КОЛЛОИДНЫЙ ЖУРНАЛ, 2007, том 69, № 4, с. 463-481
УДК 541.18
HIGH EFFICIENT CALCULATION OF THE INTERACTION ENERGIES BETWEEN DISSIMILAR DOUBLE LAYERS FOR Na2SO4 TYPE ASYMMETRIC ELECTROLYTES AT j0 > yd > 0
© 2007 Shimin Zhang*, Shizhuo Zhang**, Ye Zhang**
* College of Chemistry and Chemical Engineering, Central South University Changsha, Hunan, China ** College of Computer Scince and Technology, Beijing University of Technology, Beijing, China
Поступила в редакцию 25.08.2006 г.
Several rapidly convergent series for highly efficient calculation of the interaction energies between dissimilar double layers for Na2SO4 type electrolytes at y0 > yd > 0 are derived, the accurate numerical results are given and several approximate expressions are obtained for y0 < 1. The number of the series terms required to obtain the interaction energies with six significant digits using the series derived is not more than 2 when the dimen-sionless surface potential of double layers y0 < 20. The interaction energies between dissimilar double layers for NaCl or Na2SO4 type electrolytes are remarkably affected by the value of yd, but the interaction energies for Na2SO4 type electrolytes are hardly affected by the value of y0. The present results can also be applied to CaCl2 type electrolytes at y0 < yd < 0.
1. INTRODUCTION
The interactions between two double layers for symmetric electrolytes were studied for a long time [1-10]. For the interactions between two double layers for asymmetric electrolytes, only few investigations were reported [1112], and the accurate numeral results have never been obtained. The interaction between two similar double layers for Na2SO4 or CaCl2 type asymmetric electrolytes at positive surface potential has been investigated [13-14], and the results obtained are very accurate. In this paper, the interaction between dissimilar double layers for Na2SO4 type electrolytes at y0 > yd > 0 is investigated. The series for highly efficient calculation of the interaction energies are derived and the accurate numeral results and approximate expressions are presented. The results obtained can also be applied to CaCl2 type electrolytes at y0 < yd < 0. Moreover, the relative magnitudes of the interaction energies between dissimilar double layers for Na2SO4 and NaCl type electrolytes are compared.
2. INTERACTION BETWEEN TWO PLATES
For convenience we define the following dimension-less variables:
к = J&n n0 z2 e2/(ekT ), % = кх, %d = к d, y = ze ф/( kT), y0 = zety0/( kT), yd = zeqd/( kT), V = к V/( 2 n0 kT ), p ' = p/( 2 n0 kT),
(1) (2)
(3)
(4)
(5)
where k is Debye parameter, n0 is the electrolyte concentration in the bulk solution, z = min(z+, |z_|), e is the proton charge, £ is the dielectric constant of solution, k is the Bolt-zmann constant, T is the absolute temperature, is the di-mensionless distance and x is the distance from left plate, £d is the dimensionless distance, d is the distance between two plates, y is the dimensionless potential, 9 is the potential between two plates, y0 is the dimensionless surface potential, 90 is the surface potential on left plate, yd is the dimensionless surface potential, 9 is the surface potential on right plate, V is the dimensionless interaction energy, V is the interaction energy between two plates, p is the di-mensionless interaction force, and p is the interaction force between two plates.
With the aid of an analytic method (see Appendices A, C, D and E) the y-£, C-£d, p-£d and V'-£d dependences were obtained for Na2SO4 type asymmetric electrolytes at y0 > yd > 0 which are shown in Figs. 1-5. In Fig. 3, C is the constant of integration (see Appendix A). The maximum of p in Fig. 4 is
I 1 / 2yd , T -yd 1 \
'max = 2 ( e +2e -3 )
(6)
(see Appendix D).
Figure 4 shows that when < , p < 0 and p is an attractive force; when > , p > 0 and p is a repulsive force.
For %d < %'d , we have (see Appendices B and F)
Уо
^d = J
yd
■ 2e-y + C
( %d< % d ),
(7)
yo
V = -SVs + e-^ 2 + ey" )3/2-3V3 +
yo
x -yJ2„ , y^3/2 1 r (C + 3 )dy
+ e (2+e ) + -J ,2 ^ -
2 yJe2y + 2e-y + C (8)
yd
Fig. 1. The potential between two plates for 0 < £,d < -(e2yd + 2e-yd) < C <
y y0
yd
ye
Fig. 2. The potential between two plates for < ^ <
-(e2yd + 2e yd ) < C < -3.
