J. Phys. Chem. 1992, 96, 2365-2371
2365
Electric Polarization of Rodllke Polyions Investigated by Monte Carlo Simulations M. Yoshida, K. Kikuchi, T. Maekawa, and H. Watanabe* Department of Chemistry, The University of Tokyo, Komaba, Meguro, Tokyo 159, Japan (Received: August 13, 1991)
Polarization of associated and free counterionsof a rodlike polyion by an external electric field was investigated by the Metropolis Monte Carlo simulations. The field was applied along the symmetry axis of the polyion and the induced dipole moment was evaluated as the displacement of the center of gravity of the counterions. It was found that the induced dipole moment increases linearly with increasing field strength in the low-field region but reaches a maximum and then begins to decrease with further increase of the field strength. The decrease of the induced dipole moment is closely related to a destruction of the ion atmosphere of the free ions by strong external fields and the destruction begins from the peripherals of the atmosphere. The polarizability calculated at a constant counterion concentration increases proportionally to the cube of polyion length. The induced dipole moment arising from the free counterions depends on counterion concentration,Le., the size of cell. Further investigation of the nature of the induced dipole moment of polyelectrolyte seems to be required. With increasing the Manning’s parameter E, the association of the counterions increases exponentiallybut the average contribution of the associated counterions to the polarizability decreases. The competitive association of divalent counterions and its contribution to the induced dipole moment were worked out. The induced dipole moment seems to increase with Z2, where Z is the valence of counterion. Applicability of the Metropolis MC method to such a nonequilibrium physical process was discussed.
Introduction When a polyelectrolyte molecule in solution is placed in an external electric field, the molecule tends to orient the molecular axis according to the field direction. In the case of a rigid polyelectrolyte which has no permanent dipole moment, the induced dipole moment along the long axis of the molecule is thought to be the origin of the torque. In dielectric and electrooptical measurements, the deformation of angular distribution of the molecular axes from the homogeneous distribution by the application of an external electric field is observed. In common procedures, the induced dipole moment of the polyelectrolyte in solution is regarded to be proportional to the field strength, to the extent that the applied field is not so high. By taking the induced dipole moment to be proportional to the field strength, we are able to obtain the value of “polarizability” by standard procedures, and the polarizability thus obtained is considered as a physical quantity characteristic of the examined polyelectrolyte-solvent system. Several models have been presented as the origin of the “induced dipole moment” of the polyelectrolyte in solution. Schwarz’s2 proposed a model of longitudinal movement of “bound ion” to the surface of the polyelectrolyte as the origin of the large induced dipole moment of a highly charged elongated molecule in solution. Since the mutual repulsion among the bound ions was neglected in Schwarz’s model, the distribution of the bound ions is Gaussian around the center of polyion in the absence of an external field. Mande13 obtained an expression of the induced dipole moment of a rodlike polyelectrolyte in which the charges on the polyion are discretely distributed forming potential wells for the bound counterions. Also the mutual repulsion among the bound ions was neglected. Both the Schwarz’s and Mandel’s theories resulted in P dependence of the induced dipole moment, where I is the polyion length. O’Konski and Krause4 treated theoretically the problem of anisotropic polarization of the elongated polyion in terms of the surface conductivity of polyion. It was assumed that the bound counterions contribute to the surface conduction. The theory was extended to treat the polyion which is not highly charged so that has no bound counteri~n.~ Oosawa6 treated the electric polarization as the thermal fluctuation in the concentration of the counterion bound to a rodlike (1) (2) (3) (4) (5) 413.
Schwarz, G. Z . Phys. 1956, 145, 563. Schwarz, G. Z . Phys. Chem. 1959, 19, 286. M,andel, M. Mol. Phys. 1961,4, 489. 0 Konski, C. T.; Krause, S.J. Phys. Chem. 1970, 74, 3243. Krause, S.;Zvilichovsky,B.; Galvin, M. E. J. Biophys. Soc. 1980, 29,
(6) Oosawa, F. Biopolymers 1970, 9, 677.
