- Email: [email protected]
Electric field driven fractal growth dynamics in polymeric medium
JID:PLA AID:22777 /SCO Doctopic: Nonlinear science
[m5G; v 1.134; Prn:26/08/2014; 9:26] P.1 (1-8)
Physics Letters A ••• (••••) •••–•••
1
Contents lists available at ScienceDirect
2
67 68
3
Physics Letters A
4 5
69 70 71
6
72
www.elsevier.com/locate/pla
7
73
8
74
9
75
10
76
11 12 13 14 15 16
Electric field driven fractal growth dynamics in polymeric medium
77
Anit Dawar, Amita Chandra ∗
79 80 81
Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India
82
17 18
83
a r t i c l e
i n f o
84
a b s t r a c t
19 20 21 22 23 24
85
Article history: Received 19 May 2014 Received in revised form 5 August 2014 Accepted 15 August 2014 Available online xxxx Communicated by Z. Siwy
25 26 27 28 29
78
Keywords: Fractals Diffusion limited aggregation Polymer electrolyte composites Electric field
30
This paper reports the extension of earlier work (Dawar and Chandra, 2012) [27] by including the influence of low values of electric field on diffusion limited aggregation (DLA) patterns in polymer electrolyte composites. Subsequently, specified cut-off value of voltage has been determined. Below the cut-off voltage, the growth becomes direction independent (i.e., random) and gives rise to ramified DLA patterns while above the cut-off, growth is governed by diffusion, convection and migration. These three terms (i.e., diffusion, convection and migration) lead to structural transition that varies from dense branched morphology (DBM) to chain-like growth to dendritic growth, i.e., from high field region (A) to constant field region (B) to low field region (C), respectively. The paper further explores the growth under different kinds of electrode geometries (circular and square electrode geometry). A qualitative explanation for fractal growth phenomena at applied voltage based on Nernst–Planck equation has been proposed. © 2014 Elsevier B.V. All rights reserved.
86 87 88 89 90 91 92 93 94 95 96
31
97
32
98
33 34
99
1. Introduction
35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59
Pattern formation, i.e., the non-equilibrium growth phenomenon, has attracted considerable interest of scientists from various disciplines in the past decade. Such far from equilibrium growth studies include a wide range of problems encompassing life sciences, physics, biology, chemistry, computer science, image compression and geology which produce complex geometries of fractal or dendrite character and chaotic patterns [1–4]. Aggregation is one of the most familiar phenomena in physical and chemical processes. Many efforts have been made to develop models for fractal growth and aggregation processes [2]. Among these models, the most well known one is diffusion limited aggregation (DLA). The DLA model was proposed in 1981 by Witten and Sander as a computer algorithm [5]. In this model, aggregation process begins with fixing one particle at the center of coordinates in 2 dimensions and follows the formation of cluster by releasing particles from infinity. The particles undergo random walk due to Brownian motion until they hit any particle belonging to the growing cluster and stick to it. The DLA model has since attained a paradigmatic status due to its simplicity and presents the fundamental way for many diffusive systems such as effect of uniform drift and surface diffusion in electrodeposition [6,7], dielectric breakdown (lightening) [8], viscous fingering [9,10], growth mechanism in thin film [11], crystal morphology [12] and solidification (snowflakes) [13]. Many extensions of DLA model have
60 61 62 63 64 65 66
*
Corresponding author. Tel.: +91 11 27662295; fax: +91 11 27667061. E-mail address: [email protected] (A. Chandra).
http://dx.doi.org/10.1016/j.physleta.2014.08.016 0375-9601/© 2014 Elsevier B.V. All rights reserved.
been developed taking into account the processes involving concentration [14], particle drift [15], surface tension [16], sticking probability [17], and heterogeneous surface [18]. In polymer science, polymer growth is a fractal process [19]. Since the last 20 years, fractal growth in polymer electrolyte has attracted much more attention [20–27]. The observation of fractal pattern formation opens new avenues for detailed studies of growth due to the better understanding of the phenomenon without the use of complex theories. Chandra and Chandra [20,21] were first to report the technique to grow fractal patterns in polymer electrolyte under bias-free conditions. They reported the fractal growth in ion conducting polymer due to random walk of ions and their subsequent aggregation around nucleation centers. Recently, another group of researchers [22–25] also cultured fractals in various ion conducting polymer electrolyte membranes experimentally. They also did theoretical modeling of the experimentally cultured fractals. In our earlier work [26], polymer electrolyte (PEO:NH4 I) dispersed with Al2 O3 as a seed particle was used for obtaining the growth under bias free conditions. The growth of fractals was found to be due to the random walk and subse+ + − quent aggregation of the mobile species [I− 3 and NH4 (via NH4 I3 )]. On approaching the nucleation center, the random walkers stuck and formed an aggregate leading to DLA of different sizes and varying dimensionality. In another work [27], we reported the effect of electric field on large size fractal growth in polymer electrolyte composites by varying the voltage from 2 V to 8 V. The application of electric field to the polymer electrolyte composites caused ordered pattern formation in the electric field direction. The growth probability would not be the same in all the directions
100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
JID:PLA
AID:22777 /SCO Doctopic: Nonlinear science
2
1 2 3 4 5 6 7 8 9 10 11 12
[m5G; v 1.134; Prn:26/08/2014; 9:26] P.2 (1-8)
A. Dawar, A. Chandra / Physics Letters A ••• (••••) •••–•••
and the pattern formation was completely governed by convection which was due to the applied voltage. At higher voltages (i.e., at 6 V and 8 V), there was screening effect on the ions which prohibited the pattern to grow in the electric field direction. In the present work, electric field induced aggregation in polymer electrolyte composites has been studied under the influence of low values of electric field and extends the earlier reported work [27]. Specified threshold level has been determined. Below the threshold, the growth becomes random and above the threshold, the growth is governed by diffusion, convection and migration. Different kinds of electrode geometries have been used to simulate the growth under external field.
