- Email: [email protected]
Growth instability in diffusion controlled polymerization
Solid State Communications, Printed in Great Britain.
Vol. 60, No. 9, pp. 757-761,
GROWTH INSTABILITY J.H. Kaufman,
0038-1098/86 $3.00 + .OO Pergamon Journals Ltd.
1986.
IN DIFFUSION
CONTROLLED
POLYMERIZATION
O.R. Melroy, F.F. Abraham and A.I. Nazzal
IBM Almaden Research Laboratory,
San Jose, CA 95 193, USA
(Received 18 July 1986; in revised form 23 July 1986 by R.H. Silsbee) We report the observation of a sharp transition in the structure of polypyrrole prepared via diffusion controlled polymerization in a quasi-two dimensional electrochemical cell. As the oxidation potential is increased, the fractal dimension drops sharply from values near 2 (compact structures) to 1 as the growth becomes dendritic. At higher potentials, where the polymerization becomes diffusion limited, a continuum of structures is observed as the dendrites become more irregular and the fractal dimension increases to 1.74 + 0.01. The mean branch angle was also measured and found to decrease from 90” at the point of dendritic growth, to 45” at high potential. INTRODUCTION DIFFUSION LIMITED AGGREGATION is a process wherein successive particles moving in a space with Euclidean dimensionality E undergo a random walk until they interact with (and stick to) a growing fractal [ 1, 21. When an aggregate is produced via DLA, the resultant structure is tree-like and exhibits dilation symmetry over some range of lengths [l-6]. For example, the density of such an object is found to scale with radius to a power, D, known as the fractal dimension [6]. Because these objects are highly irregular, they can not be characterized by a single Euclidean dimension. A better description of the structure is provided, perhaps, by the fractal dimension which may be computed from the ensemble average correlation function. For random fractals produced by computer simulations on a square lattice, a typical value for the fractal dimension is 1.657 + 0.004 [l]. The apparent universality of scaling behavior characteristic of (DLA) has stimulated much interest in random aggregation phenomenon [l-6]. However, DLA is only the limiting case of a more general process, diffusion controlled aggregation (DCA). Unlike DLA, in DCA the free particle concentration at the growing surface is nonzero and as a result the aggregation process depends not only on the random motion of the aggregating particles, but also on the energetics of the deposition process. To understand the origins of the scaling behavior characteristic of DLA, it is desirable to conduct a series of experiments to systematically study the continuum of diffusion controlled aggregation conditions. Electrochemical deposition provides a convenient method for preparing aggregates under a variety of diffusion controlled conditions since the particle 757
concentration near the surface of the aggregate is determined by the applied potential [8]. As the deposition (oxidation) potential is increased, the surface concentration of the aggregating species decreases (and the concentration gradient increases). It is well-known that the magnitude of this gradient influences the structure of the aggregate. Mullins and Sekerka [9] demonstrated that growth of an aggregate into a sufficiently large gradient typically results in dendrite formation, while growth into a small gradient produces compact At even higher potentials, aggregation structures. becomes limited by diffusion (i.e. the surface concentration of aggregating particles approaches zero). In this regime electrochemical deposition approximates DLA in that the distance between aggregating particles near the surface is large and they behave like the random walkers of DLA simulations. In this Communication, we investigate the variations in structure obtained by polymerization of pyrrole under a continuum of deposition conditions. Experiments were designed to minimize effects other than diffusion. We report a growth instability where the polymer structure changes from a compact morphology to a dendritic morphology. As potential is increased further, the dendritic phase crosses over to a random aggregate phase (see Fig. 1) and the polymerization becomes limited by diffusion (the surface concentration of pyrrole monomer approaches zero). As in the case of DLA, the mass of these fractals scales with radius as M(r) a rD where, D is the fractal dimension [ 1, 21. EXPERIMENTAL Experiments to demonstrate the effects of depositing charged precursers were preformed in a thin layer cell
DIFFUSION
CONTROLLED
POLYMERIZATION
Vol. 60, No. 9
(cl Fig. 1 (a-c). Aggregates of polypyrrole prepared DCP at 0.9, 1 .5 and 6.0 volts (DIP) respectively.