0022-3654/92/2096-2365$03.00/0
polyion. He obtained as the polarizability CY equal to (p2)E*/kBT, where ( p 2 ) E = ois the mean square of dipole in the absence of external field and kBTis the thermal energy. Oosawa’s theory is regarded as a two-phase model and results in P dependence of the polarizability. van Dijk et al.’ elaborated the Oosawa’s two-phase model by taking into consideration the counterion exchange between the bound and the free ions. The influence of exchange is remarkable when the rate of exchange is high; the value of polarizability is strongly reduced and becomes proportional to 1. Another kind of model for the origin of the electric polarization of the polyelectrolyte is the deformation of the counterion atmosphere by the application of an external electric Hogan et al.IOproposed an anisotropic ion flow model for the explanation of the field strength dependence of electric dichroism of DNA fragment. The model can be regarded as an extension of the deformation of the counterion atmosphere and predicted well their experimental results. The model predicts I* dependence of the apparent dipole moment, which is in agreement with their experimental results. Stellwagen” measured the length dependence of the induced dipole moment of monodisperse DNA fragments by the electrooptical measurements and obtained l2 dependence. Other electrooptical m e a s ~ r e m e n t son ~ ~DNA . ~ ~ also resulted in an I* dependence of the induced dipole moment. It was observed that, at very high field strength, however, the induced dipole moment tends to saturate and the limiting dipole moment is substantially independent of the length of DNA.”J3 Rau and Charney14 calculated the induced dipole moment originating from the polarization of the Debye-Huckel ion atmosphere. The calculated polarizability is proportional to p @ 1 1 ” / K ‘ ‘ 2 when KI < 10, where Z is the valence of counterion, Q is the charge density on polyion, and K is the Debye-Huckel shielding parameter. Rau and CharneyI5 extended their calculation to strong fields and found that the induced dipole moment originating from the polarization of the Debye-Huckel ion atmosphere reaches a maximum value and then begins to decrease
-
(7) van Dijk, W.; van der Touw, F.; Mandel, M. Macromolecules 1981, 14, 792. (8) Schwarz, G. J. Phys. Chem. 1962, 66, 2636. (9) Pollack, M. J. Chem. Phys. 1965, 43, 908. (10) Hogan, M.; Dattagupta, N.; Crothers, D. M. Proc. Natl. Acad. Sci. U.S.A. 1978, 75, 195. ( 1 1) Stellwagen, N. C. Biopolymers 1981, 20, 399. (12) Elisa, J. G.; Eden, D. Macromolecules 1981, 14, 410. (13) Diekmann, S . ; Hillen, W.; Jung, M.; Wells, R. D.; Porschke, D. Biophys. Chem. 1982, 15, 157. (14) Rau, D. C.; Charney, E. Biophys. Chem. 1981, 14, 1. (15) Rau, D. C.; Charney, E. Macromolecules 1983, 16, 1653.
0 1992 American Chemical Society
2366 The Journal of Physical Chemistry, Vol. 96, No. 5, 1992 with increasing field strength. They attributed the effect of the strong electric field to the Wien effect. In a short review, CharneyI6 presented some experimental verifications of the theoretical models from the electric dichroism measurements on hyaluronic acid solutions. In another report,I7 we will present some results about the counterion distribution around a cylindrical polyion of finite length and its deformation by an application of an external electric field by the Metropolis Monte Carlo simulations. From the ion distribution, contour m a p of electrostatic potential around the finite polyion were calculated. One of the remarkable features observed by the M C simulations is that the number of the associated counterions is almost independent of the location along the longitudinal direction of polyion, except at both ends of the polyion. At the ends of the polyion, the association sharply decays. In the present study, in order to clarify the origin and the nature of “induced dipole moment” of the polyelectrolyte in solution, Metropolis Monte Carlo calculations under external electric fields were carried out. Although the radial distribution of the counterion density (RDCD) decays smoothly with the distance from the polyion surface, it seems convenient to classify the counterions into “associated” and “free” ions, where the associated ion means the counterion whose center happens to be within a distance of ion diameter from the polyion surface and the free ion means the other counterion which can be regarded as the ion which is governed by Debye-Huckel potential. The induced dipole moment was estimated as the displacement of the center of gravity of counterions from the unperturbed position and the polarizability was defined as the induced dipole moment thus obtained divided by the field strength. The induced dipole moment increases proportionally with applied field strength in the low-field region, but reaches a maximum with further increase of the field strength and even decreases with a further increase of the field strength. This abnormal behavior was analyzed from the deformation of the counterion distribution. Length dependence of the polarizability was evaluated and the polarizability calculated at a constant counterion concentration increases proportional to 13. The dependence of the polarizability in the low-field region on the Manning parameter was elucidated. It was also investigated how the coexisting divalent counterions contribute to the dipole moment.
