x

m 1 M

m1

and the prime on a function means its derivative with respect to its argument. ACKNOWLEDGMENT This research was supported by the Ministry of Science and Technology of the Republic of China (Taiwan) under the Grant MOST106-2221-E-002-167-MY3.

REFERENCES (1) Hsieh, T. H.; Keh, H. J. Magnetohydrodynamic Effects on a Charged Colloidal Sphere with Arbitrary Double-Layer Thickness. J. Chem. Phys. 2010, 133, 134103. (2) Satoh, A. Sedimentation Behavior of Dispersions Composed of Large and Small Charged Colloidal Particles: Development of New Technology to Improve the Visibility of Small Lakes and Ponds. Env. Eng. Sci. 2015, 32, 528-538. (3) Adachi, Y. Sedimentation and Electrophoresis of a Porous Floc and a Colloidal Particle Coated with Polyelectrolytes. Curr. Opin. Colloid Interface Sci. 2016, 24, 72-78. (4) Gopmandal, P. P.; Bhattacharyya, S.; Barman, B. Effect of Induced Electric Field on Migration of a Charged Porous Particle. Eur. Phys. J. E 2014, 37, 104. ACS Paragon Plus Environment 12

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(5) Khair, A. S. Strong Deformation of the Thick Electric Double Layer around a Charged Particle during Sedimentation or Electrophoresis. Langmuir 2018, 34, 876-885. (6) Booth, F. Sedimentation Potential and Velocity of Solid Spherical Particles. J. Chem. Phys. 1954, 22, 1956-1968. (7) Stigter, D. Sedimentation of Highly Charged Colloidal Spheres. J. Phys. Chem. 1980, 84, 27582762. (8) Ohshima, H.; Healy, T. W.; White, L. R.; O’Brien, R. W. Sedimentation Velocity and Potential in a Dilute Suspension of Charged Spherical Colloidal Particles. J. Chem. Soc., Faraday Trans. 2 1984, 80, 1299-1317. (9) Levine, S.; Neale, G.; Epstein, N. The Prediction of Electrokinetic Phenomena within Multiparticle Systems II. Sedimentation Potential. J. Colloid Interface Sci. 1976, 57, 424-437. (10) Ohshima, H. Sedimentation Potential in a Concentrated Suspension of Spherical Colloidal Particles. J. Colloid Interface Sci. 1998, 208, 295-301. (11) Keh, H. J.; Ding, J. M. Sedimentation Velocity and Potential in Concentrated Suspensions of Charged Spheres with Arbitrary Double-Layer Thickness. J. Colloid Interface Sci. 2000, 227, 540-552. (12) Carrique, F.; Arroyo, F. J.; Delgado, A. V. Sedimentation Velocity and Potential in a Concentrated Colloidal Suspension: Effect of a Dynamic Stern Layer. Colloids Surfaces A 2001, 195, 157-169. (13) Hermans, J. J. Sedimentation and Electrophoresis of Porous Spheres. J. Polymer Sci. 1955, 18, 527-533. (14) Liu, Y. C.; Keh, H. J. Sedimentation Velocity and Potential in a Dilute Suspension of Charged Porous Spheres. Colloids Surfaces A 1998, 140, 245-259. (15) Keh, H. J.; Chen, W. C. Sedimentation Velocity and Potential in Concentrated Suspensions of Charged Porous Spheres. J. Colloid Interface Sci. 2006, 296, 710-720. (16) Ohshima, H. Sedimentation Potential and Velocity in a Concentrated Suspension of Soft Particles. J. Colloid Interface Sci. 2000, 229, 140-147. (17) Chiu, Y. S.; Keh, H. J. Sedimentation Velocity and Potential in a Concentrated Suspension of Charged Soft Spheres. Colloids Surfaces A 2014, 440, 185-196. (18) Yeh, Y. Z.; Keh, H. J. Sedimentation Velocity and Potential in Dilute Suspensions of Charged Porous Shells. J. Phys. Chem. B 2018, 122, 10393-10400. (19) Ninham, B. W.; Parsegian, V. A. Electrostatic Potential between Surfaces Bearing Ionizable Groups in Ionic Equilibrium with Physiologic Saline Solution. J. Theor. Biol. 1971, 31, 405428. ACS Paragon Plus Environment 13