(a)
(b)
via
constructed with two glass plates. On one plate a 10 micron layer of silver was evaporated except for a one inch diameter circular region in the center. This region constitutes the electrochemical cell. A 1Omil hole was drilled through the center of the other plate and a silver wire was inserted, epoxied, and sheared flush with a razor blade. A second 1Omm hole was drilled 1 cm from the first to allow contact to the electrolyte with a microreference electrode. In the first experiment the electrolyte was 0.1 M AgNO a aqueous with 0.1 M KNOB as supporting electrolyte. Since the potassium ions are not depleted at the active electrode, the supporting electrolyte serves to screen the electric field. In the second experiment the supporting electrolyte was omitted. In both cases the reference electrode was Ag/AgCl with 3 M KNOa. Samples were prepared at - 0.5 V vs Ag/AgCl. Samples of polypyrrole were prepared in a similar thin layer cell consisting of two glass plates separated by a 2000W gold ring (counter electrode) evaporated on one plate and a 1Omil Kapton spacer. A 10 mm hole was drilled through the center of the other plate through which a 5 mm gold wire was inserted, epoxied, and sheared flush with a razor blade. The solution used was 0.1 M pyrrole/O.l M silver toluenesulfonate/acetonitrile. No attempt was made to dry or degass the electrolyte. The counter reaction is the deposition of silver. Since no measurements were made of concentration dependence, we chose not to use a reference electrode. As a result, voltages reported apply only to this particular cell configuration. Note that the pyrrole experiment also differs from
Vol. 60, No. 9
DIFFUSION
CONTROLLED
the metal deposition experiment described above in the boundary condition at the counter electrode. For the silver deposition, for each ion deposited, one is added to solution at the outer edge of the cell. In the case of the polymer, there is no source of pyrrole at the counter electrode. To determine the fractal dimension of the structures produced, video images of prepared samples were acquired with a standard video camera. The maximum resolution of these images was 512 x 512. Images were digitized and processed with the IBM IAX image processing software. Integrated density-density correlation function and other analyses were performed on an IBM 4381. The correlation function was carried out by setting a threshold intensity for the fractals, and producing a binary image. The two dimensional autocorrelation function was computed (in the frequency domain) and radiometrically averaged. The inverse fourier transform of this function is identical to the density-density correlation function. This procedure is preferable to computing the all-points correlation function in the space domain, which consumes excessive c.p.u. time. The density-density correlation function, which approximates the ensemble average correlation function, determines the fractal dimension of an object [6]. In addition to measuring the fractal dimension, the branch angle distribution function was measured for each image. The angle between branches was found to depend strongly on structure. A photograph of the polymer was placed on a graphics tablet and three points (vertex and two branches) were recorded at various branchpoints of the image. Since most angles correspond to side branches (as opossed to tip splitting), the measured angle is always the acute angle. There should, of course, exist a distribution of congruent angles between 90 and 180”. Each distribution was fitted to a Gaussian to compute the mean branch angle. We applied both of these techniques to samples prepared at different fxed potentials.