67 68 69 70 71 72 73 74 75 76 77 78
13 14
79
2. Experimental setup
80
15 16
81
2.1. Sample preparation
82
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
83
Samples were prepared by using the solution-cast technique. In this technique, the host polymer (Poly (ethylene oxide) (PEO), Sigma-Aldrich, Mol. Wt. ∼ 6 × 105 ) and salt (NH4 I (Sigma Aldrich, 99.999% trace metals basis)) were weighed in desired weight ratio (i.e., 50:50 wt.%). They were dissolved in distilled methanol thoroughly for ∼6–7 hr at 40 ◦ C. A small amount of activated Al2 O3 (neutral, 5 wt.% (Sigma-Aldrich, pH = 7.0 ± 0.5 (in H2 O))) was dispersed in the above PEO:NH4 I complexed solution. The highly viscous mixture of PEO:NH4 I (+Al2 O3 (neutral)) obtained after mixing was poured in polypropylene Petri-dishes (diameter = 7.5 cm) fitted with different electrode arrangements (viz. parallel, square and circular electrode geometry). The applied potential difference V across the electrode was varied from 0 to 2 V. The parallel electrodes were separated by a distance of 30 mm.
32 33
3. Results and discussion
34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
The physical scenario envisaged is the effect of external field on large size fractal growth in polymer electrolyte. The main transport mechanisms involved in the growth after application of electric field are diffusion, migration and electro-convective motion. Initially, the polymer electrolyte (PEO:NH4 I) dispersed with Al2 O3 (neutral) is electrically neutral everywhere with uniform concentration of cations and anions. (a) Fig. 1 shows the large size fractal growth after application of 1.5 V between two parallel electrodes. On application of voltage between the electrodes, the cations move towards the cathode where they discharge and act as a collection center for H2 . The cations (i.e., protons, as the polymer electrolyte (PEO:NH4 I) is predominantly a proton conductor) which migrate towards the cathode leave behind them a zone of higher pH (i.e., basic front) and near the cathode there is creation of an acidic front. Anions move towards the anode where they pile up since they cannot exit the solution. The whole Petri-dish containing the viscous polymer electrolyte composite solution under the influence of electric field is kept in a high humidity environment (R.H. ∼70%–80%). High humidity environment gives slow drying rate, provides enough time for aggregation and the right reducing environment for aggregation to take place. Earlier work [27] has reported that after ∼1–2 days, dense branched morphological patterns appear near the anode due to the presence of high concentration of anions near the anode while chain-like growth takes place in the middle part which protrudes in the electric field direction (i.e., towards cathode). Near the cathode, when the patterns meet the acidic front, the ordered chain-like growth forms an envelope around the cathode. At the cathode, growth also changes to dendritic patterns. A well defined angle is then observed between the main branches and the side branches of the dendrites. However, the orientation of the branches is only local with no long range correlation and their distribution
84 85 86 87 88 89
Fig. 1. Fractal growth after application of 1.5 V (i.e., E = 0.5 V/cm).
90 91
in all the directions results in a radial isotropic pattern maintaining an elliptical envelope. The physical model is described by the Nernst–Planck equation for the concentration of cations and anions in the polymer electrolyte subject to diffusion, convection and migration fields. The conventional large size fractal growth in polymer electrolyte composite dispersed with seed particle does not include electric field since the growth there is completely governed by diffusion of ions in the polymer electrolyte without any external bias [26]. The growth is a typical Laplacian growth as the system is completely under thermal disorder and DLA patterns at different places of different kinds appear. In the current work, it is critical to understand the configuration of the fractal growth having different patterns at different places (i.e., near anode, in the middle part, near cathode) when an electric field is applied. Here, a qualitative explanation for the phenomenon by using the Nernst–Planck equation is being given. Generally, the flux of ions in an electrolyte is completely described by the Nernst–Planck equation. The net flux of ions ( j ± ) is therefore, the diffusion term (− D ± ∇ c ± ), the migration term ( KzeT D ± c ± ∇Φ ) and the convection term (c ± u) [28]. B
j± = − D ± ∇ c± +
ze KBT
D ± c ± ∇Φ + c ± u
(1)
where D ± is the diffusion coefficient of ions, c ± their concentrations, Φ electric potential, u is the fluid velocity and ∇ c± the concentration gradient. Consider the Petri-dish containing two parallel electrodes and assume that the aggregation is proceeding slowly in quasi stationary state governed by diffusion, convection and migration in an electric field. The component of electric field parallel to the electrodes will be zero (i.e., the electric field in the x-direction, E x = 0). Thus, the second term denotes only migration in the electric field in the y-direction. When the voltage is switched on, a depletion zone of anions occurs near the cathode and that of cations near the anode. Meanwhile, an electro-convective motion sets in between the two parallel electrodes. The speed of an ion in the viscous mixture of polymer electrolyte composite is given by its mobility (μc for the cation, μa for anion) times the electric field (E = 13.5 = 0.5 V/cm). So, the growth speed of the deposit is equal to the anion’s mobility times the field. Thus, the migration term and the convection term play key roles in aggregation which is not
92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
JID:PLA AID:22777 /SCO Doctopic: Nonlinear science
[m5G; v 1.134; Prn:26/08/2014; 9:26] P.3 (1-8)
A. Dawar, A. Chandra / Physics Letters A ••• (••••) •••–•••
3
1
67
2
68
3
69
4
70
5
71
6
72
7
73
8
74
9
75
10
76
11
77
12
78
13
79
14
80
15
81
16
82
17
83
18
84
19
85
20
86
21 22
87
(a)
(b)
88
23
89
24
90
25
91
26
92
27
93
28
94
29
95
30
96
31
97
32
98
33
99
34
100
35
101
36
102
37
103
38
104
39
105
40
106
41
107
42
108
43 44
(c)
(d)
109 110
45
111
46
112
47
113
48
114
49
115
50
116
51
117
52
118
53
119
54
120
55
121
56
122
57
123
58
124
59
125
60
126
61
127
62
128
63
129
64 65 66
130
(e) Fig. 2. Fractal growth in polymer electrolyte at (a) 1 V, (b) 0.90 V, (c) 0.85 V, (d) 0.80 V, and (e) 0.50 V.