Applicability of Metropolis MC Method to Nonequilibrium Physical Processes The Metropolis MC method was found by Metropolis et a1.’* as a method suited for electronic computers to carry out a many-dimensional integral over the configurational space. A Markov chain of states is constructed and the elements of the transition matrix are calculated to generate a trajectory in the phase space which samples a representative portion from the canonical ensemble. Thus the method has been regarded to be applicable only to the study of the equilibrium state. The polyion-unterion system arrives at a stationarily flowing state under the homogeneous electric field. So that the applicability of the Metropolis M C method to such a nonequilibrium state needs some elaborations. To the authors knowledge, there is no established verification of the applicability of the Metropolis MC method to non-equilibrium physical processes. In this section, we briefly show that the Metropolis MC method is not to be considered just as a computing technique but it contains a description of the physical diffusion processes. More details and some demonstrative application will be reported elsewhere. l9 (16) Charney, E. In Dynamic Behavior of Macromolecules, Liquid Crystals and Biological Systems by Optical and Electro-Optical Methods; Watanabe, H., Ed.; Hirokawa Publishing Co.: Tokyo 1989; p 163. (17) Watanabe, H.; Yoshida, M.; Kikuchi, K.; Maekawa, T. In Proceedings of the 6th Symposium on Colloid and Molecular Electrooptics; Jennings, B. R., Ed.; IOP Publishing Ltd.: Bristol, 1992, in press. (18) Metropolis, A. W.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H.; Teller, E. J . Chem. Phys. 1953, 21, 1087. (19) Kikuchi, K.; Yoshida, M.; Maekawa, T.; Watanabe, H. Chem. Phys. Lett. 1991, 185, 3 3 5 .
Yoshida et al. To facilitate the discussion, we use the one-dimensional nomenclature. In the Metropolis procedures, we fmt place n particles of the system in any codiguration. Then we move the n particles in succession by giving a uniform random displacement along the coordinate direction such as X+X+hu
(1)
where h is the maximum allowed displacement and u is a random number between -1 and 1. After the displacement, the particle is equally likely to be anywhere within a line element of length 2h centered at its original position. We then calculate the change in potential energy of the system AU caused by the move. If AU < 0 we allow the move unconditionally, and if AU > 0 we allow the move with probability exp(-AU/kBT). On the computer let there be a large but finite number W of new possible positions for the particle; the particle remains on the same position with the probability
Here we consider the Metropolis MC as a physical process and assume all the particles in their new positions in the “time interval” At. Since only small steps occur in the process, we can describe it by the Fokker-Planck equation
where P(X,t) is the probability density, X i s the space coordinate, t is the time, and U is the potential function. Let us calculate the mean displacement (AX)during a small time interval At. Assuming the maximum allowed displacement h is small, AU is written as AU = avhu
ax
(4)
For convenience assuming that aU/aX < 0, we have to the second order of h
That is, the particle gains an average drift proportional to the force 4 U / a X exerted by the surroundings during the time interval At. Similarly, the mean square displacement ( (AX)2) calculated to the same order of h is
Thus the mean square displacement is independent of the force field within the approximation. If we define the diffusion constant D by
D=--((AX)’)
- -h2- 1 2Ar 6 At the mean displacement per unit “time” is written as
(7)
or by using the Einstein’s relation D/kBT = l/{, where {is the friction constant as
(9) We therefore obtain the Fokker-Planck equation -P(xJ) =at
{
div ((grad v) P(x,t)J + DV2P(x,t)
The Journal of Physical Chemistry, Vol. 96, No. 5, 1992 2361
Electric Polarization of Rodlike Polyions
=t
.1
0
c
E=0.05
-
.1
E
0
X
E=0.15
t l 7
J
F i p e 1. The model.
It has a form of the diffusion equation and in fact it is the diffusion equation for the Brownian particle in the system characterized by the phenomenological constants D and {. We have applied the Metropolis MC method for the diffusion process of the harmonically bound particles with U = a / 2 , where K is the force constant and compared the results with the analytical solution given by Uhlenbeck and OmsteinZo
- exP(-ZKt/OJ/K
Models and Computing Methods The model used in this work is illustrated in Figure 1 . In the following, length is expressed in nanometers. A cylindrical polyion is centered at a rectangular parallelepiped whose edges are in the x , y , and z directions of the Cartesian coordinate, the origin is at the center of polyion, and the z axis is along the symmetry axis of the cylinder. We express the distance from the symmetry axis by r = (xz + Y * ) ' / ~ .n negative charges were set with equal separation of b on the symmetry axis of the cylinder whose radius is a (=OS). The length of the cylinder is 1 = (n l)b 2u. The dimension of the rectangular parallelepiped in the x and y directions is C and in the z direction is L. L is taken equal to 1 + 12 except for the case of changing concentration. For the standard system (C = 12, L = 24), the polyion concentration was 4.8 X lo4 M and the average counterion concentration was 4.8 X n X lo4 M. The rectangular parallelepiped is the unit cell of the system and the movement of n counterions is confined inside the unit cell. The counterion cannot penetrate into the cylinder. The choice of mtangdar parallelepiped as the unit cell enabled us to assemble 27 identical unit cells without any vacancy among them. In the present case where the extemal field was applied along the z axis, the counterions flow along the z axis. To take into account the interactions among the flowing counterions beyond the cell walls, the central cell was sandwiched by two equivalent cells in the z direction as illustrated in Figure 1. The diameter of the counterions was taken equal to d (=0.3). Therefore, two ions cannot approach closer than d. (20) Uhlenbeck, G. E.;Omstein. L. S.Phys. Rev. 1930, 36,823.