C
0
-3
C
Fig. 3. The C-td curve for yo > yd > 0, Cm
2 yd -yd = -( e + 2e ).
yo
JVe2y + 2e-y + Cdy (td<td).
yd
Substituting C = -3 into Eqs. (7) and (8) we obtain the coordinates of the maximum point in Fig. 5:
[Vo /-> yd , 2 Ve - 1 (V3e + V2 + e ) id, max = -7-ln ---■;=-- , (9)
^ (T^+T^T0)
Fmax = - 6V3 + 2(2 + eVdW 1+2e-yd. (10)
Figure 5 and Eq. (10) show that when the flat plate type colloid particles aggregate, the colloid particles must get over an energy peak, and the energy peak heighten with increasing yd.
For td ^ td , we have (see Appendices B and F)
td = f(yo) + f(yd) (td ^td),
where
yo
f(yo ) = J
7e2 y + 2e-y + C '
where
f(yd ) = J , 2 "y ,
IJe2' + 2e-y + C
se
V = g (yo ) + g ( yd ) (td ^td), g ( yo ) = -3V3 + e-y°/2( 2 + eyo )3/2 +
(11)
(12)
(13)
(14)
+ 1 J ( C + 3 )dy - ye2y + 2e-y + Cdy, 2J e2 y + 2e-y + C I
g ( yd ) = -^V3 + e-yd /2( 2 + eyd )3/2 +
(15)
yd
yd
(16)
+ 1 J ( C + 3 )dy--y e2 y + 2e-y + C dy.
2U e2 y + 2e-y + C i
In what follows we calculate td and V by the series expansion.
y
0
t
t
d
0
t
t
f
t
d
p' У
0
Fig. 4. The curve for yo > yd > 0.
2.1. Interaction for < ^d
2.1.1. The case of C > -3. Let w2 = e-y, then Eq. (7) becomes
Уо
^ = l
-Jbd( w2)
Fig. 5. The V-£,d curve for yo > yd > 0.
Equation (21) takes the form
-Vadul V1 -
yd
Vl+2 bw2Jb -2b w + w
(17)
td = J
u
d ,p2 2(1 + 8b3- A2)u
where b = -a/2 and a is a root of the cubic equation a3 + + Ca + 2 = 0:
where
a = -
18 r
where
r=
9r4 + 3Cr2 + C2'
3JlWl + (C/JT3
(18)
(19)
a =
P =
4b3( 2 + A,) + (1 + A)2
4 bA
Let
Equation (19) shows that -3 < C < Equation (18) can also be rewritten as
4 b3 (2 + ■A) + (1 + -A)2'
14 b 3 (2 " - A ) + ( 1 - - A ) 2
¡4b 3 ( 2 - + A)- + (1 + A)2
= зД+7Т7(С/37 -3^1 + 71 + (С/3)3. (20) then Eq. (25) takes the form
u0
- adu
So -2 < a < 0, 0 < b < 1. We preferably use Eq. (18) be cause Eq. (20) is not convenient for very large C. Intro ducing a parameter A into Eq. (17), we obtain
td = J
Уо
td = J *
-Jbd (w2)
y^TA
1+2bw2
(21)
A
b - 2b
2 2. 4
w + w
Let
or
or
(25)
1+2 bw 1+u
A
1 - u
2 1 / 1 , 1 \ , A 1 w = -"""""" (1 + A) + """""-
2b b 1 - u
u=
1 + 2 b w2 - A 1+2 bw2 + A.
(22)
(23)
(24)
л/1 - u2 Vp2 + u2 Equations (26) and (27) are as follows
8b
a = -
(1+ A)( 3 + A)'
p = aV b (1- b3).
Equation (24) shows that |и| < 1. Equation (30) can also be rewritten as
(26)
(27)
(28)
(29)
(30)
(31)
1 = 1(1 ab
+ лЛ + 8 b3V 3 + л/1 + 8 b
> 1. (32)
From Eqs. (31) and (32) we know that 0 < a < 1 and 0 < p < 1. Let
2 x 0 2y
u = ГГ5' P = T*'
1 + x 1 - у
(33)
a
466 or
x =
1 + 7Ï —
l> Y
i + Vi + p2
(34)
V ' = -^T3 + e-y°/2( 2 + e yo )3/2-373 + + e-V2(2 + eyd)3/2 We2yd + 2e-yd + C -
yo
Equation (34) shows that |x| < 1 and 0 < y < 1. So Eq. (29) becomes
-Je2 y° + 2e-y° + C +
1 f ( 3 - C ) dy
2 J /
e2y + 2e-y + C
= - J
( 1 - y2 ) Tâdx = 7y2 + x2 A/i + y2 x2
yo
J-
yd'
3w dy
(35)
,, (2m)!(-1 )my2
-( 1-y )Va\ *—m V
o 4m( m! )2
m = 0 v 7
;7e2 y + 2e-y + C ' By comparing Eqs. (7) and (35), we obtain _dy_ _ ( 1 - y2 ) 7â dx
where
I = f x2mdx _ (2m)!