RESULTS AND DISCUSSION When charged species are deposited on an electrode, and no buffer ions (supporting electrolyte) are present, a large electric field will exist in the depletion region near the electrode surface. Since particle motion in this depletion region is important in determining structure, it is necessary to understand the types of interactions which can occur. To study the effects of electric fields, silver aggregates were prepared under identical conditions except that in one case supporting electrolyte was added to provide some screening of the electric field. The applied potential was - 0.5 V (vs Ag/Ag/Cl)
POLYMERIZATION
759
which is well below the thermodynamic potential for electrolysis of water (about - 0.7V at pH 7). At this potential, with supporting electrolyte, the current was diffusion limited but no stirring of the solution or gas evolution occurred. The aggregates produced are shown in Fig. 2(a and b). In Fig. 2(a) the field was screened and the deposition controlled by diffusion. The object is similar to a random aggregate and has fractal dimension 1.75 f 0.03. The structure shown in Fig. 2(b) is obviously different (fractal dim. 1.81 f 0.03). In this case the only screening is provided by the silver salt and silver ions are depleted near the growing surface. In order to study aggregation as a function of electrode potential, it is first necessary to consider the meaning of the different potentials in an electrochemical cell. When the cell potential is measured using only two electrodes, the aggregate (working electrode) and the counter electrode, the measured voltage is the sum of the potential drops at the working electrode, the counter electrode, and the IR drop in the cell. In the absence of supporting electrolyte, this last term will change with time as the charged species in the cell are depleted and the cell resistance increases. In the three electrode configuration, a reference electrode is positioned near the working electrode (outside the depletion region). Negligible current flows through the reference electrode. The voltage between the reference and aggregate reflects the potential drop at the working electrode with only a small contribution due to resistive loss in the cell. In the silver experiments described above, this potential was - 0.5 V (vs Ag/AgCl) in both cases while the potential between the aggregate and counter electrode was - 1 .I V in the cell without supporting electrolyte and - 1 .O V in the cell with supporting electrolyte. Though the potential measured using a two electrode configuration may be roughly proportional to the half cell potential at the aggregate, it is only the latter which determines the local electrochemistry. For example, if the potential at the aggregate much exceeds the thermodynamic potential for the electrolysis of water (- 0.7V vs Ag/AgCl at pH 7), the reaction at the aggregate will be predominantly the evolution of gas (resulting in stirring of the solution) with only a fraction of the net current resulting in deposition. Recently [ 10, 111 a pair of experiments were reported in which ionic species were deposited without supporting electrolyte. No reference electrode was used and applied voltages exceeded that for electrolysis of water as evidenced by the evolution of gas. Variations in structure were studied as a function of applied voltage and concentration. The changes in structure observed in these experiments were affected by convection, stirring from gas evolution, diffusion, and migration (electric fields). The results were further complicated by the fact
760
DIFFUSION
CONTROLLED
POLYMERIZATION
Vol. 60, No. 9
that without a reference electrode, the applied potential is not a meaningful number when the concentration is varied over orders of magnitude. To gain insight into the fundamental phenomena which influence diffusion controlled aggregation, we chose to confine the following studies to the deposition of neutral species with supporting electrolyte under conditions which minimize effects other than diffusion. The electrochemistry was first studied using a reference electrode to insure electrolyte integrity at all potentials applied. Potential dependence was then studied using a two electrode configuration. Though not ideal, this was necessary as acetonitrile is a volatile solvent which tended to evaporate through the hole for the reference electrode (the cell would dry up during long runs making the growth of large aggregates difficult at low potential where growth is slow). As a result, potentials sited are only proportional to the working electrode potential.
1.0
1.8 D 1.6
1.0I 0
I
fl
I 2
Voltage
I 4
I
I 0.0 6
Fig. 3. The voltage dependence of the fractal dimension and the mean branch angle exhibit a growth instability. The structure changes from compact to dendritic to fractal.
(b) Fig. 2(a, b). Silver aggregates prepared with and without supporting electrolyte respectively (See text).