131 132
JID:PLA
AID:22777 /SCO Doctopic: Nonlinear science
[m5G; v 1.134; Prn:26/08/2014; 9:26] P.4 (1-8)
A. Dawar, A. Chandra / Physics Letters A ••• (••••) •••–•••
4
1
67
2
68
3
69
4
70
5
71
6
72
7
73
8
74
9
75
10
76
11
77
12
78
13
79
14
80
15
81
16
82
17
83
18
84
19
85
20
86
21 22
87
(a)
(b)
88
23
89
24
90
25
91
26
92
27
93
28
94
29
95
30
96
31
97
32
98
33
99
34
100
35
101
36
102
37
103
38
104
39
105
40
106
41
107
42 43
108
(c)
(d)
109
44
110
45
111
46
112
47
113
48
114
49
115
50
116
51
117
52
118
53
119
54
120
55
121
56
122
57
123
58
124
59
125
60
126
61
127
62
128
63 64 65 66
129
(e) Fig. 3. Fractal growth in circular electrode geometry: (a) without bias, (b) after few hours of applied voltage, (c) after 24 hours of applied voltage, (d) after 2–3 days, and (e) after complete growth.
130 131 132
JID:PLA AID:22777 /SCO Doctopic: Nonlinear science
[m5G; v 1.134; Prn:26/08/2014; 9:26] P.5 (1-8)
A. Dawar, A. Chandra / Physics Letters A ••• (••••) •••–•••
5
1
67
2
68
3
69
4
70
5
71
6
72
7
73
8
74
9
75
10
76
11
77
12
78
13
79
14
80
15
81
16
82
17
83
18
84
19
85
20
86
21
87
22 23
88
(a)
(b)
89
24
90
25
91
26
92
27
93
28
94
29
95
30
96
31
97
32
98
33
99
34
100
35
101
36
102
37
103
38
104
39
105
40
106
41
107
42
108
43
109
44
110
45
111
(c)
46 47 48
112
Fig. 4. Fractal growth using circular electrode geometry with polarity in reverse direction: (a) after 24 hours of applied voltage, (b) after complete growth, and (c) magnified image.
49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
113 114 115
considered in bias-free aggregation [26]. This is to say, the transport of anions in the aggregation process around the seed particles (Al2 O3 , neutral) is determined by all the three; diffusion, convection and migration terms. One should consider the contribution of these three terms in the aggregation process that leads to different kinds of patterns (i.e., DBM to chain-like growth to dendritic growth) at a fixed distance (i.e., field) from the anode and the cathode. The third term (i.e., convection term) generally includes natural convection (driven by density difference), forced convection (driven by pressure gradient) and local convection (driven by electric field). The forced convection can be neglected due to absence of any external force. The natural convection arises due to the high humid environment. The local convection arises because of the electro-convective motion that sets in. After application of the electric field, the concentration of anions gradually starts increasing at the anode. After some time, the concentration of anions is highest at the anode end then at
the middle part followed by the cathode end. Thus, the electroconvection is higher near the anodic part as the concentration of anions is high which changes the shape of the electrode from planar to irregular with sharp corners (since electric field intensity is larger at sharp corners and force acting on particle is also the strongest). Since the whole system is in a high humidity environment, as the volatile component starts evaporating, the concentration of anions starts increasing at the anode which makes the convection term more dominating. Thus, besides the migration term, electro-convection term also plays an important role in the formation of the DBM near the anode. The convection is higher near the anode part where probability of anions attaching to the aggregate becomes larger and the DBM forms. Other workers [29] have also demonstrated that in electrochemical deposition process, the migration term ( KzeT D ± c ± ∇Φ ) could induce DBM, particularly B in high field region which is depicted in Fig. 1. In the middle part (constant field region), the concentration of anions is low as com-
116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
JID:PLA
AID:22777 /SCO Doctopic: Nonlinear science
[m5G; v 1.134; Prn:26/08/2014; 9:26] P.6 (1-8)
A. Dawar, A. Chandra / Physics Letters A ••• (••••) •••–•••
6
1
67
2
68
3
69
4
70
5
71
6
72
7
73
8
74
9
75
10
76
11
77
12
78
13
79
14
80
15
81
16
82
17
83
18
84
19
85
20 21
86
(a)
(b)
87
22
88
23
89
24
90
25
91
26
92
27
93
28
94
29
95
30
96
31
97
32
98
33
99
34
100
35
101
36
102
37
103
38
104
39
105
40
106
41
107
42 43
(c)
(d)
108 109
44
110
45
111
46
112
47
113
48
114
49
115
50
116
51
117
52
118
53
119
54
120
55
121
56
122
57
123
58
124
59
125
60
126
61
127
62
128
63 64 65 66
129
(e)
(f)
Fig. 5. Fractal growth using square electrode geometry: (a) without bias, (b) after 2 minutes of applied voltage, (c) after 24 hours, (d) after 3 days, (e) after 4–5 days, and (f) after complete growth.
