The Journal of Physical Chemistry, Vol. 83, No. 22, 1979 2911
Dielectric Saturation Effects (28) A. Johansson, P. Kollman, and S.Rothenberg, Chem. Phys. Left., 18, 123 (1972). (29) R. Ditchfield, D. P. Miller, and J. A. Pople, quoted by R. Dltchfield in “Critical Evaluationof Chemical and Physical Structural Information”, D. R. Lide and M. A. Paul, Ed., National Academy of Sciences, Washington, D.C., 1974.
(30) D. C. Pan and L. C. Allen, J. Chem. Phys., 46, 1797 (1967). (31) A. D. Bucklngham, 0.Rev. Chem. SOC., 13, 183 (1959). (32) C. H. Townes and A. L. Schawlow, “Microwave Spectroscopy”, McGraw-Hili, New York, 1955. (33) C. W. Kern and M. Karplus, J . Chem. Phys., 40, 1374 (1964). (34) R. M. Garvey and F. C. DeLucia, J. Mol. Spectrosc., 50, 38 (1974).
Effects of Dielectric Saturation on Planar Double Layer Interactions for No Added Salt Hiroshi Maeda” and Fumlo Oosawat Department of Chemistry, Faculty of Science, Nagoya University, Nagoya, 464, Japan (Received October 10, 1978; Revised Manuscript Received February 12, 1979) Publication costs assisted by Nagoya University
The effect of dielectric saturation of water on the interaction between electric double layers on two charged plates is theoretically investigated in the absence of low molecular weight salts. The dielectric constant of water, e , is given as a function of the electric displacement, D, instead of the electric field, E, by e = (to + 3aD2)/(1 aD2)or in shorter form by = eo/(l + aD2),which is derived on the basis of the assumption that the self-free energy of polarized dielectrics is expressed as (up2+ bP)where P is the dielectric polarization. By using this expression for e, we solved the Poisson-Boltzmann equation analytically,and the potential, the field, and the counterion concentration were obtained as functions of the distance from the plates for various values of the charge density on the plates. The electric energy, entropy, and free energy are also calculated, and the effect of dielectric saturation is estimated for various values of the parameter a. The counterion activity cannot exceed a certain limit as the charge density increases and the limit decreases with increasing distance between the two plates. The electric field is made small by the dielectric saturation except in the vicinity of the plates where the field is made large. The electric entropy is decreased by the dielectric saturation (its absolute value is increased),while the electric energy is little influenced. The electric displacement is made smaller everywhere except at the surface of the plate. The effect of dielectric saturation becomes less significant with increasing distance between the two plates.
+
Introduction The effect of dielectric saturation on the interaction between planar electric doouble layers in low molecular weight salts has been examined theoretically by several a~thors.l-~ Both Sparnaay2and Levine and Bell3expressed the dielectric constant of water, t, in a power series of the electric field, E, and assumed the form
a’E2 (1) The value of the coefficient a’ used by them was 3 X or 3 X lo-’ (cm2/esu)2,respectively. With t given by eq 1, however, the one-dimensional Poisson-Boltzmann equation cannot be solved analytically. Therefore, Sparnaay gave an approximate solution by a perturbation method and Levine and Bell solved the equation by numerical integration. For the study of the interaction between highly charged plates in water, it is important to find an expression for t which is applicable to high field strengths and convenient for analytical integration of the Poisson-Boltzmann equation. Grahamel proposed the following form of t as a function of E with only one parameter b to cover high field strengths. e = 3 + [(to - 3)b/Eb1I2] arctan (t~l/~E) (2) If the right-hand side is expanded in a power series of E, the coefficient a’ in eq 1 is found to be equivalent to (eO - 3)b/3. He pointed out that eq 2 agrees well with the e
to -
Institute of Molecular Biology, Faculty of Science, Nagoya University.
0022-365417912083-2911$01 ,0010
result of the theory of Booth,4 if b is set at 1.08 X (cm2/esu)2.Unfortunately, however, eq 2 is not convenient for integration of the Poisson-Boltzmann equation. Most of the theories on dielectric saturation have been concerned with derivation of the dielectric constant as a function of ESp7The basic idea of this work is to express the dielectric constant as a function of electric displacement, D,and to solve the Poisson-Boltzmann equation for planar double layers.