In Fig. 3 the voltage dependance of the averaged fractal dimension for a number of samples is shown. The data, plotted as D vs V, begins at a minimum voltage of 0.8 V which is near the threshold voltage for oxidation of pyrrole monomer in our cell. The most remarkable feature in the data is the sharp transition observed at about 1.5 volts where the growth changes from compact (Fig. l(a)) to dendritic (Fig. l(b)). At higher voltages a continuum of structures is observed as the growth becomes more irregular and the fractal dimension increases asymptotically to about 1.7. The left hand side of this transition is predicted by the instability of a compact electrode growing into a sufficiently large gradient via DCA. When the concentration (or potential) gradient is large enough to overcome stabilizing forces like capillarity (Gibbs Thomson), a planar electrode is known to become unstable [9, 121, and irregularities on
Vol. 60, No. 9
DIFFUSION
CONTROLLED
the surface formed by (e.g.) fluctuations, will propagate. In the absence of crystalline anisotropy, such tips or dendrites will grow normally from the surface [9, 121 and the mean branch angle will be at 90 degrees. The mean branch angle for our aggregates is also plotted in Fig. 3. At 1.5 volts (above which distinct angles can be measured), the mean branch angle decreases from about 90 to 42 f 7 degrees at high voltage. This same distribution function was measured for a diffusion limited aggregate made by computer simulation. Since the simulation is on a square lattice, all microscopic angles are 90”. However, when the aggregate is defocused, and the angles measured on scales greater than ten lattice constants, the distribution was found to peak at 43 f 6 degrees. This agreement in no way implies that polypyrrole forms a square lattice. TEM of the polymer branches indicate the material formed is completely amorphous and should not exhibit crystalline anisotropy. The peak at 45” may indicate that branches at low angles are subject to recombination with the principle branch, and at high angles (near 90’) are subject to screening effects. Note that on formal grounds [6] the data in Fig. 3 must be bounded by the Euclidean dimension E = 2, and the topological dimension Dt = 1. The discontinuity in Fig. 3 demonstrates that for our polymerization conditions there exists a diffusion profile where compact growth is unstable with respect to dendrite formation. We stress again that the actual half cell potential corresponding to this instability is not the applied potential in our two electrode cell. The structures obtained in the diffusion limited regime are more dense than those obtained in DLA simulations. Similar structures have been reported by Sawada et al. [lo] (who refer to some of their aggregates as “homogeneous”), and Crier et al. [ 111. The increase in density is probably due to the fact that before aggregation begins, the electrochemical cell is completely filed by a “gas” of particles whereas in many simulations the space is empty except for a single random walker. In the limit of high initial particle concentration, the aggregate will resemble DLA on small scales. On large scales, however, the shape of the aggregate reflects the radial gradient. This effect is more pronounced in cells where the aggregating particles are replaced at the counter electrode (so the concentration there actually increases) and the growth is allowed to proceed into that region. In the case of the pyrrole aggregates, the total particle number is futed and the concentration of particles decreases as the aggregate grows. This latter situation may more closely approximate the models of DLA. We believe that understanding the observed growth instability is critical to any complete theory of DLA. Since the random fractal is the limiting case of the
761
POLYMERIZATION
diffusion controlled aggregate, the correct theory of DCA should predict the transition from fractal to dendrite as well as dendrite to compact. The scale at which this transition occurs will be case dependent, but the transition itself should be universal. For example, aging in dielectrics exposed to high electric fields results in a transition from a uniform space charge density to a branched defect structure [ 13, 141. This transition from compact to dendritic space charge distribution may reflect the same universal process which occurs in DCA, though the scale of the process is particular to the dielectric. CONCLUSION In conclusion, we have studied the electrochemical polymerization of pyrrole under conditions which range from diffusion controlled to diffusion limited. The results demonstrate that as the oxidation potential is raised, a growth instability occurs and the structure changes from compact to dendritic. As the surface concentration of aggregating particles decreases, the growth becomes disordered. Random fractals produced under ideal DLA conditions are the limiting case of this “diffusion controlled” process. Acknowledgements - We thank A. Kapituhrik M. Flickner for many useful discussions.
and
REFERENCES 1. 2.
3. 4.
2:
;:
9. 10. 11.
;:: 14.