Figure 2. Effect of applied external fields on the counterion association. The extemal fields are applied along the z axis. Numerals on the figures are the percentage of associated counterions.
(12)
The agreement between the results of the simulation and the analytical solution was indeed excellent. Thus we have shown that the physical significance of the Metropolis MC lies in the dynamic description of the system on the coarsegrained time scale and the relevance of the Metropolis MC method for the elucidation of the electric polarization in the polyelectrolyte system has been established.
- +
0
n counterions were generated at random in the accessible space of the unit cell. Each counterion was moved in turn in a random direction over a random distance uh, where u is a random number between 0 and 1. The Metropolis decision was made for each move. The value of the Manning t parameter,
and the mean square displacement by
((MI2) = kBn1
.1
5
ez/(4?rcobDkBT)
(13)
where e is the proton charge, Q is the permittivity of vacuum, and D is the dielectric constant of water, by taking e = 1.6 X 10-19C, TO = 8.854 X 1O-I2Fm, D = 78.3, kB = 1.38 X T = 298 K, 1s
7.15 x 10-"/b = lB/b
(14)
Thus the Bjerrum length IB is 0.715 and t = 1 when b = 0.715. In the present work, parameters are fixed on n = 16, 1 = 12, and b = 0.715 so that the value of [ is unity excepting the case of varying the value of E. For the change of E, only the value of b was varied. The distribution of the counterions is expressed by following integral and differential RDCD p(z,ro) =
fo
a
p'(z,r) =
f+b/2 z-b/2
p(z,r)21rr dr dz
(15)
S2+b/2 p(z,r)2ur dz 2-b
f2
where p(z,r) is the number density of counterions in a volume element of dr in radial direction and thickness b in the z direction. For an isolated polyion of infinite length, p(z,ro) 1 for ro =. p'(z,r) is the radial distribution function centered at z and in the thickness of b. The number of associated counterions in the thickness of b in z direction is defined as the value of p(z,l.O).
- -
Results Figure 2 demonstrates the effect of the applied external electric field on the counterion association. The applied field strengths are given in the figure. The unit of E is that the potential energy difference of the displacement of unit charge by lB(r0.715) along the direction of E is equal to kBTat T = 298 K. This means that the field strength is measured by the units of B j e r " length and thermal energy. It is clearly seen that the distribution of the associated counterions shifts according to the field gradient. It is interesting to note that the associated counterions are stripped away from the polyion by the external field. It is the more
2368
The Journal of Physical Chemistry, Vol. 96, No. 5, 1992
:f
4 2
Yoshida et al. -35
0 A
0 A
0 A
0
A
0
A
0
A
8
--
0
0
0
0
0
O
0
::qo
0
0
C
I
-3 n
k W e .25
CF
1
.2 -15 __
--
0
2
4
6
4
6
r
-25 m
k
W
4
remarkable the larger the field strength is. The ratios of the associated counterions are indicated in each figure. The induced dipole moment mtotl,defined as the displacement of the center of gravity of the counterions can be divided into two parts, massoand mfree,where massois the dipole moment arising from the associated counterions and mfreeis the dipole moment = marising from the unassociated counterions. Of course, qou
.2
CF -15 .1
0
2
r
+ mfree.
