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Double-layer interaction between cylinders
439
Chapter 18. cylinders
Double-layer interaction between
1. I N T R O D U C T I O N
In this chapter we deal with the double layer interaction between two parallel infinitely long charged cylinders. We start with the linear superposition approximation (LSA) based on the effective surface potential of a single cylinder [1]. We then discuss the exact solution to the linearized Poisson-Boltzmann equation for two parallel cylinders [2, 3]. 2.
LINEAR
SUPERPOSITION A P P R O X I M A T I O N C Y L I N D E R - C Y L I N D E R INTERACTION
FOR
In chapter 5 we have developed the method of the linear superposition approximation (LSA) for sphere-sphere interaction applicable for large separations. We apply this method to the interaction between two parallel plates. Consider two parallel interacting cylindrical colloidal particles 1 and 2 of radii al and a2 having surface potentials lpo 1 and ~)o2, respectively, at separation R=H+al+a2 between their axes in a general electrolyte solution. According to the method of Brenner and Parsegian [4], the asymptotic expression for the interaction energy V(R) per unit length is given by V(R) = cr~n~q22(R) -- cro~p~(R)
(18.1)
where C~eff, (i = 1, 2) is the charge density per unit length of the hypothetical cylinder of infinitesimal thickness that would produce the same asymptotic potential as produced by cylinder i and ~p~(R)(i= 1, 2) is the asymptotic form of the unperturbed potential at a large distance R from the axis of cylinder ], when each cylinder exists separately. The hypothetical line charge of density oeff~gives rise to a potential ~i(R) - ~ Qe~ K0(KR) 2arer~o
(18.2)
Chapter 18
440
where K0(r.R) is the 0 the order Bessel function of the second kind. The asymptotic expression for the unperturbed potential of cylinder i (i =1, 2) at a large distance R from the axis of cylinder i may be expressed as =
(18.3)
r"
Here Yi is the scaled effective surface potential, which reduces to the scaled surface potential yoi in the low potential limit. Comparison of Eq. (18.2) with (18.3) leads to crce~ = 2 z e r e °{ (kT'~ 7 ]Y/
Ko(
1
(18.4) )
By combining Eqs. (18.1), (18.3), and (18.4), we find that the doublelayer interaction energy per unit length between two parallel cylinders at large separations is given by
V ( R ) = 2~'Er eo
YY2
Ko(ifal)Ko(tCa2)
(18.5)
Equation (18.5) becomes in the low ~Po,limit (kT 2
V(R)=
Z~ErEo(T)
K0(
)
/Pol/J3o2 Ko(Kal)Ko(Ka2)
(18.6)
which agrees with Brenner and Parsegian's result [3] and also with the leading term of the exact expression for the double-layer interaction energy between two parallel cylinders per unit length (given later by Eq. (18.10)). An approximate expression for the scaled effective surface potential Y for a cylinder of radius a having a surface potential ~Po immersed in a symmetrical electrolyte solution of valence z and bulk concentration n is given below [1].