T.A. Witten Jr. & L.M. Sander, Phys. Rev. Lett. 47,140o (1981). See e.g., Kinetics of Aggregation and Gellation, (edited by F. Family and D.P. Landau), NorthHolland, Amsterdam (1984). M. Matsushita, M. Sano, Y. Hayakawa, H. Honjo & Y. Sawada, Phys. Rev. Lett. 53,286 (1984). S.R. Forrest & T.A. Witten Jr., J. Phys. A. 12, L109 (1979). R.M. Brady & R.C. Ball, Nature 309,225 (1984). See e.g., B.B. Mandelbrot, The Fractal Geometry Of Nature, Freeman, San Francisco, (1983). J.H. Kaufman et aZ.,Phys. Rev. Lett. (in press). See e.g., A.J. Bard & L.R. Faulkner, Electrochemical Methods, Fundamentals and Applications ch.s 1, 4, 6. John Wiley and Sons, New York (1980). W.W. Mullins & R.F. Sekerka, J. Appl. Phys. 34, 323 (1963); 35,444 (1964). Y. Sawada, A. Dougherty & J.P. Gollub, Phys. Rev. Lett. 56,126O (1986). D. Grier, E. Ben-Jacob, R. Clarke & L.M. Sander, Phys. Rev. Lett. 56, 1264 (1986). J.S. Langer,Rev. Mod. Phys, 52, 1 (1980). L. Niemeyer, L. Pietronero & H.J. Wiesmann, Phys. Rev. Lett. 52,1033 (1984). P. Pfluger, H.R. Zeller & J. Bernasconi, Phys. Rev. Lett. 53,94 (1984).
Vol. 60, No. 9, pp. 757-761,
GROWTH INSTABILITY J.H. Kaufman,
0038-1098/86 $3.00 + .OO Pergamon Journals Ltd.
1986.
IN DIFFUSION
CONTROLLED
POLYMERIZATION
O.R. Melroy, F.F. Abraham and A.I. Nazzal
IBM Almaden Research Laboratory,
San Jose, CA 95 193, USA
(Received 18 July 1986; in revised form 23 July 1986 by R.H. Silsbee) We report the observation of a sharp transition in the structure of polypyrrole prepared via diffusion controlled polymerization in a quasi-two dimensional electrochemical cell. As the oxidation potential is increased, the fractal dimension drops sharply from values near 2 (compact structures) to 1 as the growth becomes dendritic. At higher potentials, where the polymerization becomes diffusion limited, a continuum of structures is observed as the dendrites become more irregular and the fractal dimension increases to 1.74 + 0.01. The mean branch angle was also measured and found to decrease from 90” at the point of dendritic growth, to 45” at high potential. INTRODUCTION DIFFUSION LIMITED AGGREGATION is a process wherein successive particles moving in a space with Euclidean dimensionality E undergo a random walk until they interact with (and stick to) a growing fractal [ 1, 21. When an aggregate is produced via DLA, the resultant structure is tree-like and exhibits dilation symmetry over some range of lengths [l-6]. For example, the density of such an object is found to scale with radius to a power, D, known as the fractal dimension [6]. Because these objects are highly irregular, they can not be characterized by a single Euclidean dimension. A better description of the structure is provided, perhaps, by the fractal dimension which may be computed from the ensemble average correlation function. For random fractals produced by computer simulations on a square lattice, a typical value for the fractal dimension is 1.657 + 0.004 [l]. The apparent universality of scaling behavior characteristic of (DLA) has stimulated much interest in random aggregation phenomenon [l-6]. However, DLA is only the limiting case of a more general process, diffusion controlled aggregation (DCA). Unlike DLA, in DCA the free particle concentration at the growing surface is nonzero and as a result the aggregation process depends not only on the random motion of the aggregating particles, but also on the energetics of the deposition process. To understand the origins of the scaling behavior characteristic of DLA, it is desirable to conduct a series of experiments to systematically study the continuum of diffusion controlled aggregation conditions. Electrochemical deposition provides a convenient method for preparing aggregates under a variety of diffusion controlled conditions since the particle 757
concentration near the surface of the aggregate is determined by the applied potential [8]. As the deposition (oxidation) potential is increased, the surface concentration of the aggregating species decreases (and the concentration gradient increases). It is well-known that the magnitude of this gradient influences the structure of the aggregate. Mullins and Sekerka [9] demonstrated that growth of an aggregate into a sufficiently large gradient typically results in dendrite formation, while growth into a small gradient produces compact At even higher potentials, aggregation structures. becomes limited by diffusion (i.e. the surface concentration of aggregating particles approaches zero). In this regime electrochemical deposition approximates DLA in that the distance between aggregating particles near the surface is large and they behave like the random walkers of DLA simulations. In this Communication, we investigate the variations in structure obtained by polymerization of pyrrole under a continuum of deposition conditions. Experiments were designed to minimize effects other than diffusion. We report a growth instability where the polymer structure changes from a compact morphology to a dendritic morphology. As potential is increased further, the dendritic phase crosses over to a random aggregate phase (see Fig. 1) and the polymerization becomes limited by diffusion (the surface concentration of pyrrole monomer approaches zero). As in the case of DLA, the mass of these fractals scales with radius as M(r) a rD where, D is the fractal dimension [ 1, 21. EXPERIMENTAL Experiments to demonstrate the effects of depositing charged precursers were preformed in a thin layer cell
DIFFUSION
CONTROLLED
POLYMERIZATION
Vol. 60, No. 9
(cl Fig. 1 (a-c). Aggregates of polypyrrole prepared DCP at 0.9, 1 .5 and 6.0 volts (DIP) respectively.