Figure 3 is a plot of mtotl(0),masso(a), and mfree(A)versus external field strength. It is noted that mtotlincreases almost proportionally with the increase of the applied field strength in the low-field region. But it arrives at a maximum at about 0.2 of E and begins to decrease with further increase of the field strength. It is also noted that the contribution of massoto mtotl is much less than that of mfree.massoseems to keep its maximum value beyond E 0.2. Since the number of associated counterions decreases with the increase of field strength, average contribution per single associated ion to m-, which is indicated by 0, increases linearly up to E 0.2, and then deviates downward but still increasing up to E 0.5. Thus we see that the decrease of mtotl in the high-field strength is mainly due to the decrease of mfree. Figure 4a demonstrates the effect of external field on the total differential RDCD defined as
n
k
W
m @
--
Numerals on the curves are the values of E. The total differential RDCD, qT'(r), reduces in the near region and increase in the remote region with the increase of E. That shows quantitatively how the counterions are carried away from the vicinity of the central polyion to the remote region by the external field. To see the field effect on the total differential RDCD in more detail, we present in Figure 4, b and c, how the differential RDCDs in the -1/2 to 1/2 (region A), defined as qA'(r) =
and in the -L/2 to -1/2
-112
-il
.05 0
0
I
I
I
I
I
I
, 4
2
I
;
I
1 6
r Figure 4. Effect of external field on differential RDCD, qT'(r)(a, top). Numerals on the curves are the values of E. Parts b (middle) and c (bottom) are differential RDCDs in the region A and region B, respectively.
k
W
6
p(z,r)27rr dz
+ 1/2 to L/2 (region B), defined as r
are affected by the presence of external field, respectively. In Figures 4, b and c, regions A and B are illustrated by shadow in the cell, respectively. The figures more clearly indicate that the external field carries the counterions to the remote region. Figure 5 shows plots of mav(r)= m(r)/v(r),where m(r) is the dipole moment due to the counterions in the cylindrical volume of -1/2 to 1/2 in the z direction and a to r in the radial direction (volume S)and v(r) is the number of counterions in the volume S, for different values of E. mav(r)is the average contribution of counterions in the volume S to mtotl. As is seen, except the associated counterions, ma"@)does not depend on r for the weak field but decreases with the increase of r when the field strength
Figure 5. Plots of mav(r)= m(r)/v(r),where v(r) is the average counterion density in the volume S, versus r for different values of E. ma"is the average contribution of the counterion in the volume S to mtotl.
is high. This corresponds to the decrease of mfreeat high fields. Figure 6a,b illustrates the contour lines of the electrostatic potential along z axis for E = 0.05, 0.15, and 0.25 at r = 1 (a) and r = 3 (b). As is shown, the contour lines are declined by the bias of the external field, and the central polyion creates a deep potential well at r = 1 for the associated ions. The ions in the well may move to the lower end of the declined potential well and may escape occasionally by thermal agitation. This corresponds to the drift of associated counterions to one side. The central polyion is pictured in the bottom. At r = 3 the well is very shallow
The Journal of Physical Chemistry, Vol. 96, No. 5, 1992 2369
Electric Polarization of Rodlike Polyions
c Q,
-2
-4 -4
0
a
-A L
"C
L e
-A
I
-6
I
I
1 1-1
I
2
I
I
I
I
I
I
I
4
3
In n Figure 8. Length dependences of induced dipole moment on the polyion length plotted as In m versus In n. 0 and 0 are mtoIland m, for constant counterion concentration and 0 and are mtoIl and m, for
constant cell size, respectively. 6.,,,
-
-4
Figure 6. Contour lines of electrostatic potential along the z axis for E = 0.1, 0.3, and 0.5 at r = 1 (a, top) and r = 3 (b, bottom).
'I
$
11
0
A
0
0 0
0
0 A
8
A
0
2
4
-
A
A
0 0
0
f2 0 O
0 O
Figure 9. Plots of m,,l (01, mdival (A) and mtotl(0) versus charge fraction of divalent counterions.
I
t
*2rF
8
I-
0
.l5I 1 .1
.8
-O
0
t
.05
0b 0
315
10
4
2
1
c
5
3
0
t Figure 7. (a, top) Plots of mtoIl (0) and ma, (0)against 5. (b, bottom) Plots of na, (m) and ma"= mauo/nauo( 0 )versus 5.
so that the counterions may flow almost freely. Figure 7a presents plots of m,otl(0)and ma,, (0) against [. We see that mtodincreases linearly with F in low values o f t but tends to saturate at high values of F. Since the number of associated counterions na, (m) increases with the increase of [ as is presented in Figure 7b, ma, = m,,,/n,,, ( 0 )decreases with the increase of 4. Figure 8 presents length dependencies of mIotl(0)and ma,,, (0) for 5 = 1 polyion plotted as In m versus In n. The size of cell was varied so as to keep the counterion concentration a constant. As is seen, mlotlincreases with 13. ma, also increases with l3 a t low n but deviates downwards from it at large value of n. It was
-4
0 0
.2
.6
.4
.8
1
n2 Figure 10. Plot of the number fraction of divalent counterions in the associated counterions, ndiv,a,, versus number fraction of divalent counterions in the system.