Double-layer interaction between cylinders
y= 1
Zl+{,_
441
8y
(18.7)
where fl is given by Eq. (1.110), ~, = tanh(yo/4), and yo=ze~po/kT is the scaled surface potential of the cylinder. Equation (18.7) can also be obtained directly from Eq. (1.114). Note that D given by Eq. (1.115) is related to Y as Y=8D/z(l+fl). For !¢a >>1, Eq. (18.7) reduces to Y= 4y/z = 4tanh(yo/4)/z, which is the effective surface potential of a plate (Eq. (14.132)). By substituting Eq. (18. 8) into Eq. (18.5), we obtain the following expression for the interaction energy between two similar cylinders of radius a carrying surface potential ~Po[ 1]: V(R)= 128srereo( kT'~2
y2 K°(tcR) t ze ) [1 + { 1 - ( 1 - ~2)y2}1/212X2o(xa)
(18.8)
Equation (18.8) can be generalized to the case of two dissimilar cylinders 1 and 2 of radii al and a2 carrying surface potentials ~Poland ~Po2,respectively, v(R) =
x
128 :rereoY1~'2(k T~ ze )2 [1 + {1- (1-/~2)y2}1/2][1 + {1- (1-/~2)y2}1/2 ]
K° (tca )
(18.9)
where fli = Ko(tcai)/KlOcai) (i = 1, 2). 3. E X A C T SOLUTION TO BOLTZMANN EQUATION P A R A L L E L CYLINDERS
THE LINEARIZED POISOONFOR TWO INTERACTING
As in the case of the linearized Poisson-Boltzmann equation, the cylindrical Poisson-Boltzmann equation for the system of two interacting
Chapter18
442
parallel cylinders can also be solved exactly. The results are given below [2]. Consider a cylinder of radius al carrying a constant surface potential ~Pol (cylinder 1) and a cylinder of radius a2 carrying a constant surface potential ~Po2(cylinder 2), separated by a distance R between their axes, immersed in an electrolyte solution. We first treat the case where the surface potentials of cylinders 1 and 2 both remain constant during interaction independent of R. The result is with V~(R) = 2~re eo~PogPo2
Ko(~)
Xo(~al )Ko(~2 )
K2 (ta~1 ._
+~e,.eo~2:
(2)K~ (r.R)
1)K] (rR) K02~~ 2 ) ~_~_=G,( 1
+2~/~re o/Pol/Poz
Ko(~c~)Ko(~:c~)
oo
x ~ EG,,(2IGm(1)K(xR)K+.,(v.R)K(KR) ?I=-oo m=-oo
1 .#.,,..Jr. Y't~rEo~)ol~)o2
Ko (~u~)Ko(~) , n~,..., .~v)}
/71= --oo /12= --o¢
n2v ~--uo
×K.~(~R)r~ (~)
Double-layer interaction between cylinders
E E''"
r
,~. . . . ~. . . . ~=-~[/~2 (ta:tl
+
2
443
' n2'" ""' n2,'-2)G2,'-1(2)
K2~-'~a2)Iaz(rfi,nz,'",nzv-z)G2,-~(1)
] (18.10)
with I~,( nl , n . . . .. .2._2 ) = X .,+.. ( , : R ) X
+~ ( ~ ) × . . .
× K.~ ~+.2v, (,,'R)Kr~_,+,,~ (,,:R)
×G,~ ( 2)(7,.(1) x ... x G,~q( 2)G, ~.(1)
G,,(i)=
I,(;¢a,), K,(ra,)
(i = 1, 2)
(18.11)
(18.12)
where L12 is obtained from L2t by the interchange G,(1)~--,G,(2) in Eq (18.11). The leading term of V~(R) and V(R) agrees with Eq. (16.61). It can be shown that the interaction energy V~(R) per unit length between cylinder 1 (of relative permittivity ~1) and cylinders 2 (of relative permittivity ~2) at constant surface charge density is obtained by by the interchange Gn(i) ~--,H,,( i), which is given by
H,,(i)=
K,(lca~)_(epi[n/~a,)K (tca,) , (i= 1,2)
(18.13)
It can also be shown that when cylinder 1 has a constant surface potential and cylinder 2 has a constant surface charge density, the interaction energy VV"is given by Eq. (18.10) with G~(2) replaced by/-/,(2) (with G,(1) kept constant).