(a)
(b)
via
constructed with two glass plates. On one plate a 10 micron layer of silver was evaporated except for a one inch diameter circular region in the center. This region constitutes the electrochemical cell. A 1Omil hole was drilled through the center of the other plate and a silver wire was inserted, epoxied, and sheared flush with a razor blade. A second 1Omm hole was drilled 1 cm from the first to allow contact to the electrolyte with a microreference electrode. In the first experiment the electrolyte was 0.1 M AgNO a aqueous with 0.1 M KNOB as supporting electrolyte. Since the potassium ions are not depleted at the active electrode, the supporting electrolyte serves to screen the electric field. In the second experiment the supporting electrolyte was omitted. In both cases the reference electrode was Ag/AgCl with 3 M KNOa. Samples were prepared at - 0.5 V vs Ag/AgCl. Samples of polypyrrole were prepared in a similar thin layer cell consisting of two glass plates separated by a 2000W gold ring (counter electrode) evaporated on one plate and a 1Omil Kapton spacer. A 10 mm hole was drilled through the center of the other plate through which a 5 mm gold wire was inserted, epoxied, and sheared flush with a razor blade. The solution used was 0.1 M pyrrole/O.l M silver toluenesulfonate/acetonitrile. No attempt was made to dry or degass the electrolyte. The counter reaction is the deposition of silver. Since no measurements were made of concentration dependence, we chose not to use a reference electrode. As a result, voltages reported apply only to this particular cell configuration. Note that the pyrrole experiment also differs from
Vol. 60, No. 9
DIFFUSION
CONTROLLED
the metal deposition experiment described above in the boundary condition at the counter electrode. For the silver deposition, for each ion deposited, one is added to solution at the outer edge of the cell. In the case of the polymer, there is no source of pyrrole at the counter electrode. To determine the fractal dimension of the structures produced, video images of prepared samples were acquired with a standard video camera. The maximum resolution of these images was 512 x 512. Images were digitized and processed with the IBM IAX image processing software. Integrated density-density correlation function and other analyses were performed on an IBM 4381. The correlation function was carried out by setting a threshold intensity for the fractals, and producing a binary image. The two dimensional autocorrelation function was computed (in the frequency domain) and radiometrically averaged. The inverse fourier transform of this function is identical to the density-density correlation function. This procedure is preferable to computing the all-points correlation function in the space domain, which consumes excessive c.p.u. time. The density-density correlation function, which approximates the ensemble average correlation function, determines the fractal dimension of an object [6]. In addition to measuring the fractal dimension, the branch angle distribution function was measured for each image. The angle between branches was found to depend strongly on structure. A photograph of the polymer was placed on a graphics tablet and three points (vertex and two branches) were recorded at various branchpoints of the image. Since most angles correspond to side branches (as opossed to tip splitting), the measured angle is always the acute angle. There should, of course, exist a distribution of congruent angles between 90 and 180”. Each distribution was fitted to a Gaussian to compute the mean branch angle. We applied both of these techniques to samples prepared at different fxed potentials.