A Phenomenological Approach to the Dielectric Saturation Effect Let us consider a large dielectric which is placed between two parallel charged plates and polarized in a constant external field, Eo. The free-energy density is given by the following expression:
F = (1/47r)JD 0 E dD
(3)
This free-energy density is composed of three terms: F = Fo Fint+ F, (4) The first term, Fo, is the free-energy density in a reference state where the dielectric is removed from the external field; this is given by E 2 / 8 ~ .The second term, Fht, which is due to the interaction between the electric field and the polarized dielectric is equal t o -Ed‘, where P is the dielectric polarization. The work necessary to polarize the dielectric is called self-energy and denoted here as F,. According to Landau’s expansion: this free-energydensity F, is expressed in a power series of P as follows:
+
0 1979 American Chemical Society
The Journal of Physical Chemistty, Vol. 83, No. 22, 1979
2912
F, = aP2 + b P + ..,
(5) If only the first term is retained, the dielectric constant eo at zero field strength is obtained. The effect of dielectric saturation is included in higher order terms. If the first and second terms are taken into calculation, we have
Eo2/& - E$
( 1 / 4 7 ) S D E dD 0
+ aP2 + bP
H. Maeda and F. Oosawa 80
60
(6)
In the present system we can set Eo = D. Differentiation of eq 6 with respect to D gives (dP/dD)(-D 2aP 4 b P ) = 0 (7)
+
+
Since dP/dD is generally nonvanishing, we have (8) -D + 2aP 4 b P = 0 Introducing the relation P = (1- ~ - l ) p / 4 ainto eq 8 and expanding the equation in a power series of t-l, we obtain (1 - a / 2 - bD2/16r3) + t-'(a/2 + 3bD2/16a3)+ OW2) = 0 (9) which, after rearrangement, gives E = (eo + 3aD2)/(1 + aD2) (loa)
20
+
+ aD2)
(10)
= a / ( a - 2a)
(11)
= b / 8 a 2 ( a- 27)
(12)
c0/(l
where E0
0 1 0
I
I
I
1
2
3
[ (ESU
/
CM2)
4
x
Figure 1. Proposed dependence of the dielectric constant, E, on field strength, E. Solid curves are calculated according to eq 10. Values of a (cm2/esul2(from top to bottom): and IO-". 2x Open circles represent the result of Grahame-Booth, accordlng to eq 2 with b = 1.08 X 10". Two dashed curves represent the dependence glven by eq 1. Values of a' (cm2/esu)2: (a) 3 X lo-' (Sparnaay), (b) 3 X lo-' (Bell and Levine).
and Thus, the dielectric constant is obtained as a function of electric displacement, D, instead of E. If eo is very much larger than unity, the simplified form eq 10 gives a good approximation of eq loa. The error arising from the use of eq 10 in place of eq 10a is not greater than 10% for the value of aD2 smaller than 3, which corresponds to about 20 of E. Therefore, in the case of planar double layer, eq 10 can be used everywhere, if the condition a D 2 5 3 is satisfied where Do is the dielectric displacement at the surface of charged plates. This means that Do should be smaller than about 1.7 X lo6, 5.5 X lo5, or 1.7 X lo5 (esu/cm2) if a is W2,W1,or (cm2/esu)2,respectively. In other words, the surface charge density should be smaller than 0.03, 0.01, or 0.003 (elementary charges per A2), respectively. At higher charge densities, however, the difference between eq 10 and 10a becomes appreciable. In the limit of high field strengths, the dielectric constant must approach a value similar to that of nonpolar liquids. Equation loa gives a reasonable value of E = 3 as the limit, but eq 10 gives a wrong value. When the dielectric constant is given by eq 10 or loa, the Poisson-Boltzmann equation for planar double layers can be solved analytically. Particularly, if eq 10 is applied, all thermodynamic quantities for the system can be obtained in the form of analytical functions, as shown in later sections. The dependence of the dielectric constant on the field strength obtained above is compared with the results of other theories in Figure 1. Equation 10 shows approximate agreement with the result of Booth-Grahame at a = 1X and with the expression of Sparnaay at low field strength with a = 1 X
Solutions of the One-Dimensional Poisson-Boltzmann Equation Let us consider an infinite series of plates parallel to each other. The thickness of each plate is I and the
0
d12
d
X
Figure 2. Schematic representation of the system.