130 131 132
JID:PLA AID:22777 /SCO Doctopic: Nonlinear science
[m5G; v 1.134; Prn:26/08/2014; 9:26] P.7 (1-8)
A. Dawar, A. Chandra / Physics Letters A ••• (••••) •••–•••
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
pared to the anode. The contribution of the migration and the convection term cannot be negligible, hence, in the middle part, chain-like growth is observed guided in the direction of the electric field due to the electro-convective motion that sets between the two parallel electrodes. At the cathode (low field region), cations form an acidic zone. Contribution of migration and electro-convection term near the cathode is low. Chain-like growth which is protruding towards the cathode, meets the acidic front which turns the growth into dendritic type of growth. The aggregation form an envelope around the cathode by making a well defined angle between the main branches and the side branches of the dendrities. (b) At lower voltages {i.e., at 1 V (E = 13.0 = 0.333 V/cm) and 0.9 3
at 0.9 V (E = = 0.30 V/cm)}, Fig. 2(a)–(b), the growth looses its directionality in the middle part. This is because the strength of the electric field is not enough to guide the growth. (c) At further lower voltages {i.e., at 0.85 V (E = 0.385 = 0.283 V/cm), at 0.8 V (E =
0.8 3
= 0.266 V/cm) and at 0.50 V
(E = 0.350 = 0.166 V/cm)}, Fig. 2(c)–(e), the growth becomes random. The system behaves like the bias-free aggregation case as reported earlier [26]. The contributions of the migration and the electro-convection terms are negligible. The system is completely under thermal disorder. The aggregates are ramified structures with an outer boundary that remains circularly symmetric (especially at low voltages). Thus, from the above, one can conclude that growth below 0.9 V (i.e., threshold voltage) is random giving rise to different sizes of DLA at different places in the Petri dish while the growth above 0.9 V is due to diffusion, convection and migration. The growth in the middle part for applied voltage 1.5 V, 1.0 V, and 0.9 V further confirms the theoretical model developed by ZhiJie Tan et al. [30], as discussed in earlier work [27], which gives a statistical mechanism for computer simulation that includes the influence of external field in DLA. They have shown that in DLA under electric field, with increasing field, the structural transition occurs that lead to a change in the pattern from pure DLA to chain-like directional growth (along the electric field, see Fig. 1 of Ref. [30]). They attributed this structural transition to the change of dominating interaction from thermal force to field induced drag which causes anisotropy within the system. The appearance of other patterns (DBM and dendritic growth) is attributed to the anisotropy within a medium due to the effect of voltage that produces convection in the medium. This experimental observation can be compared with the computer simulated effect of voltage on the growth morphology [31]. After determining the threshold voltage (i.e., 0.90 V) above which structural pattern formation takes place in the polymer electrolytes, different electrode geometries have been used. Fig. 3, Fig. 4 and Fig. 5 show the growth of polymer electrolyte composites under circular and square electrode geometry. (i) Circular geometry. After application of electric field between the centre and the circumference of the circle (above threshold, (E = 33.0 = 1.00 V/cm)), the cations start piling up at the cathode and anions start piling up at the anode. The ionic transport near the anode could involve a mixture of diffusion, bulk convective motion, and field driven motion which leads to DBM. After that, the growth in the circular electrode geometry is under the influence of two main driving forces namely, the Brownian motion of the ions and the radial ionic motion due to the applied electric field. The growth consists of two components, the random motion and the directional radial movement which leads to chain-like DLA growth with dense branches which protrude in the electric field direction towards the cathode. Thus, the transition occurs from DBM growth to chain-like growth as the voltage is varied. Growth halts near the cathode due to the presence of the acidic front. There is
7
no transition from the chain-like growth to the dendritic pattern. The experimental observations can be seen in Fig. 3. In the case of reverse polarity, there is no formation of DBM pattern since there is no accumulation of the anions near the anode. After application of the electric field, anions uniformly distribute around the circumference of the circle and the cations towards the cathode. So, an acidic zone is created near the center. Only chain-like growth pointing towards in the electric field direction is observed as seen in Fig. 4. (ii) Square geometry. After application of electric field (above threshold, (E = 23.0 = 0.666 V/cm)), there is separation of ions and the anions pile up at the anode near the two electrodes. Cations pile up at the cathode which leads to H2 bubbles as shown in Fig. 5(b). The experimental observation for growth under square geometry can be seen from Fig. 5(a)–(f). Due to aggregation of anions near the anode, Fig. 5(c), the electrode geometry effectively changes from planar to irregular. Hence, the DBM patterns in region ‘A’, Fig. 5(e), and chain-like growth start protruding towards the electric field direction, region ‘B’. Dendritic patterns grow near the cathode, region ‘C’, Fig. 5(f). Further studies on the optimization based on free-morphing parameters by means of recent optimization strategies such as Constructal theory [32–34] are underway.
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90
4. Conclusions
91 92
It has been seen that the fractal growth between two electrodes depends only on the distance separating the two electrodes, i.e., the electric field per unit length. For the present system the threshold has been found to be 0.30 V/cm. Below the specified threshold, there is no growth. Above the specified threshold, fractal growth kinetics is a combined effect of many competing processes, i.e., Brownian motion due to thermal agitation (high humid environment), electro-convective motion and very high local electric field near the tips of the aggregate. Thus, the combined effect of diffusion, electro-convection and migration make the aggregates undergo morphological transition from DBM to chain-like growth to dendritic pattern. The ion diffusion phenomenon that leads to large size fractal growth after the application of electric field has been analyzed on the basis of Nernst–Planck equation. Voltages greater than the threshold value have been chosen to simulate the growth using circular electrode geometry and square electrode geometry. This is a step ahead for a better understanding of morphogenesis of fractal growth kinetics in ion conducting polymer electrolyte without the use of complicated equations and theories.
93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113
Acknowledgements
114 115
The financial support by Delhi University’s R&D grant is gratefully acknowledged. One of the authors (A.D.) thanks UGC (University Grants Commission) for grant of Junior Research Fellowship.
116 117 118 119
References
120 121
[1] B.B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco, 1982. [2] T. Vicsek, Fractal Growth Phenomenon, 2nd ed., World Scientific, Singapore, 1992. [3] M. Ben Amar, P. Pelcé, P. Tabeling (Eds.), Growth and Form: Nonlinear Aspects, NATO ASI Ser., Ser. B: Phys., vol. 276, Plenum, New York, 1991. [4] L.P. Kadanoff, J. Stat. Phys. 39 (1985) 267. [5] T.A. Witten, L.M. Sander, Phys. Rev. Lett. 47 (1981) 1400. [6] S.C. Hill, J.I.D. Alexander, Phys. Rev. E 56 (1997) 4317. [7] M. Castro, R. Cuerno, A. Sanchez, Phys. Rev. E 62 (2000) 161. [8] l. Niemeyer, L. Pietronero, H.J. Wiesmann, Phys. Rev. Lett. 52 (1984) 1033. [9] J.S.S. Hele-Shaw, Nature 58 (1898) 34. [10] L. Paterson, J. Fluid Mech. 113 (1981) 513. [11] Z.Y. Zhang, M.G. Lagally, Science 276 (1997) 377. [12] V.A. Bogoyavlenskiv, N.A. Chernova, Phys. Rev. E 61 (2000) 1629.