RESULTS AND DISCUSSION When charged species are deposited on an electrode, and no buffer ions (supporting electrolyte) are present, a large electric field will exist in the depletion region near the electrode surface. Since particle motion in this depletion region is important in determining structure, it is necessary to understand the types of interactions which can occur. To study the effects of electric fields, silver aggregates were prepared under identical conditions except that in one case supporting electrolyte was added to provide some screening of the electric field. The applied potential was - 0.5 V (vs Ag/Ag/Cl)
POLYMERIZATION
759
which is well below the thermodynamic potential for electrolysis of water (about - 0.7V at pH 7). At this potential, with supporting electrolyte, the current was diffusion limited but no stirring of the solution or gas evolution occurred. The aggregates produced are shown in Fig. 2(a and b). In Fig. 2(a) the field was screened and the deposition controlled by diffusion. The object is similar to a random aggregate and has fractal dimension 1.75 f 0.03. The structure shown in Fig. 2(b) is obviously different (fractal dim. 1.81 f 0.03). In this case the only screening is provided by the silver salt and silver ions are depleted near the growing surface. In order to study aggregation as a function of electrode potential, it is first necessary to consider the meaning of the different potentials in an electrochemical cell. When the cell potential is measured using only two electrodes, the aggregate (working electrode) and the counter electrode, the measured voltage is the sum of the potential drops at the working electrode, the counter electrode, and the IR drop in the cell. In the absence of supporting electrolyte, this last term will change with time as the charged species in the cell are depleted and the cell resistance increases. In the three electrode configuration, a reference electrode is positioned near the working electrode (outside the depletion region). Negligible current flows through the reference electrode. The voltage between the reference and aggregate reflects the potential drop at the working electrode with only a small contribution due to resistive loss in the cell. In the silver experiments described above, this potential was - 0.5 V (vs Ag/AgCl) in both cases while the potential between the aggregate and counter electrode was - 1 .I V in the cell without supporting electrolyte and - 1 .O V in the cell with supporting electrolyte. Though the potential measured using a two electrode configuration may be roughly proportional to the half cell potential at the aggregate, it is only the latter which determines the local electrochemistry. For example, if the potential at the aggregate much exceeds the thermodynamic potential for the electrolysis of water (- 0.7V vs Ag/AgCl at pH 7), the reaction at the aggregate will be predominantly the evolution of gas (resulting in stirring of the solution) with only a fraction of the net current resulting in deposition. Recently [ 10, 111 a pair of experiments were reported in which ionic species were deposited without supporting electrolyte. No reference electrode was used and applied voltages exceeded that for electrolysis of water as evidenced by the evolution of gas. Variations in structure were studied as a function of applied voltage and concentration. The changes in structure observed in these experiments were affected by convection, stirring from gas evolution, diffusion, and migration (electric fields). The results were further complicated by the fact
760
DIFFUSION
CONTROLLED
POLYMERIZATION
Vol. 60, No. 9
that without a reference electrode, the applied potential is not a meaningful number when the concentration is varied over orders of magnitude. To gain insight into the fundamental phenomena which influence diffusion controlled aggregation, we chose to confine the following studies to the deposition of neutral species with supporting electrolyte under conditions which minimize effects other than diffusion. The electrochemistry was first studied using a reference electrode to insure electrolyte integrity at all potentials applied. Potential dependence was then studied using a two electrode configuration. Though not ideal, this was necessary as acetonitrile is a volatile solvent which tended to evaporate through the hole for the reference electrode (the cell would dry up during long runs making the growth of large aggregates difficult at low potential where growth is slow). As a result, potentials sited are only proportional to the working electrode potential.
1.0
1.8 D 1.6
1.0I 0
I
fl
I 2
Voltage
I 4
I
I 0.0 6
Fig. 3. The voltage dependence of the fractal dimension and the mean branch angle exhibit a growth instability. The structure changes from compact to dendritic to fractal.
(b) Fig. 2(a, b). Silver aggregates prepared with and without supporting electrolyte respectively (See text).