distance between two surfaces facing each other is d , as shown in Figure 2. Positive charges produced by dissociation of counterions are distributed on the plates with a uniform density u(esu/cm2). The potential distribution is symmetric in the space between two adjacent plates and we need the solution only between x = 0 and x: = d / 2 . The Poisson equation is written in terms of the protonic charge, eo,the counterion concentration, C (ions/cm3),and the electric potential, #, as follows: dLl/dx = -d(e d#/dx)/dx = -4reoC(x) (13) According to the Boltzmann law, we found that the counterion concentration is related to the potential by the equation C(x) = n exp[dx) - 4 ( d / 2 ) 1 (14) where 4 = eo#/kT, k is the Boltzmann constant, T i s the absolute temperature, and n is the counterion concentration at the midpoint, x = d / 2 . Then, the following Poisson-Boltzmann equation is obtained: dD/dx = -(4rneo) exp[4 - 4 ( d / 2 ) ] (15) The applicability of this Poisson-Boltzmann equation for the present system will be discussed again in Appendix A. Differentiation of eq 15 with respect to x gives d2D/dx2 = (dD/dx) (d4 /dx) = -(eo/kT)(D/4 (dD/dr) (16) Integration of eq 16 between x = d / 2 and x gives
The Journal of Physical Chemistry, Vol, 83,No. 22, 1979 2913
Dielectric Saturation Effects dD/dx
+ ( e o / k T ) J0
D
E dD = [ d D / d ~ I , = d /=~ -4aneo
(17) where we used the boundary condition that D = 0 at x = d/2. Equation 16 or 17 can be solved if E or D/t is given as a function of D. If eq 10 is introduced, we have dD/dx + (eo/cOkT)[D2/2+ aD4/4 + (tokT/eo)' K'] = 0 (18)
4
8
t 6
h
U
Y Y
3
Y
where the Debye-Huckel parameter K is defined by K'
= 4anez/~~kT
(19)
Equation 18 can be written in terms of the dimensionless quantities X = 2x/d, Y = DIDd (Dd = 2eokT/eod),and Y+ = [(I B1/2)/aDd2]1/2
Y- = [(I - B1/2)/aDd2]1/2 as follows: dY/dX = -(aDd2/4)(In
+ Y+')(P + Y?)
[arctan Y + J 1 + cos w '1' + J-Y ~-COSW arctan Y J 1 -cos w J - Y l+cosw In 2JY cos w 52 2J In - 2 J Y COS w + S2
-1 +
+
for Y = J 25
[
(20)
1/2]7+ sin t
~
+
1 + cos w 1- cos w
]+
T-
I
I
I
I
I
100
sin t = p(1 - X)/2 (23) 2J
Here t = (1/2) arctan p, w = arctan [P/((l + P')l/' - l)], The function g(J,Y) is defined and J = (1+ P2)1/4/a1/2Dd. by g(J,Y) = -a(J > Y) and 3a (J < Y). Details of calculation are given in Appendix B. The value of B or /3 is determined from the boundary condition at x = 0: D(x=O) = Do = 4 7 ~ ~ (24) When B tends to zero, the solutions are transformed into the following equation: al/'D/(l + aD2) arctan (al/'D) = (1 - X)/(2a1/'Dd) (25) Numerical calculation based on the above solutions is carried out in the following sections, by setting to = 78, T = 300 K, and a = (cm'/esu)', unless stated otherwise. From Figure 1, it is very probable that the value of a is between and 10-ll.
+
Counterion Activity and Electric Potential Once the value of the parameter B is determined from the boundary condition, eq 24, the Debye-Huckel pa-
I 500
d(lo-Bcm) Figure 3. Limiting values of Kdagainst interplate distance d.