122 123 124 125 126 127 128 129 130 131 132
JID:PLA
8
1 2 3 4 5 6
[13] [14] [15] [16] [17] [18] [19]
7 8 9 10 11
[20] [21] [22] [23] [24]
AID:22777 /SCO Doctopic: Nonlinear science
[m5G; v 1.134; Prn:26/08/2014; 9:26] P.8 (1-8)
A. Dawar, A. Chandra / Physics Letters A ••• (••••) •••–•••
J.S. Langer, Rev. Mod. Phys. 52 (1980) 1. R.F. Voss, Phys. Rev. B 30 (1984) 334. P. Meakin, Phys. Rev. B 28 (1983) 5221. R. Tao, M.A. Novotny, E. Kashi, Phys. Rev. A 38 (1988) 1019. R.F. Voss, J. Stat. Phys. 36 (1984) 861. M. Nazzarro, F. Nieto, A.J. Ramirez-Pastor, Surf. Sci. 497 (2002) 275. C. Huo, X.H. Reo, B.P. Liu, Y.R. Yang, S.X. Rong, J. Appl. Polym. Sci. 90 (2003) 1463. A. Chandra, S. Chandra, Phys. Rev. B 49 (1994) 633. A. Chandra, Solid State Ion. 86 (1996) 1437. S. Amir, N.S. Mohamed, S.A. Hashim Ali, Cent. Eur. J. Phys. 8 (2010) 150. S. Amir, N.S. Mohamed, S.A. Hashim Ali, Adv. Mater. Res. 93 (2010) 35. S. Amir, S.A. Hashim Ali, N.S. Mohamed, Ionics 17 (2011) 121.
[25] [26] [27] [28] [29] [30] [31] [32]
S. Amir, S.A. Hashim Ali, N.S. Mohamed, Phys. Scr. 84 (2011) 045802. A. Dawar, A. Chandra, Commun. Nonlinear Sci. Numer. Simul. 18 (2013) 959. A. Dawar, A. Chandra, Phys. Lett. A 376 (2012) 3604. V. Fluery, J. Kaufmann, B. Hibbert, Phys. Rev. E 48 (1993) 3831. F. Sagues, M.Q. Lopez-Salvans, J. Claret, Phys. Rep. 337 (2000) 97. Z.-J. Tan, X.-W. Zou, W.-B. Zhang, Z.-Z. Jin, Phys. Lett. A 268 (2000) 112. W. Huang, D. Brynn Hibbert, Physica A 233 (1996) 888. M.R. Hajmohammadi, S. Poozesh, S.S. Nourazar, Proc. Inst. Mech. Eng., Part D, J. Proc. Mech. Eng. 226 (2012) 324–336. [33] M.R. Hajmohammadi, S. Poozesh, A. Campo, S.S. Nourazar, Energy Conversation and Management 66 (2013) 33–40. [34] M.R. Hajmohammadi, M. Rahmani, A. Campo, O.J. Shariatzadeh, Energy 68 (2014) 609–616.
67 68 69 70 71 72 73 74 75 76 77
12
78
13
79
14
80
15
81
16
82
17
83
18
84
19
85
20
86
21
87
22
88
23
89
24
90
25
91
26
92
27
93
28
94
29
95
30
96
31
97
32
98
33
99
34
100
35
101
36
102
37
103
38
104
39
105
40
106
41
107
42
108
43
109
44
110
45
111
46
112
47
113
48
114
49
115
50
116
51
117
52
118
53
119
54
120
55
121
56
122
57
123
58
124
59
125
60
126
61
127
62
128
63
129
64
130
65
131
66
132
[m5G; v 1.134; Prn:26/08/2014; 9:26] P.1 (1-8)
Physics Letters A ••• (••••) •••–•••
1
Contents lists available at ScienceDirect
2
67 68
3
Physics Letters A
4 5
69 70 71
6
72
www.elsevier.com/locate/pla
7
73
8
74
9
75
10
76
11 12 13 14 15 16
Electric field driven fractal growth dynamics in polymeric medium
77
Anit Dawar, Amita Chandra ∗
79 80 81
Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India
82
17 18
83
a r t i c l e
i n f o
84
a b s t r a c t
19 20 21 22 23 24
85
Article history: Received 19 May 2014 Received in revised form 5 August 2014 Accepted 15 August 2014 Available online xxxx Communicated by Z. Siwy
25 26 27 28 29
78
Keywords: Fractals Diffusion limited aggregation Polymer electrolyte composites Electric field
30
This paper reports the extension of earlier work (Dawar and Chandra, 2012) [27] by including the influence of low values of electric field on diffusion limited aggregation (DLA) patterns in polymer electrolyte composites. Subsequently, specified cut-off value of voltage has been determined. Below the cut-off voltage, the growth becomes direction independent (i.e., random) and gives rise to ramified DLA patterns while above the cut-off, growth is governed by diffusion, convection and migration. These three terms (i.e., diffusion, convection and migration) lead to structural transition that varies from dense branched morphology (DBM) to chain-like growth to dendritic growth, i.e., from high field region (A) to constant field region (B) to low field region (C), respectively. The paper further explores the growth under different kinds of electrode geometries (circular and square electrode geometry). A qualitative explanation for fractal growth phenomena at applied voltage based on Nernst–Planck equation has been proposed. © 2014 Elsevier B.V. All rights reserved.