In Fig. 3 the voltage dependance of the averaged fractal dimension for a number of samples is shown. The data, plotted as D vs V, begins at a minimum voltage of 0.8 V which is near the threshold voltage for oxidation of pyrrole monomer in our cell. The most remarkable feature in the data is the sharp transition observed at about 1.5 volts where the growth changes from compact (Fig. l(a)) to dendritic (Fig. l(b)). At higher voltages a continuum of structures is observed as the growth becomes more irregular and the fractal dimension increases asymptotically to about 1.7. The left hand side of this transition is predicted by the instability of a compact electrode growing into a sufficiently large gradient via DCA. When the concentration (or potential) gradient is large enough to overcome stabilizing forces like capillarity (Gibbs Thomson), a planar electrode is known to become unstable [9, 121, and irregularities on
Vol. 60, No. 9
DIFFUSION
CONTROLLED
the surface formed by (e.g.) fluctuations, will propagate. In the absence of crystalline anisotropy, such tips or dendrites will grow normally from the surface [9, 121 and the mean branch angle will be at 90 degrees. The mean branch angle for our aggregates is also plotted in Fig. 3. At 1.5 volts (above which distinct angles can be measured), the mean branch angle decreases from about 90 to 42 f 7 degrees at high voltage. This same distribution function was measured for a diffusion limited aggregate made by computer simulation. Since the simulation is on a square lattice, all microscopic angles are 90”. However, when the aggregate is defocused, and the angles measured on scales greater than ten lattice constants, the distribution was found to peak at 43 f 6 degrees. This agreement in no way implies that polypyrrole forms a square lattice. TEM of the polymer branches indicate the material formed is completely amorphous and should not exhibit crystalline anisotropy. The peak at 45” may indicate that branches at low angles are subject to recombination with the principle branch, and at high angles (near 90’) are subject to screening effects. Note that on formal grounds [6] the data in Fig. 3 must be bounded by the Euclidean dimension E = 2, and the topological dimension Dt = 1. The discontinuity in Fig. 3 demonstrates that for our polymerization conditions there exists a diffusion profile where compact growth is unstable with respect to dendrite formation. We stress again that the actual half cell potential corresponding to this instability is not the applied potential in our two electrode cell. The structures obtained in the diffusion limited regime are more dense than those obtained in DLA simulations. Similar structures have been reported by Sawada et al. [lo] (who refer to some of their aggregates as “homogeneous”), and Crier et al. [ 111. The increase in density is probably due to the fact that before aggregation begins, the electrochemical cell is completely filed by a “gas” of particles whereas in many simulations the space is empty except for a single random walker. In the limit of high initial particle concentration, the aggregate will resemble DLA on small scales. On large scales, however, the shape of the aggregate reflects the radial gradient. This effect is more pronounced in cells where the aggregating particles are replaced at the counter electrode (so the concentration there actually increases) and the growth is allowed to proceed into that region. In the case of the pyrrole aggregates, the total particle number is futed and the concentration of particles decreases as the aggregate grows. This latter situation may more closely approximate the models of DLA. We believe that understanding the observed growth instability is critical to any complete theory of DLA. Since the random fractal is the limiting case of the
761
POLYMERIZATION
diffusion controlled aggregate, the correct theory of DCA should predict the transition from fractal to dendrite as well as dendrite to compact. The scale at which this transition occurs will be case dependent, but the transition itself should be universal. For example, aging in dielectrics exposed to high electric fields results in a transition from a uniform space charge density to a branched defect structure [ 13, 141. This transition from compact to dendritic space charge distribution may reflect the same universal process which occurs in DCA, though the scale of the process is particular to the dielectric. CONCLUSION In conclusion, we have studied the electrochemical polymerization of pyrrole under conditions which range from diffusion controlled to diffusion limited. The results demonstrate that as the oxidation potential is raised, a growth instability occurs and the structure changes from compact to dendritic. As the surface concentration of aggregating particles decreases, the growth becomes disordered. Random fractals produced under ideal DLA conditions are the limiting case of this “diffusion controlled” process. Acknowledgements - We thank A. Kapituhrik M. Flickner for many useful discussions.
and
REFERENCES 1. 2.
3. 4.
2:
;:
9. 10. 11.
;:: 14.
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