Chapter18
444
Consider the case where ~¢a1>>1 and ~¢a2>)1. For the constant surface potential case, W becomes
VaP(R ) : 2 2 ~ Er,%~13o1~3o2 K~ e-2~[ 2
] KazR
e-r..//
4~ 2
I ,¢,~R]
-~~GG---~ t~po,al~R_az +~po2~~_a 1 +O(e -3,'u)
(18.14)
For the constant surface charge density case, if el and ~2 are finite, we obtain
e-~[ ~po21al.[~Rr2_.~I " R
VR-ch{1 er ;rx~(R-~)J
"]'~Er ~oT
1
(18.1 Further, i f H <
V * ( R ) = 2 2~G~Po'g)o2
--
I trail2 -~4 al +a2 e
I /~21a2 -214/(~)ol 2 + q)~Z) "~'~gr/7° al + 0 2
V°(R)
e
=
2~-~
.[ Ka, ch -,,n e,eo~Pol~Po 211/~ + 42 e
(1 8.1 6)
Double-layer interaction between cylinders
445
I tcala2e_2~ilpo211 2 ep21 1 + 1 ) +'~-~ere° a l + a 2 ' -~ e sea1 ~ca2
2 ~dl
+
1
+~'~2 1- /--~ e ~tct 1 teaz
(18.17)
Equations (18.16) and (18.17) agree with the results obtained via Derjaguin's approximation (see Ref. 5 for the case of identical cylinders at constant surface potential). Equation (18.17) shows that the next-order curvature correction to Derjaguin's approximation is of the order of 1 / ~ , . (i = 1, 2), as in the case of sphere-sphere interaction (see Chapter 17). 4. I N T E R A C T I O N B E T W E E N A C Y L I N D E R A N D A P L A T E
One can also obtain the interaction energy between a cylinder and a hard plate, both having constant surface potential for the case where the cylinder axis is perpendicular to the plate surface. The result is e-r(H+a2 )
V*(H)
= 2,,~rgo/Pol/Po2
at". . . .
2-2x(H+a2) Jt ~reoq,, .o~e
Ko(~C~) G,(2)
~0 {x0(,:a2)}2 e-~( H +a2) ~
~ r~ o~ ol ~ ~
Ko( ~ ) ~.jPo= ._
+
fi,0)G,(2)
Chapter 18
446
2f3,,,,G,,(2)G,~(2)
. . . 2 -2,,(~+~=) -J~. . .er~oq'ole
n=-co gt=--~
+~GGlPo22 {Ko(~a2)} 1 z ~ ._? o "/3"°G"(2)
+3~'ErEo~PollPo2
g0(K.a
2 )
_
nm(~On "k
=_
fl.o)G.(2)Gm(2)+ "'"
e-,,-(.~+a~) ,~, .. _ 1)"-~ --~gr~:°q'°'~P'~ K-o( ~ 2 ) "~°-'~"2"-~ ~=-
xfl.,.~ fl~.~ ""fl.~_,.~ (flo., + fl.~o)G., (2)G.2 (2)"" G.,v (2)
22
n1 =-oo n2 =-oo
~ =-oo
x/3,,,,~ fl,,=,~ x.-.x/~,,v_l~ G,,1(2)G,, 2(2)--. G,,..(2) +at GG~Po22
1
{K0(Ka2)}
.... ~ ....
.
.
.
.
.
1)~ -1 .
xfl.,~ fl.2r~"" fl.~_,.~flo., ~.~oG., (2)G.~ (2)" "- G~ (2) + ""
(18.18)
with r e x p [ _ 2 ~ 2 + _ h . 2 _ ( H + a2)] { ~
_ _ £ exp[-2Kt~/.t_7l(H - + c~ )] T.(t )Tm(t)dr
+K2 )
/'~'S + s¢2)
(18.19)
Double-layer interaction between cylinders
447
where T~(x) is the n th order Tchebycheffs polynomial. When plate a and cylinder 2 have constant surface charge densities, the interaction energy V~ is obtained by Eq. (18.14) with G~(2) and/3.m replaced by Hn(2) and -y.m, respectively, where )',mis defined by Y,,,, = _ £
eqk- ~
+ h"2 exp[-2.4-k "S"+ h"2 (_H + ai) ]
elk + e~-k7 + 2
4 k 2 + K2
j7~1El/~.e1~/77-1 exp[-2~:t(H + a2)]T~(t)Tm(t)dt.