noted that mtotldepends on the concentration, Le., the available space in the unit cell for the counterions, whereas ma,, does not depend on the concentration. In the figure, 0 and stand for mtaland m,, respectively, obtained at a constant cell size so that the counterion concentration increased with increase of n. Figure 9 shows the contribution of coexisting divalent counterions to mtotl(0). mmonoval (0) and mdival(A) are the dipole moment arising from monovalent and divalent counterions, respectively. The abscissa f2 = 2n2+/(n+ + 2n2+) is the charge fraction of divalent ion where n+ and n2+ are the number of monovalent and divalent ions, respectively. As is seen, the contribution of divalent ions becomes dominant and mtollincreases with the increase of f2. Figure 10 is a plot of n2,asso= n2+a,,,/(n+asso+ n2+asso),where n+- and n2+- are the number of the associated monovalent and divalent ions, respectively, versus n2 = n2+/(n2++ n+) and shows
2370 The Journal of Physical Chemistry, Val. 96, No. 5, I992
a.
E
Figure 11. A demonstrative illustration of ion atmosphere deformation and destruction by increase of field strength. how the fraction of divalent counterions in the associated ions increases with the increase of divalent number fraction.
Discussion When an external electric field is applied to a polyion-counterion system, angular distribution of the polyion axis reaches a steady-state at equilibrium in the field. The counterions feel the Coulomb forces form the polyion and other counterions together with the gradient of external field and can establish an equilibrium distribution around the polyion. At the same time, the polyion and the counterions will flow along the field gradient and attain a stationary state. Thus we face a very complicated situation where the rotation of polyion axis and the ion flow is coupled to elucidate fully the electric polarizability observed by the electrooptical effect of the polyelectrolyte. In the present study, to simplify the problem, the external field was applied along the z axis of an isolated polyion, so that the polyion rotation was not taken into consideration. More elaborate elucidation of the problem will be appear in a following paper. The induced dipole moment evaluated as the displacement of the center of the gravity of counterions with respect to the central polyion was divided into two components, ma, and mfree. The decrease of m,/n, divided by E (average contribution of single associated ion to the polarizability) with the increase of field strength indicates that the repulsion among the accumulated counterions on the end part of polyion counteracts against the polarization. At the same time, the number of associated countenons decreases with the increase of the field strength. Therefore, the decrease of polarizability due to the associated ions arises from two factors; one is the decrease of the contributing counterions and the other is the decrease of the polarizability due to the repulsion among the accumulated counterions at the end of polyion. The fact that the associated counterions are stripped away from the polyion by the external field indicates that a kind of Wien effect is taking place resulting in the increase of nonbound ions. The dramatic decrease of mfm at high field strength is a result of the destruction of the ion atmosphere. This destruction of the ion atmosphere starts from the remote region and progresses more and more toward the core polyion with the increase of field strength. The destruction of the ion atmosphere means that the counterions begin to flow almost freely and their distribution becomes homogeneous (Figure 11). This destruction of the ion atmosphere may result in an increase of the mobility of each counterion along the field gradient and this may be another cause of the Wien effect. The decrease of induced dipole moment at high field strength was predicted theoretically by Rau and Chamey.I5 We understand that the Rau and Charney’s theory is a simplification to the Debye-Huckel approximation of a more general theory by Fixmanz1in which he calculated the torque on a charged cylinder (21) Fixman, M. Macromolecules 1980, 13, 711.
Yoshida et al. in a weak, steady external field by solving an equation for the ion flux. The generalized forces of the flux are the concentration gradient and the electrostatic field gradient. The electrostatic field is a mean field given by the Poisson-Boltzmann equation. Rau and Charney attributed the decrease of the induced dipole moment at high field strength to the destruction of the ion atmosphere. Our results are in agreement with Rau and Chamey’s theoretical results in the sense that the ion atmosphere is destroyed by the strong external electric field. The destruction of the ion atmosphere means that the Debye-Huckel potential around the central polyion is taken over by the strong external field. We should be careful, however, to discuss the stationary and dynamic physical phenomena caused by the external electric field under which flow of ions is taking place by any solution of PB equation. As is expected, the number of the associated counterions exponentially increases with the increase of 6 so that ma, also incream. However, the average contribution of a single associated ion to the polarizability decreases with the increase of 6. This is probably due to the repulsion among the associated ions and could be a main cause of the decrease of the polarizability with the increase of 6. The induced dipole moment evaluated as the displacement of the center of the counterions is proportional to P,in accordance to the theoretical prediction of Schwarz1%2 and Mande1,j when the length 1 was varied to keep the counterion concentration constant. However, mfra depends on the size of cell, i.e., the counterion concentration, so that when the length 1 was varied at constant cell size, mfmvaries with P . The experimental results in the Ken region suggest that the induced dipole moment varies with P.10Jk16 With this subject, there seems remaining at least two points to be clarified: the first is the definition of the induced dipole moment of polyion-counterion system, and the second is the effective field strength in conducting solutions which is regarded as equivalent to the applied field between two electrodes. In a following paper, the problem of the definition of the induced dipole moment of polyionaunterion system will be fully discussed. When the divalent counterions coexist with the monovalent counterions, the divalent ions push away the monovalent ions from the vicinity of polyion. The theoretical prediction of z2 dependence of induced dipole m ~ m e n t , ~ ,where ~ , ’ 2 is the valence of the counterion, is in agreement with the present results.