For negative values of B, after introducing a positive quantity P defined by $3 = B1i2,we obtain the following solutions: for Y # J
1
I
I
0
of a (cm*/esu)*:(a) 0, (b) 10-l2, (c) lo-", and (d) lo-''.
where B = 1 - aDd2(Kd)'. Integration of eq 20 can be performed in different ways depending on the sign of B. For positive values of B, we obtain (l/Y-) arctan (Y/Y-) - (l/Y+) arctan (Y/Y+)= B1/'(l - X)/2 (21)
-1
I
2
Values
rameter, K, and the counterion concentration, n, at the midpoint, d/2, are both obtained. The latter is directly related to the counterion activity. At a given distance d, the product Kd increases as the surface charge density, u, increases. There is, however, an upper limit, which can be derived from eq 21-23 where X and Yare set equal to zero and Yo (= Do/Dd),respectively. The result is
( B > 0)
[(~d)'/2~'][1 d/'DdKd] = 1
(26)
and = 4a2a1/'Dd[sin (0.5 arctan (Dd2K2d2- 1)'/2]]2 (aDaK2d2- 1)Kd ( B < 0) (27)
In Figure 3, the limit of Kd is plotted against the distance, d, at various values of a. Even without dielectric saturation, an upper limit exists, which is given by 2l/'a, independently of d. Therefore, the upper limit of the counterion activity is a characteristic property of charged plates. If dielectric saturation is taken into account, the limit of Kd becomes smaller, depending on d. At large distances, it converges to 2l/%r irrespective of a. As pointed out in the previous section, the validity of eq 10 is not guaranteed at large values of u. Nevertheless, we have just examined the limit of Kd for infinite u. Actually, however, the limit calculated above is reached by values of ./eo of about 2 X 1014for a = and about 5 X 1013for a = 10-lo,in esu units. Therefore, the limit calculated is practically available in spite of apparent theoretical inconsistency. The activity coefficient of counterions y is calculated as n / ( C ) , where ( C ) is the average concentration of counterions, which is given by 2u/eod. Some of the results are given in Table I. The effect of dielectric saturation is small except at short distances (d < 20 A). The electric potential difference between the surface and the midpoint A$o can be calculated from eq 15 and 17, as follows: A$
$(x=O) - $(x=d/2) = In [(-1/4?meo) X
[
( d D / d ~ ) , = ~=]In 1 + (4ank59-l JDo(D/c) dD]= 1n [1
0
(rod)-'0 0 ( 1 / 2 + aDo2/4)] (28)
Some of the results are given in Table 11. The effect of dielectric saturation on the potential difference is more
2914
H. Maeda and F. Oosawa
The Journal of Physical Chemistry, Vol. 83, No. 22, 1979
TABLE I: Activity Coefficient of Counterions ua 0.001 0.002 0.005 0.01 l o - A Interplate Distance a = lo-'' 0.929 0.862 0.676 0.443 a=o 0.929 0.867 0.717 0.550 ratio 1.00 0.99 0.94 0.80 50-8 Interplate Distance a = lo-" 0.715 0.543 0.299 0.162 a=o 0.716 0.550 0.319 0.186 ratio 1.00 0.99 0.94 0.88 a = lo-" a=O
ratio
0.03 0.161 0.279 0.58 0.0552 0.0692 0.80
TABLE 11: Potential Difference between and Midpoint, A G O oa 0.001 0.002 0.005 10-8. Interplate Distance a = lo-" 0.220 0.444 1.23 a=o 0.216 0.417 0.942 ratio 1.02 1.06 1.31 a = 10'"
a=o
ratio
100-A Interplate Distance 0.548 0.367 0.177 0.0926 0.550 0.371 0.186 0.101 1.00 0.99 0.95 0.92
0.0312 0.0356 0.88
a = lo-'' o=o
ratio
1000-A Interplate Distance 0.101 0.0525 0.0215 0.0108 0.101 0.0526 0.0216 0.0109 ratio 1.00 1.00 1.00 1.00 a Surface charge density (protonic chargel8 '). expressed in (cm'/esu)'. a = lo-" a=o
0.00360 0.00366 0.98 b a
is
0.01
0.03
2.73 1.63 1.67
6.60 3.22 2.05
5.28 4.12 1.28
9.27 6.19 1.50
Distance 4.53 6.53 4.12 5.41 1.10 1.21
10.5 7.54 1.39
50-8 Interplate Distance 0.955 1.69 3.34 0.942 1.63 2.92 1.01 1.03 1.14 100-A Interplate 1.64 2.65 1.63 2.57 1.01 1.03
Surface
Distance 8.94 11.0 15.0 8.57 9.91 12.1 ratio 1.04 1.11 1.24 a , b See the corresponding footnotes t o Table I. a = lo-" a=o
1000-8 Interplate 5.43 6.83 5.41 6.74 1.00 1.01
4
3
, . 8
. mu
1
3
1 2.0
.