--
147 ,
(18.20)
47-1
When plate 1 has a constant surface potential and cylinder 2 has a constant surface charge density, the interaction energy VV'°is given by Eq. (18.14) with Gn(2) replaced by Hn(2). When plate 1 has a constant surface charge density and cylinder 2 has constant surface potential, the interaction energy is given by Eq.(18.14) with ~nmreplaced by -Y,,m. Finally we compare the image interactions between a hard cylinder and a hard plate with the usual image interaction between a line charge and a plate by taking the limit of ra2 --" 0 for the case where the surface charge density of plate 1 is always zero 0Pol = 0). In this limit, we have
f Q2
I 4~re~-~ K°(2KH)'
(e p, = 0)
V"(H)= ~
I
(18.21)
O2 --"
Ko(2~H),
(Ep, = (x))
where we have introduced the total charge Q - 2~racr on cylinder 2 per unit length (i.e., the line charge density of cylinder 2), which is related to the unperturbed surface potential ~Po2of cylinder 2 by
448 ~Poz=
Chapter 18
o 2argrgo tfa 2
1(1(~ca2)
(18.22)
Equation (18.17) is the screened image interaction between a line charge and an uncharged plate, both immersed in an electrolyte solution of Debye-H0ckel parameter ~¢. We see that in the former case (epl = 0) the interaction force is repulsion and the latter case (epl = ~) attraction. Further, in the absence of electrolytes (x ~ 0), we can show from E q (18.17) that the interaction force -OV"/OH per unit length between plate 1 and cylinder 2 with a2 ---- 0 is given by
_OV o _ Q2 ( e r _ e p , ) l . OH 4areeo ~ Er + •pl } H
(18.23)
which exactly agrees with usual image force per unit length between a line charge and an uncharged plate [6].
REFERENCES
[1] H. Ohshima. Colloid Polym. Sci., 274 (1996) 1176. [2] H. Ohshima, J. Colloid Interface Sci., 200 (1998) 291. [3] H. Ohshima, Colloid Polymer Sci 277 (1999) 563. [4] S.L. Brenner and VA. Parsegian, Biophys. J., 14 (1974) 327. [5] M.J. Sparnaay, Recueil, 78 (1959) 680. [6] L. D. Landau and E. M. Lifshitz, Electrodynamics in Continuous Media, Pergamon, New York (1963).
Chapter 18. cylinders
Double-layer interaction between
1. I N T R O D U C T I O N
In this chapter we deal with the double layer interaction between two parallel infinitely long charged cylinders. We start with the linear superposition approximation (LSA) based on the effective surface potential of a single cylinder [1]. We then discuss the exact solution to the linearized Poisson-Boltzmann equation for two parallel cylinders [2, 3]. 2.
LINEAR
SUPERPOSITION A P P R O X I M A T I O N C Y L I N D E R - C Y L I N D E R INTERACTION
FOR
In chapter 5 we have developed the method of the linear superposition approximation (LSA) for sphere-sphere interaction applicable for large separations. We apply this method to the interaction between two parallel plates. Consider two parallel interacting cylindrical colloidal particles 1 and 2 of radii al and a2 having surface potentials lpo 1 and ~)o2, respectively, at separation R=H+al+a2 between their axes in a general electrolyte solution. According to the method of Brenner and Parsegian [4], the asymptotic expression for the interaction energy V(R) per unit length is given by V(R) = cr~n~q22(R) -- cro~p~(R)
(18.1)
where C~eff, (i = 1, 2) is the charge density per unit length of the hypothetical cylinder of infinitesimal thickness that would produce the same asymptotic potential as produced by cylinder i and ~p~(R)(i= 1, 2) is the asymptotic form of the unperturbed potential at a large distance R from the axis of cylinder ], when each cylinder exists separately. The hypothetical line charge of density oeff~gives rise to a potential ~i(R) - ~ Qe~ K0(KR) 2arer~o