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(20) Carnie, S. L.; Chan, D. Y. C. Interaction Free Energy between Plates with Charge Regulation: A Linearized Model. J. Colloid Interface Sci. 1993, 161, 260-264. (21) Pujar, N. S.; Zydney, A. L. Charge Regulation and Electrostatic Interactions for a Spherical Particle in a Cylindrical Pore. J. Colloid Interface Sci. 1997, 192, 338-349. (22) Dan, N. Interactions between Charge-Regulating Surface Layers. Langmuir 2002, 18, 3524-3527. (23) Ding, J. M.; Keh, H. J. Electrophoretic Mobility and Electric Conductivity in Dilute Suspensions of Charge-Regulating Composite Spheres. Langmuir 2003, 19, 7226-7239. (24) Philipse, A. P.; Tuinier, R.; Kuipers, B. W. M.; Vrij, A.; Vis, M. On the Repulsive Interaction Between Strongly Overlapping Double Layers of Charge-regulated Surfaces. Colloid Interface Sci. Comm. 2017, 21 10-14. (25) Ding, J. M.; Keh, H. J. Sedimentation Velocity and Potential in a Suspension of Charge Regulating Colloidal Spheres. J. Colloid Interface Sci. 2001, 243, 331-341. (26) Biesheuvel, P. M. Evidence for Charge Regulation in the Sedimentation of Charged Colloids. J. Phys.: Condens. Matter 2004, 16, L499–L504. (27) Yue, T.; Wu, X.; Chen, X.; Liu, T. A Study on the Flocculation and Sedimentation of Iron Tailings Slurry Based on the Regulating Behavior of Fe3+. Minerals 2018, 8, 421. (28) Deiber, J. A.; Piaggio, M. V.; Peirotti, M. B. Determination of Electrokinetic and Hydrodynamic Parameters of Proteins by Modeling Their Electrophoretic Mobilities through the Electrically Charged Spherical Porous Particle. Electrophoresis 2013, 34, 700-707. (29) Huang, H. Y.; Keh, H. J. Electrophoretic Mobility and Electric Conductivity in Suspensions of Charge-Regulating Porous Particles. Colloid Polym. Sci. 2015, 293, 1903-1914. (30) De Groot, S. R.; Mazur, P.; Overbeek, J. Th. G. Nonequilibrium Thermodynamics of the Sedimentation Potential and Electrophoresis. J. Chem. Phys. 1952, 20, 1825-1829. (31) van de Ven, T. G. M. Colloidal Hydrodynamics; Academic Press: London, 1989.


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Figure Captions Figure 1. Geometrical sketch for the sedimentation of a porous spherical particle. Figure 2. The velocity retardation coefficient  U 2 /(a) 4U 0 for the sedimentation of a charge regulating porous sphere versus the electrolyte concentration n1 for various values of a . The

solid and dashed curves denote the cases K   K   10 4 M and Q  A1 (no charge regulation), respectively. 2

Figure 3. The normalized velocity retardation  U 2 Q /U 0 for the sedimentation of a charge regulating porous sphere versus the electrolyte concentration n1 for a  1 : (a) K  K   108 M 2 ;

(b) K  / K   1 . 2 Figure 4. The sedimentation potential coefficient  E / A1a versus the electrolyte concentration

n1 for a dilute suspension of charge-regulating porous spheres with various values of a . The solid and dashed curves denote the cases K   K   10 4 M

and Q  A1 (no charge regulation),

respectively. Figure 5. The normalized sedimentation potential e E / kT versus the electrolyte concentration

n1 for a dilute suspension of charge-regulating porous spheres with a  1 : (a) K  K   108 M 2 ; (b) K  / K   1 .



Figure 1


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Figure 2



Figure 3a

Figure 3b


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Figure 4



Figure 5a

Figure 5b


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TOC Graphic

Q  A1  A2

Q  A1e /  2 kT

   (1  A2 /  2 )1 / 2 2

3

U  U 0  U 2 Q  O(Q ) E SED   (1   p )(  p   ) g

 Ee z Λ


Sedimentation Velocity and Potential in Dilute Suspensions of Charge

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