SURFACE CHARGE DENSITY ( PROTONIC CHARGE / ~2 )
1.2
0.0 1
1 .o
Figure 4. Electrostatic free energy, Fd,and entropy, S,, as functions of surface charge density. F,,IkT (-) and -S,,IK (---). Interplate distance d ( A ) (from top to bottom): lo3, 200, 100, 50, 20, and I O .
0
.o 1
Electrostatic Energy, Entropy, and Free Energy The electrostatic energy, Uel, is given by
/ 42
)
Flgure 5. Effects of dielectric saturation on electrostatic energy (UJ, entropy (Sei), and free energy ( F J . Ordinates represent ratios of the results with saturation effect to those without saturation. Interplate distance (A) (from top to bottom): 10, 20, 50, 100, and lo3.
In this limit, Velincreases linearly with u according to eq
Vel = _LSd12dxSDE 4a 0 0 dD
30.
= W a / e o ) ( l - 7)
(30)
Equation 30 is identical with the result obtained in the absence of dielectric saturation. The effect of dielectric saturation is included in the value of y. At large values of u, the product Kd increases and tends to a limit which is related to the limit of ( a y / e o )by the following equation: (uy/ eo)(I-.m = (E& T/8neo2d) ( ~ dr-m)
.03
SURFACE CHARGE DEYSITY ( PROTONIC CHARGE
significant than on the counterion activity.
.o 2
1
(31)
The electrostatic entropy, Sel,is given by the equation -TSel = Jdiz[C(x)kTln (C(x)/(C))]dx = --
kT(a/eo)[ln y
d/2
+ ,Ar$o] - (4a)-lL
ED dx (32)
If dielectric saturation is not taken into account, the last integral in eq 32 is reduced to 2Ue1. The results of cal-
The Journal of Physical Chemisfty, Vol. 83, No. 22, 1979 2915
Dielectric Saturation Effects
r
3
0
I
2
3
4
5
0
1
2
3
4
Imz
5
x ii~~'ctni
Figure 0. Dependence on the distance xfrom the surface. Surface charge density is 0.01 ( e o / A 2 ) .Interplate distance (A): (A) 100 and (B) 20. (a) Dielectric constant c, (b) u ' / u (see text), and (c) fieM strength E . Values of a (cm*/esu)*: lo-'' (-) and 0 (---).
culations of Se1are plotted against u for various values of d in Figure 4. The electrostatic free energy, F,1, can be calculated by the equation: Figure 7. Contours of integration for
I- (top) and I+ (bottom).
of polyelectrolyte solutions at infinite d i l ~ t i o n . ~ The results of the calculation of Fel are also shown in Figure 4. The effect of dielectric saturation can be estimated by calculation of the ratios of Uel, Sel, and F,1 to those quantities in the absence of dielectric saturation. The results are shown in Figure 5. The effect of dielectric saturation on Vel is small, although the ratio, Vel/ U,1(0), has a maximum at an intermediate value of the charge density, u. On the other hand, the effect of dielectric saturation on SEIis quite large. The value of S1, is negative and the absolute value of S1, is made larger by dielectric saturation. The ratio Se1/Se1(O) increases with increasing u. This effect comes mainly from stronger accumulation of counterions near the surface of the plates. The electric displacement, Do, at the surface x = 0 is determined by the charge density on the plate, independently of the dielectric saturation, whereas the electric field, E , at the surface is increased by dielectric saturation. However, this increase of the field is restricted near the surface, as shown in Figure 6, where the electric field and the dielectric constant are given as functions of the distance from the surface, for two different values of the distance between adjacent plates. Therefore, as described previously, the electric potential difference is very much changed by dielectric saturation, but the energy, Vel,and the counterion activity are only slightly influenced. In Figure 6, the total charge contained in the space between x = 0 and x , u*, is also shown as a function of the distance x . The cancellation of the charges on the plate by counterions is made stronger by dielectric saturation. It must be noted here that the results of the calculation at very short distances, given in Figure 6, are not quantitatively significant, because the counterion size was not taken into consideration. In general, the effect of dielectric saturation is larger for shorter distance between plates. In fact, in the limit of infinite dilution, the effect no longer exists. This is consistent with the statement that the macroscopic dielectric constant can be used for thermodynamic quantities
Acknowledgment. Numerical calculations were carried out with a FACOM 230-75 computer at the Computation Center of Nagoya University.