(18.2)
Chapter 18
440
where K0(r.R) is the 0 the order Bessel function of the second kind. The asymptotic expression for the unperturbed potential of cylinder i (i =1, 2) at a large distance R from the axis of cylinder i may be expressed as =
(18.3)
r"
Here Yi is the scaled effective surface potential, which reduces to the scaled surface potential yoi in the low potential limit. Comparison of Eq. (18.2) with (18.3) leads to crce~ = 2 z e r e °{ (kT'~ 7 ]Y/
Ko(
1
(18.4) )
By combining Eqs. (18.1), (18.3), and (18.4), we find that the doublelayer interaction energy per unit length between two parallel cylinders at large separations is given by
V ( R ) = 2~'Er eo
YY2
Ko(ifal)Ko(tCa2)
(18.5)
Equation (18.5) becomes in the low ~Po,limit (kT 2
V(R)=
Z~ErEo(T)
K0(
)
/Pol/J3o2 Ko(Kal)Ko(Ka2)
(18.6)
which agrees with Brenner and Parsegian's result [3] and also with the leading term of the exact expression for the double-layer interaction energy between two parallel cylinders per unit length (given later by Eq. (18.10)). An approximate expression for the scaled effective surface potential Y for a cylinder of radius a having a surface potential ~Po immersed in a symmetrical electrolyte solution of valence z and bulk concentration n is given below [1].
Double-layer interaction between cylinders
y= 1
Zl+{,_
441
8y
(18.7)
where fl is given by Eq. (1.110), ~, = tanh(yo/4), and yo=ze~po/kT is the scaled surface potential of the cylinder. Equation (18.7) can also be obtained directly from Eq. (1.114). Note that D given by Eq. (1.115) is related to Y as Y=8D/z(l+fl). For !¢a >>1, Eq. (18.7) reduces to Y= 4y/z = 4tanh(yo/4)/z, which is the effective surface potential of a plate (Eq. (14.132)). By substituting Eq. (18. 8) into Eq. (18.5), we obtain the following expression for the interaction energy between two similar cylinders of radius a carrying surface potential ~Po[ 1]: V(R)= 128srereo( kT'~2
y2 K°(tcR) t ze ) [1 + { 1 - ( 1 - ~2)y2}1/212X2o(xa)
(18.8)
Equation (18.8) can be generalized to the case of two dissimilar cylinders 1 and 2 of radii al and a2 carrying surface potentials ~Poland ~Po2,respectively, v(R) =
x
128 :rereoY1~'2(k T~ ze )2 [1 + {1- (1-/~2)y2}1/2][1 + {1- (1-/~2)y2}1/2 ]
K° (tca )
(18.9)
where fli = Ko(tcai)/KlOcai) (i = 1, 2). 3. E X A C T SOLUTION TO BOLTZMANN EQUATION P A R A L L E L CYLINDERS
THE LINEARIZED POISOONFOR TWO INTERACTING
As in the case of the linearized Poisson-Boltzmann equation, the cylindrical Poisson-Boltzmann equation for the system of two interacting
Chapter18
442
parallel cylinders can also be solved exactly. The results are given below [2]. Consider a cylinder of radius al carrying a constant surface potential ~Pol (cylinder 1) and a cylinder of radius a2 carrying a constant surface potential ~Po2(cylinder 2), separated by a distance R between their axes, immersed in an electrolyte solution. We first treat the case where the surface potentials of cylinders 1 and 2 both remain constant during interaction independent of R. The result is with V~(R) = 2~re eo~PogPo2
Ko(~)
Xo(~al )Ko(~2 )
K2 (ta~1 ._
+~e,.eo~2:
(2)K~ (r.R)
1)K] (rR) K02~~ 2 ) ~_~_=G,( 1
+2~/~re o/Pol/Poz
Ko(~c~)Ko(~:c~)
oo
x ~ EG,,(2IGm(1)K(xR)K+.,(v.R)K(KR) ?I=-oo m=-oo
1 .#.,,..Jr. Y't~rEo~)ol~)o2
Ko (~u~)Ko(~) , n~,..., .~v)}
/71= --oo /12= --o¢
n2v ~--uo
×K.~(~R)r~ (~)
Double-layer interaction between cylinders
E E''"
r
,~. . . . ~. . . . ~=-~[/~2 (ta:tl
+
2
443
' n2'" ""' n2,'-2)G2,'-1(2)
K2~-'~a2)Iaz(rfi,nz,'",nzv-z)G2,-~(1)
] (18.10)
with I~,( nl , n . . . .. .2._2 ) = X .,+.. ( , : R ) X
+~ ( ~ ) × . . .