y
,2.2 9^ xd y2 + x2 2
2 ^ y2 m ln ^ + - (m! ) td
7e2y + 2e-y + C 7y 2 + x^1 + y: From Eqs. (23) and (33), we have
2 4b2 , 2À x
22 x
(43)
w =
+
2m 2m - k, 2,^ Nk ^2 m - k, 2,^
Y (^o ( y / {o) - ^ ( y /td )
Y( ) (2m - k) !k!( m - k + 8k m)
k = o
(36)
1+ À b ( 1- x )
2
(44)
(m = 0, 1, 2,...),
where
Substituting Eqs. (43) and (44) into Eq. (42), we arrive at
V = -3T3 + e-y°/2( 2 + eyo )3/2-373 +
td = xd + */y2 + x2
y2
d
to xo + + x2,
(37)
(38)
, -yJ2,~ . y^3/2 . I 2yd . ~ -yd . + e (2 + e ) + Ve +2e + C -
cdx
-Je2 yo + 2e-yo + C + J
Vy2 + x2 V1 + y
22 x
+ (45)
„ _ 1- (-1 )('m-k + 1 )! _ J1, k =
°k, m = 2 =
m
o, k * m.
+
6 f À( 1 - y2)7âxdx
bJ
(39) bl( 1- x)Vy2 + x%A + y2x2'
where
In order to diminish error, Eq. (24) should be rewritten as
2bw2-. 8 b
u =
1 + 71 + 8 b3
2 bw2 + 1 + 71 + 8b3 Now we use the following relation
(40)
c = 1 (y2 C -S + fb).
The last term in Eq. (45) can be rewritten as
(46)
J
xdx
( 1-x )27y2 + x2 71 + y
22 x
+c = 7^°
e y + 2e
Cdy
e2 y + 2e + Cdy -3eydy
(41)
J
2 x 2 dx
( 1- x2 )Vy2 + x2 71 + y2 x2
+
(47)
7e2 y + 2e-y + C 7e2 y + 2e-y + C to replace 7e2y + 2e-y + Cdy in Eq. (8), we obtain
+
o
1J
( 1 + x2 ) d ( x2 )
( 1- x2 )Vy2 + x2 71 + y2 x2
x
x
d
x
o
x
d
x
x
d
x
x
d
x
d
i y<K312
The result of the calculation of the second integral on the right hand side of Eq. (47) is
"U
1JU
(1 + X2) d (x2)
V' = - 3 73 + e 2(2 + eyu) -3^3 +
+ e-V 2( 2 + eyd )3/2 + -
(1- x2 )VY2 + x2 Vl + Y2 x2
= —11-r-2 (r^WY 2 + x2V1+ Y2
(1+ y 2 )2 (1- x0
--Y2 + x ^1 + Y2x ; ).
1 - xd
Using the following relation
-7e2 yu + 2e-yu + C
6 (1- Xu)(1+ Y2 )2
x
X-
(48)
x
h 2 + x+ y 2 x;-
1 - x 0
1 - Xj
x
xh 2 + X+ Y
22
n / ^ \i/ i \ m 2m + 2
+ (1-xu)£ (2m ) ! (- 1 ) Y /,
d
"Vy2 + x2aA + Y
22 X
Y( 1- x2)
Y( 1 - x ) dx
7y 2 + x2 A/1 + Y2 x2
m = U
n
4m (m! )2
m +1
+
+
+
2 (1 + y 2 )2 x2dx
(49)
+ ry( 2 m ) ! (- 1 )m y 2m
4m ( m!) 2 m'
m=U
where
Y( 1- x2 )VY2 + x2 a/1 + Y
22 x
6 ^¿ii^xi.
6( 1+ y 2)
G = c -
the first integral on the right hand side of Eq. (47) can be rewritten as
(52)
(53)
2.1.2. The case of - (e2yd + 2e-d) < C < -3. Let
J
2 x dx
2 2 2 2 2 2
(1- x ) vy + x a/ 1+ Y x
2y* ~ -y* C = - e - 2e ,
(54)
then Eq. (7) takes the form
(1+ y 2 )2 ^ 1- xU
h 2 + xJ1+y 2
x-
y0
1- x:
2 . 2 /, . 2 2 + xW 1 + Y xd
(50)
td = J
yd
e2 y + 2e-y -e2 y* -2e-y*
yU
(55)
-y
Y2 f (1- x2) dx
22 x
yd
e ve +2e -e *-2e
(1+y 2 )2lJY*+x2J1+?
Substituting Eqs. (48) and (50) into Eq. (47), we obtain
x0
xdx
Let e y = w2, th
Для дальнейшего прочтения статьи необходимо приобрести полный текст. Статьи высылаются в формате PDF на указанную при оплате почту. Время доставки составляет менее 10 минут. Стоимость одной статьи — 150 рублей.