Appendix A. Applicability of the Poisson-Boltzmann Equation The electric potential and the counterion concentration must be determined by the condition giving a minimum of free energy. As far as the potential or the field and the concentration are assumed to be functions of the coordinate x , the distance from the plate, the energy can be expressed as eq 29 and the entropy can be expressed as eq 32. Therefore, the variation of the free energy with respect to variations of the field and the counterion concentration is given by
6F,1 = ( 4 ~ ) " & ~ ' ' dx 6[
LDE dD]+
kTSd"dx 0 [6C(x) In C(x)/(C)
= ( 4 ~ ) - ~ 1 ~ '6D ' Edx
+ 6C(x)] (Al)
+
kTSd/'[ln (C(x)/(C))+ 1]6C(x) dx (A2) 0
The variation of D is related to that of C(x) by the Poisson equation (eq 13) as follows:
6D = -4*eoS" 6C(x) dx d/2
(A3)
where the condition that D = 0 at the midpoint x = d/2 is used. The first term on the right-hand side of eq A2 can be transformed by using eq A3, as follows: ( 4 ~ ) -0~ s ~ 6D ' ' Edx = i d i z d x [ e 0 ( d $ / d x )d ~l 2'6C(x? dx'] = - i d / z e o + 6C(x) dx (A4)
Introducing eq A4 into eq A2, we obtain
2916
The Journal of Physical Chemistry, Vol. 83, No. 22, 1979
6Fe1 = Ldi2dx [kTIn ( C ( x ) / ( C ) )+ kT - eo$] 6C(x) (A51 The condition of minimum free energy requires that the integrand should be constant and independent of x. This results in the Boltzmann law of eq 14. Thus, the Boltzmann law is applicable even in the presence of dielectric saturation. Although the dielectric constant may depend on the coordinate x as a result of dielectric saturation, the distribution of counterions is always related to the potential by eq 14. The Poisson-Boltzmann equation describes a relation between the potential and the concentration in the equilibrium state. Bell and Levine’OJl gave the following expression to the chemical potential y(x) of counterions as a function of the coordinate x : y(x) = [ ( x ) + kT In C ( x ) - eo$(x) (-46) Here [ ( x ) represents the chemical potential due to short-range interactions of counterions with solvent molecules. The present treatment involves the assumption that the term [ ( x ) is independent of x. Appendix B. Solution of the Poisson-Boltzmann Equation When the Parameter B is Negative ’” With a new parameter 9, defined in the text by io = B1I2, integration of ea 20 is written as follows:
H. Maeda and F. Oosawa
Similarly
I+ = in - (1/2) In
rZ - 2 J Y COS w + SZ r? + 2 J Y cos w + SZ
1-
( Y cos 8 + J)d8 037) k“? + 2 J Y cos 0 + SZ
From eq B6 and B7
I-
+ I+ = 2ni - 2Ji
(Ycos0 + J)d0 I.“L + 2 J Y cos 8 + SZ
I - - I+ = In [(V - 2 J Y cos w
038)
+ SZ)/(V + 2 J Y cos w + SZ)] (B9)
The integral appearing in eq B8 is evaluated as follows:
(Y cos 8 + J)d0 = n/2J E1 + 2 J Y cos 0 + 52
+
(SZ - P)/2J[S1 cos w dt ((2JYt + SZ +V)(1- t2)1/2)-1 + 1
dt ((2JYt + SZ + V)(1 Jcos
- t2)1/2)-1]
w
Each term in eq B1 will be integrated separately and the two integrals, I- and I+, are defined as follows:
1’( + 3>’
dY =
(2J9-l exp[(i/2) arctan @]I-(B2)
’( + 3 ) V
= n/2J (J=
dY =
(2J9-l exp[(-i/2) arctan @]I+(B3) The integrals I- and I+ are concisely expressed in terms of a complex variable z.
Here 4 = J sin [arctan (p)/2] and 77 = J cos [arctan (/3)/2]. If we add two parts of contours of integration to each of I- or I+ represented by two semicircles of radius J, one centered at the origin and the other at a point (Y,O) as shown in Figure 7, then we get the following from Cauchy’s theorem: 2-l dz = I- = in - J; z-V=J,w-a