× K.~ ~+.2v, (,,'R)Kr~_,+,,~ (,,:R)
×G,~ ( 2)(7,.(1) x ... x G,~q( 2)G, ~.(1)
G,,(i)=
I,(;¢a,), K,(ra,)
(i = 1, 2)
(18.11)
(18.12)
where L12 is obtained from L2t by the interchange G,(1)~--,G,(2) in Eq (18.11). The leading term of V~(R) and V(R) agrees with Eq. (16.61). It can be shown that the interaction energy V~(R) per unit length between cylinder 1 (of relative permittivity ~1) and cylinders 2 (of relative permittivity ~2) at constant surface charge density is obtained by by the interchange Gn(i) ~--,H,,( i), which is given by
H,,(i)=
K,(lca~)_(epi[n/~a,)K (tca,) , (i= 1,2)
(18.13)
It can also be shown that when cylinder 1 has a constant surface potential and cylinder 2 has a constant surface charge density, the interaction energy VV"is given by Eq. (18.10) with G~(2) replaced by/-/,(2) (with G,(1) kept constant).
Chapter18
444
Consider the case where ~¢a1>>1 and ~¢a2>)1. For the constant surface potential case, W becomes
VaP(R ) : 2 2 ~ Er,%~13o1~3o2 K~ e-2~[ 2
] KazR
e-r..//
4~ 2
I ,¢,~R]
-~~GG---~ t~po,al~R_az +~po2~~_a 1 +O(e -3,'u)
(18.14)
For the constant surface charge density case, if el and ~2 are finite, we obtain
e-~[ ~po21al.[~Rr2_.~I " R
VR-ch{1 er ;rx~(R-~)J
"]'~Er ~oT
1
(18.1 Further, i f H <
V * ( R ) = 2 2~G~Po'g)o2
--
I trail2 -~4 al +a2 e
I /~21a2 -214/(~)ol 2 + q)~Z) "~'~gr/7° al + 0 2
V°(R)
e
=
2~-~
.[ Ka, ch -,,n e,eo~Pol~Po 211/~ + 42 e
(1 8.1 6)
Double-layer interaction between cylinders
445
I tcala2e_2~ilpo211 2 ep21 1 + 1 ) +'~-~ere° a l + a 2 ' -~ e sea1 ~ca2
2 ~dl
+
1
+~'~2 1- /--~ e ~tct 1 teaz
(18.17)
Equations (18.16) and (18.17) agree with the results obtained via Derjaguin's approximation (see Ref. 5 for the case of identical cylinders at constant surface potential). Equation (18.17) shows that the next-order curvature correction to Derjaguin's approximation is of the order of 1 / ~ , . (i = 1, 2), as in the case of sphere-sphere interaction (see Chapter 17). 4. I N T E R A C T I O N B E T W E E N A C Y L I N D E R A N D A P L A T E
One can also obtain the interaction energy between a cylinder and a hard plate, both having constant surface potential for the case where the cylinder axis is perpendicular to the plate surface. The result is e-r(H+a2 )
V*(H)
= 2,,~rgo/Pol/Po2
at". . . .