86 87 88 89 90 91 92 93 94 95 96
31
97
32
98
33 34
99
1. Introduction
35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59
Pattern formation, i.e., the non-equilibrium growth phenomenon, has attracted considerable interest of scientists from various disciplines in the past decade. Such far from equilibrium growth studies include a wide range of problems encompassing life sciences, physics, biology, chemistry, computer science, image compression and geology which produce complex geometries of fractal or dendrite character and chaotic patterns [1–4]. Aggregation is one of the most familiar phenomena in physical and chemical processes. Many efforts have been made to develop models for fractal growth and aggregation processes [2]. Among these models, the most well known one is diffusion limited aggregation (DLA). The DLA model was proposed in 1981 by Witten and Sander as a computer algorithm [5]. In this model, aggregation process begins with fixing one particle at the center of coordinates in 2 dimensions and follows the formation of cluster by releasing particles from infinity. The particles undergo random walk due to Brownian motion until they hit any particle belonging to the growing cluster and stick to it. The DLA model has since attained a paradigmatic status due to its simplicity and presents the fundamental way for many diffusive systems such as effect of uniform drift and surface diffusion in electrodeposition [6,7], dielectric breakdown (lightening) [8], viscous fingering [9,10], growth mechanism in thin film [11], crystal morphology [12] and solidification (snowflakes) [13]. Many extensions of DLA model have
60 61 62 63 64 65 66
*
Corresponding author. Tel.: +91 11 27662295; fax: +91 11 27667061. E-mail address: [email protected] (A. Chandra).
http://dx.doi.org/10.1016/j.physleta.2014.08.016 0375-9601/© 2014 Elsevier B.V. All rights reserved.
been developed taking into account the processes involving concentration [14], particle drift [15], surface tension [16], sticking probability [17], and heterogeneous surface [18]. In polymer science, polymer growth is a fractal process [19]. Since the last 20 years, fractal growth in polymer electrolyte has attracted much more attention [20–27]. The observation of fractal pattern formation opens new avenues for detailed studies of growth due to the better understanding of the phenomenon without the use of complex theories. Chandra and Chandra [20,21] were first to report the technique to grow fractal patterns in polymer electrolyte under bias-free conditions. They reported the fractal growth in ion conducting polymer due to random walk of ions and their subsequent aggregation around nucleation centers. Recently, another group of researchers [22–25] also cultured fractals in various ion conducting polymer electrolyte membranes experimentally. They also did theoretical modeling of the experimentally cultured fractals. In our earlier work [26], polymer electrolyte (PEO:NH4 I) dispersed with Al2 O3 as a seed particle was used for obtaining the growth under bias free conditions. The growth of fractals was found to be due to the random walk and subse+ + − quent aggregation of the mobile species [I− 3 and NH4 (via NH4 I3 )]. On approaching the nucleation center, the random walkers stuck and formed an aggregate leading to DLA of different sizes and varying dimensionality. In another work [27], we reported the effect of electric field on large size fractal growth in polymer electrolyte composites by varying the voltage from 2 V to 8 V. The application of electric field to the polymer electrolyte composites caused ordered pattern formation in the electric field direction. The growth probability would not be the same in all the directions
100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
JID:PLA
AID:22777 /SCO Doctopic: Nonlinear science
2
1 2 3 4 5 6 7 8 9 10 11 12
[m5G; v 1.134; Prn:26/08/2014; 9:26] P.2 (1-8)
A. Dawar, A. Chandra / Physics Letters A ••• (••••) •••–•••
and the pattern formation was completely governed by convection which was due to the applied voltage. At higher voltages (i.e., at 6 V and 8 V), there was screening effect on the ions which prohibited the pattern to grow in the electric field direction. In the present work, electric field induced aggregation in polymer electrolyte composites has been studied under the influence of low values of electric field and extends the earlier reported work [27]. Specified threshold level has been determined. Below the threshold, the growth becomes random and above the threshold, the growth is governed by diffusion, convection and migration. Different kinds of electrode geometries have been used to simulate the growth under external field.
67 68 69 70 71 72 73 74 75 76 77 78
13 14
79
2. Experimental setup
80
15 16
81
2.1. Sample preparation
82
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
83
Samples were prepared by using the solution-cast technique. In this technique, the host polymer (Poly (ethylene oxide) (PEO), Sigma-Aldrich, Mol. Wt. ∼ 6 × 105 ) and salt (NH4 I (Sigma Aldrich, 99.999% trace metals basis)) were weighed in desired weight ratio (i.e., 50:50 wt.%). They were dissolved in distilled methanol thoroughly for ∼6–7 hr at 40 ◦ C. A small amount of activated Al2 O3 (neutral, 5 wt.% (Sigma-Aldrich, pH = 7.0 ± 0.5 (in H2 O))) was dispersed in the above PEO:NH4 I complexed solution. The highly viscous mixture of PEO:NH4 I (+Al2 O3 (neutral)) obtained after mixing was poured in polypropylene Petri-dishes (diameter = 7.5 cm) fitted with different electrode arrangements (viz. parallel, square and circular electrode geometry). The applied potential difference V across the electrode was varied from 0 to 2 V. The parallel electrodes were separated by a distance of 30 mm.
32 33
3. Results and discussion
34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
The physical scenario envisaged is the effect of external field on large size fractal growth in polymer electrolyte. The main transport mechanisms involved in the growth after application of electric field are diffusion, migration and electro-convective motion. Initially, the polymer electrolyte (PEO:NH4 I) dispersed with Al2 O3 (neutral) is electrically neutral everywhere with uniform concentration of cations and anions. (a) Fig. 1 shows the large size fractal growth after application of 1.5 V between two parallel electrodes. On application of voltage between the electrodes, the cations move towards the cathode where they discharge and act as a collection center for H2 . The cations (i.e., protons, as the polymer electrolyte (PEO:NH4 I) is predominantly a proton conductor) which migrate towards the cathode leave behind them a zone of higher pH (i.e., basic front) and near the cathode there is creation of an acidic front. Anions move towards the anode where they pile up since they cannot exit the solution. The whole Petri-dish containing the viscous polymer electrolyte composite solution under the influence of electric field is kept in a high humidity environment (R.H. ∼70%–80%). High humidity environment gives slow drying rate, provides enough time for aggregation and the right reducing environment for aggregation to take place. Earlier work [27] has reported that after ∼1–2 days, dense branched morphological patterns appear near the anode due to the presence of high concentration of anions near the anode while chain-like growth takes place in the middle part which protrudes in the electric field direction (i.e., towards cathode). Near the cathode, when the patterns meet the acidic front, the ordered chain-like growth forms an envelope around the cathode. At the cathode, growth also changes to dendritic patterns. A well defined angle is then observed between the main branches and the side branches of the dendrites. However, the orientation of the branches is only local with no long range correlation and their distribution
84 85 86 87 88 89
Fig. 1. Fractal growth after application of 1.5 V (i.e., E = 0.5 V/cm).