Conclusion 1. Polarization of associated and free counterions of an rodlike polyion by the external electric field was investigated by the Metropolis Monte Carlo simulations. The field was applied along the symmetry axis of the polyion and the induced dipole moment was evaluated as the displacement of the center of gravity of the counterions. The counterions were classified into the associated and free ions and contributions of ma, and mfrceto mtOtlwas estimated separately. 2. The associated counterions are stripped away from the polyion by the strong external electric field so that the number of the associated counterions decreases with the increase of field strength. The average polarizability of the associated counterions slightly decreases with the increase of the field strength. The decrease indicates the effect of the repulsive interaction of the accumulated counterions to the end of polyion. 3. It was found that the induced dipole moment increases linearly with the field strength in the low-field region but arrives at a maximum and then begins to decrease with further increase of the field strength. The decrease of the induced dipole moment is strongly related to the destruction of the ion atmosphere of free ions by the strong external field and the destruction begins from the periphy of the ion atmosphere. 4. The polarizability calculated at a constant counterion concentration increases proportional to 13. 5 . The mlrcedepends on the counterion concentration, Le., the cell size. Further investigation of the nature of the induced dipole moment of polyelectrolyte seems to be needed.
J. Phys. Chem. 1992, 96, 2371-2375
6. The average contribution of single associated ion to the polarizability decreases with the increase of 4. This could be a main cause of the decrease of the total polarizability with the increase of 4.
2371
7. The induced dipole moment seems to increase with z?, where 2 is the valence of counterion. 8. The applicability of the Metropolis M C method to nonequilibrium physical processes was verified.
STM Imaging and Electronic Conductivity Mechanisms In Oxidatively Doped Poly(N-methylpyrrole) Thin Films Stephen E. Creager Department of Chemistry, Indiana University, Bloomington, Indiana 47405 (Received: August 26, 1991; In Final Form: October 17, 1991)
Scanning tunneling microscopy (STM) images, localized current vs bias potential curves, and localized current vs tip displacement (normal to the sample plane) curves have been obtained for thin films of oxid&ively doped poly(Nmethylpyrro1e) on polycrystalline gold. STM images of the polymer are quite noisy but are otherwise relatively featureless. Current vs tip displacement data taken over the polymer are characterized by currents that persist long after they would have died away at a bare metal surface; we postulate that the tip is buried in the polymer film, probably even while imaging. Current vs bias potential data acquired over polymer films reveal ohmic behavior at low bias but exponentially rising currents at high bias. This behavior is described in terms of an electric-field-drivenelectron-hopping process where the field is probably localized around the STM tip.
Introduction The analytical chemistry of surfaces is being revolutionized by the scanning tunneling microscope (STM) and the family of related scanned probe microscopes.'-5 A particularly interesting and important subdiscipline of STM involves the study of molecular and macromolecular surface structures. Fascinating images rich in structure, some at atomic resolution, have been reported;6 yet fundamental questions remain about how electrons are transported through these materials and how image contrast is generated. Electronically conducting polymers based on pyrrole, aniline, and thiophene have received considerable attention as electrochemically active materials, in part because of their high conductivity and in part because of their relative stability in air.7-9 Polymer films based on these monomers have been subjected to preliminary STM characteri~ation,'*'~ mostly with the goal of (1) Hansma, P. K.; Tersoff, J. T. J . Appl. Phys. 1986, 61, R1. (2) Chiang, S.;Wilson, R. J. Anal. Chem. 1987,59, 1267A. (3) Gould, S.A. C.; Drake B.; Prater, C. B.;Weishorn, A. L.; Manne, S.;
Hansma, H. G.; Massie, J.; Longmire, M.; Elings, V.; Dixon Northern, B.; Mukergee, B.;Peterson, C. M.; Stoeckenius, W.; Albrecht, T. R.; Quate, C. F. J. Vac. Sci. Technol. A 1990, 8, 369. (4) Wickramasinghe, H. K. Sci. Am. 1989, 260, 98. (5) Baldeschwieler, J. D.; Gill, J. M.; West, P. E. Am. Lab. (Fairfield, Conn) 1991, 23 (2), 34. (6) Driscoll, R. J.; Youngquist, M. G.; Baldeschwieler, J. D. Nature 1990, 346, 294. (7) Evans, G. P. In Advances in Electrochemical Science and Engineering Gerischer, H., Tobias, C. W., Eds.; VCH: New York, 1990; Chapter 1. (8) Heinze, J. Top. Curr. Chem. 1990, 152, 1. (9) Frommer, J. E.; Chance, R. R. In Encyclopedia oJPolymer Science and Engineering Wiley: New York, 1986; Vol. 5, p 462.