2-2x(H+a2) Jt ~reoq,, .o~e
Ko(~C~) G,(2)
~0 {x0(,:a2)}2 e-~( H +a2) ~
~ r~ o~ ol ~ ~
Ko( ~ ) ~.jPo= ._
+
fi,0)G,(2)
Chapter 18
446
2f3,,,,G,,(2)G,~(2)
. . . 2 -2,,(~+~=) -J~. . .er~oq'ole
n=-co gt=--~
+~GGlPo22 {Ko(~a2)} 1 z ~ ._? o "/3"°G"(2)
+3~'ErEo~PollPo2
g0(K.a
2 )
_
nm(~On "k
=_
fl.o)G.(2)Gm(2)+ "'"
e-,,-(.~+a~) ,~, .. _ 1)"-~ --~gr~:°q'°'~P'~ K-o( ~ 2 ) "~°-'~"2"-~ ~=-
xfl.,.~ fl~.~ ""fl.~_,.~ (flo., + fl.~o)G., (2)G.2 (2)"" G.,v (2)
22
n1 =-oo n2 =-oo
~ =-oo
x/3,,,,~ fl,,=,~ x.-.x/~,,v_l~ G,,1(2)G,, 2(2)--. G,,..(2) +at GG~Po22
1
{K0(Ka2)}
.... ~ ....
.
.
.
.
.
1)~ -1 .
xfl.,~ fl.2r~"" fl.~_,.~flo., ~.~oG., (2)G.~ (2)" "- G~ (2) + ""
(18.18)
with r e x p [ _ 2 ~ 2 + _ h . 2 _ ( H + a2)] { ~
_ _ £ exp[-2Kt~/.t_7l(H - + c~ )] T.(t )Tm(t)dr
+K2 )
/'~'S + s¢2)
(18.19)
Double-layer interaction between cylinders
447
where T~(x) is the n th order Tchebycheffs polynomial. When plate a and cylinder 2 have constant surface charge densities, the interaction energy V~ is obtained by Eq. (18.14) with G~(2) and/3.m replaced by Hn(2) and -y.m, respectively, where )',mis defined by Y,,,, = _ £
eqk- ~
+ h"2 exp[-2.4-k "S"+ h"2 (_H + ai) ]
elk + e~-k7 + 2
4 k 2 + K2
j7~1El/~.e1~/77-1 exp[-2~:t(H + a2)]T~(t)Tm(t)dt.
--
147 ,
(18.20)
47-1
When plate 1 has a constant surface potential and cylinder 2 has a constant surface charge density, the interaction energy VV'°is given by Eq. (18.14) with Gn(2) replaced by Hn(2). When plate 1 has a constant surface charge density and cylinder 2 has constant surface potential, the interaction energy is given by Eq.(18.14) with ~nmreplaced by -Y,,m. Finally we compare the image interactions between a hard cylinder and a hard plate with the usual image interaction between a line charge and a plate by taking the limit of ra2 --" 0 for the case where the surface charge density of plate 1 is always zero 0Pol = 0). In this limit, we have
f Q2
I 4~re~-~ K°(2KH)'
(e p, = 0)
V"(H)= ~
I
(18.21)
O2 --"
Ko(2~H),
(Ep, = (x))
where we have introduced the total charge Q - 2~racr on cylinder 2 per unit length (i.e., the line charge density of cylinder 2), which is related to the unperturbed surface potential ~Po2of cylinder 2 by
448 ~Poz=
Chapter 18
o 2argrgo tfa 2
1(1(~ca2)
(18.22)
Equation (18.17) is the screened image interaction between a line charge and an uncharged plate, both immersed in an electrolyte solution of Debye-H0ckel parameter ~¢. We see that in the former case (epl = 0) the interaction force is repulsion and the latter case (epl = ~) attraction. Further, in the absence of electrolytes (x ~ 0), we can show from E q (18.17) that the interaction force -OV"/OH per unit length between plate 1 and cylinder 2 with a2 ---- 0 is given by
_OV o _ Q2 ( e r _ e p , ) l . OH 4areeo ~ Er + •pl } H
(18.23)
which exactly agrees with usual image force per unit length between a line charge and an uncharged plate [6].
REFERENCES
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