90 91
in all the directions results in a radial isotropic pattern maintaining an elliptical envelope. The physical model is described by the Nernst–Planck equation for the concentration of cations and anions in the polymer electrolyte subject to diffusion, convection and migration fields. The conventional large size fractal growth in polymer electrolyte composite dispersed with seed particle does not include electric field since the growth there is completely governed by diffusion of ions in the polymer electrolyte without any external bias [26]. The growth is a typical Laplacian growth as the system is completely under thermal disorder and DLA patterns at different places of different kinds appear. In the current work, it is critical to understand the configuration of the fractal growth having different patterns at different places (i.e., near anode, in the middle part, near cathode) when an electric field is applied. Here, a qualitative explanation for the phenomenon by using the Nernst–Planck equation is being given. Generally, the flux of ions in an electrolyte is completely described by the Nernst–Planck equation. The net flux of ions ( j ± ) is therefore, the diffusion term (− D ± ∇ c ± ), the migration term ( KzeT D ± c ± ∇Φ ) and the convection term (c ± u) [28]. B
j± = − D ± ∇ c± +
ze KBT
D ± c ± ∇Φ + c ± u
(1)
where D ± is the diffusion coefficient of ions, c ± their concentrations, Φ electric potential, u is the fluid velocity and ∇ c± the concentration gradient. Consider the Petri-dish containing two parallel electrodes and assume that the aggregation is proceeding slowly in quasi stationary state governed by diffusion, convection and migration in an electric field. The component of electric field parallel to the electrodes will be zero (i.e., the electric field in the x-direction, E x = 0). Thus, the second term denotes only migration in the electric field in the y-direction. When the voltage is switched on, a depletion zone of anions occurs near the cathode and that of cations near the anode. Meanwhile, an electro-convective motion sets in between the two parallel electrodes. The speed of an ion in the viscous mixture of polymer electrolyte composite is given by its mobility (μc for the cation, μa for anion) times the electric field (E = 13.5 = 0.5 V/cm). So, the growth speed of the deposit is equal to the anion’s mobility times the field. Thus, the migration term and the convection term play key roles in aggregation which is not
92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
JID:PLA AID:22777 /SCO Doctopic: Nonlinear science
[m5G; v 1.134; Prn:26/08/2014; 9:26] P.3 (1-8)
A. Dawar, A. Chandra / Physics Letters A ••• (••••) •••–•••
3
1
67
2
68
3
69
4
70
5
71
6
72
7
73
8
74
9
75
10
76
11
77
12
78
13
79
14
80
15
81
16
82
17
83
18
84
19
85
20
86
21 22
87
(a)
(b)
88
23
89
24
90
25
91
26
92
27
93
28
94
29
95
30
96
31
97
32
98
33
99
34
100
35
101
36
102
37
103
38
104
39
105
40
106
41
107
42
108
43 44
(c)
(d)
109 110
45
111
46
112
47
113
48
114
49
115
50
116
51
117
52
118
53
119
54
120
55
121
56
122
57
123
58
124
59
125
60
126
61
127
62
128
63
129
64 65 66
130
(e) Fig. 2. Fractal growth in polymer electrolyte at (a) 1 V, (b) 0.90 V, (c) 0.85 V, (d) 0.80 V, and (e) 0.50 V.
131 132
JID:PLA
AID:22777 /SCO Doctopic: Nonlinear science
[m5G; v 1.134; Prn:26/08/2014; 9:26] P.4 (1-8)
A. Dawar, A. Chandra / Physics Letters A ••• (••••) •••–•••
4
1
67
2
68
3
69
4
70
5
71
6
72
7
73
8
74
9
75
10
76
11
77
12
78
13
79
14
80
15
81
16
82
17
83
18
84
19
85
20
86
21 22
87
(a)
(b)
88
23
89
24
90
25
91
26
92
27
93
28
94
29
95
30
96
31
97
32
98
33
99
34
100
35
101
36
102
37
103
38
104
39
105
40
106
41
107
42 43
108
(c)
(d)
109
44
110
45
111
46
112
47
113
48
114
49
115
50
116
51
117
52
118
53
119
54
120
55
121
56
122
57
123
58
124
59
125
60
126
61
127
62
128
63 64 65 66
129
(e) Fig. 3. Fractal growth in circular electrode geometry: (a) without bias, (b) after few hours of applied voltage, (c) after 24 hours of applied voltage, (d) after 2–3 days, and (e) after complete growth.
130 131 132
JID:PLA AID:22777 /SCO Doctopic: Nonlinear science
[m5G; v 1.134; Prn:26/08/2014; 9:26] P.5 (1-8)
A. Dawar, A. Chandra / Physics Letters A ••• (••••) •••–•••
5
1
67
2
68
3
69
4
70
5
71
6
72
7
73
8
74
9
75
10
76
11
77
12
78
13
79
14
80
15
81
16
82
17
83
18
84
19
85
20
86
21
87
22 23
88
(a)
(b)
89
24
90
25
91
26
92
27
93
28
94
29
95
30
96
31
97
32
98
33
99
34
100
35
101
36
102
37
103
38
104
39
105
40
106
41
107
42
108
43
109
44
110
45
111
(c)
46 47 48
112
Fig. 4. Fractal growth using circular electrode geometry with polarity in reverse direction: (a) after 24 hours of applied voltage, (b) after complete growth, and (c) magnified image.