(IO) (a) Yang, R.; Dalsin, K. M.; Evans, D. F.; Christensen, L.; Hendrickson, W. A. J . Phys. Chem. 1989, 93,511. (b) Yang, R.; Evans, D. F.; Christensen, L.; Hendrickson, W. A. J. Phys. Chem. 1990, 94, 6117. (c) Yang, R.; Naoi, K.; Evans, D. F.; Smyrl, W. H.; Hendrickson, W. A. Lnnamui; 1991, 7, 556. (11) (a) Caple, G.; Wheeler, B. L.; Swift, R.; Porter, T. L.; Jeffers, S.J. Phys. Chem. 1990, 94, 5639. (b) Porter, T. L.; Lee, C. Y.; Wheeler, B. L.; Caple, G. J. Vac.Sci. Technol. A 1991, 9, 1452. (c) Porter, T. L.; Dillingham, T. R.; Lee, C. Y.; Jones, T. A.; Wheeler, B. L.; Caple, G. Synrh. Mer. 1991, 40, 187. (12) (a) Bonnell, D. A.; Angelopoulos, M. Synrh. Mer. 1989,33, 301. (b) Jeon, D.; Kim, J.; Gallagher, M. C.; Willis, R. F.; Kim, Y.-T. J . Vac. Sci. Technol. B 1991, 9, 1154. (13) Fan, F. F.; Bard, A. J. J. Electrochem. SOC.1989, 136, 3216.
structurally characterizing nucleation sites for the oxidative electropolymerization process. An advantage of these materials for fundamental STM studies of organic materials is that their electronic conductivity is documented and fairly well understood, in contrast with materials like DNA that are wide-band-gap insulators with no obvious states near the Fermi energy that can help transport charge. The nature of the interaction of these materials (and most other organic materials) with an STM tip is furthermore not well understood but is likely very important if their STM images are to be properly interpreted. We have initiated an STM study of the polymer poly(Nmethylpyrrole), prepared by electrochemical oxidation of the monomer N-methylpyrrole at a polycrystalline gold electrode. This monomer was chosen because of its ease of purification, the modest potential required to form polymers, the known air stability of the polymer in both the oxidized and reduced states, and the existing body of knowledge on the properties of the polymer, particularly involving oxidative doping and electronic conductivity.I4J5 The goal of this study is to investigate fundamental aspects of electron tunneling and conduction at the junction between a tunneling tip and a thin film of an organic conductor. To accomplish this, we supplement the conventional STM imaging experiments with data from localized current vs bias potential and current vs tip displacement measurements, acquired with the tip poised over a specific location on a sample surface. We report herein experiments on poly(N-methylpyrrole) films which reveal that the STM tip is very likely buried in the polymer, sometimes up to 100 A deep, probably even while imaging. The observed weak dependence of current on tip displacement over the polymer renders the conventional metal-insulator-metal formalism for electron tunneling inappropriate for such a junction. We have adopted a model based on electron hopping between localized states driven by a large electric field to explain the measured current vs bias potential data and postulate that this electric field is localized in a region of bulk polymer immediately surrounding the STM tip. (14) (a) Diaz, A. F.; Castillo, J. I.; Logan, J. A.; Lee, W. Y. J. Electroanal. Chem. 1981, 129, 115. (b) Diaz, A. F.; Castillo, J.; Kanazawa, K. K.; Logan, J. A.; Salmon, M.; Fajardo, 0. J. Electroanal. Chem. 1982, 133, 233. (15) (a) Asavapiriyanont, S.;Chandler, G. K.; Gunawardena, G. A.; Pletcher, D. J. Electroanal. Chem. 1984, 177, 245. (b) Downard, A. J.; Pletcher, D. J . Electroanal. Chem. 1986, 206, 139.
0022-3654/92/2096-237 1%03.00/0 0 1992 American Chemical Society