49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
113 114 115
considered in bias-free aggregation [26]. This is to say, the transport of anions in the aggregation process around the seed particles (Al2 O3 , neutral) is determined by all the three; diffusion, convection and migration terms. One should consider the contribution of these three terms in the aggregation process that leads to different kinds of patterns (i.e., DBM to chain-like growth to dendritic growth) at a fixed distance (i.e., field) from the anode and the cathode. The third term (i.e., convection term) generally includes natural convection (driven by density difference), forced convection (driven by pressure gradient) and local convection (driven by electric field). The forced convection can be neglected due to absence of any external force. The natural convection arises due to the high humid environment. The local convection arises because of the electro-convective motion that sets in. After application of the electric field, the concentration of anions gradually starts increasing at the anode. After some time, the concentration of anions is highest at the anode end then at
the middle part followed by the cathode end. Thus, the electroconvection is higher near the anodic part as the concentration of anions is high which changes the shape of the electrode from planar to irregular with sharp corners (since electric field intensity is larger at sharp corners and force acting on particle is also the strongest). Since the whole system is in a high humidity environment, as the volatile component starts evaporating, the concentration of anions starts increasing at the anode which makes the convection term more dominating. Thus, besides the migration term, electro-convection term also plays an important role in the formation of the DBM near the anode. The convection is higher near the anode part where probability of anions attaching to the aggregate becomes larger and the DBM forms. Other workers [29] have also demonstrated that in electrochemical deposition process, the migration term ( KzeT D ± c ± ∇Φ ) could induce DBM, particularly B in high field region which is depicted in Fig. 1. In the middle part (constant field region), the concentration of anions is low as com-
116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
JID:PLA
AID:22777 /SCO Doctopic: Nonlinear science
[m5G; v 1.134; Prn:26/08/2014; 9:26] P.6 (1-8)
A. Dawar, A. Chandra / Physics Letters A ••• (••••) •••–•••
6
1
67
2
68
3
69
4
70
5
71
6
72
7
73
8
74
9
75
10
76
11
77
12
78
13
79
14
80
15
81
16
82
17
83
18
84
19
85
20 21
86
(a)
(b)
87
22
88
23
89
24
90
25
91
26
92
27
93
28
94
29
95
30
96
31
97
32
98
33
99
34
100
35
101
36
102
37
103
38
104
39
105
40
106
41
107
42 43
(c)
(d)
108 109
44
110
45
111
46
112
47
113
48
114
49
115
50
116
51
117
52
118
53
119
54
120
55
121
56
122
57
123
58
124
59
125
60
126
61
127
62
128
63 64 65 66
129
(e)
(f)
Fig. 5. Fractal growth using square electrode geometry: (a) without bias, (b) after 2 minutes of applied voltage, (c) after 24 hours, (d) after 3 days, (e) after 4–5 days, and (f) after complete growth.
130 131 132
JID:PLA AID:22777 /SCO Doctopic: Nonlinear science
[m5G; v 1.134; Prn:26/08/2014; 9:26] P.7 (1-8)
A. Dawar, A. Chandra / Physics Letters A ••• (••••) •••–•••
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
pared to the anode. The contribution of the migration and the convection term cannot be negligible, hence, in the middle part, chain-like growth is observed guided in the direction of the electric field due to the electro-convective motion that sets between the two parallel electrodes. At the cathode (low field region), cations form an acidic zone. Contribution of migration and electro-convection term near the cathode is low. Chain-like growth which is protruding towards the cathode, meets the acidic front which turns the growth into dendritic type of growth. The aggregation form an envelope around the cathode by making a well defined angle between the main branches and the side branches of the dendrities. (b) At lower voltages {i.e., at 1 V (E = 13.0 = 0.333 V/cm) and 0.9 3
at 0.9 V (E = = 0.30 V/cm)}, Fig. 2(a)–(b), the growth looses its directionality in the middle part. This is because the strength of the electric field is not enough to guide the growth. (c) At further lower voltages {i.e., at 0.85 V (E = 0.385 = 0.283 V/cm), at 0.8 V (E =
0.8 3
= 0.266 V/cm) and at 0.50 V
(E = 0.350 = 0.166 V/cm)}, Fig. 2(c)–(e), the growth becomes random. The system behaves like the bias-free aggregation case as reported earlier [26]. The contributions of the migration and the electro-convection terms are negligible. The system is completely under thermal disorder. The aggregates are ramified structures with an outer boundary that remains circularly symmetric (especially at low voltages). Thus, from the above, one can conclude that growth below 0.9 V (i.e., threshold voltage) is random giving rise to different sizes of DLA at different places in the Petri dish while the growth above 0.9 V is due to diffusion, convection and migration. The growth in the middle part for applied voltage 1.5 V, 1.0 V, and 0.9 V further confirms the theoretical model developed by ZhiJie Tan et al. [30], as discussed in earlier work [27], which gives a statistical mechanism for computer simulation that includes the influence of external field in DLA. They have shown that in DLA under electric field, with increasing field, the structural transition occurs that lead to a change in the pattern from pure DLA to chain-like directional growth (along the electric field, see Fig. 1 of Ref. [30]). They attributed this structural transition to the change of dominating interaction from thermal force to field induced drag which causes anisotropy within the system. The appearance of other patterns (DBM and dendritic growth) is attributed to the anisotropy within a medium due to the effect of voltage that produces convection in the medium. This experimental observation can be compared with the computer simulated effect of voltage on the growth morphology [31]. After determining the threshold voltage (i.e., 0.90 V) above which structural pattern formation takes place in the polymer electrolytes, different electrode geometries have been used. Fig. 3, Fig. 4 and Fig. 5 show the growth of polymer electrolyte composites under circular and square electrode geometry. (i) Circular geometry. After application of electric field between the centre and the circumference of the circle (above threshold, (E = 33.0 = 1.00 V/cm)), the cations start piling up at the cathode and anions start piling up at the anode. The ionic transport near the anode could involve a mixture of diffusion, bulk convective motion, and field driven motion which leads to DBM. After that, the growth in the circular electrode geometry is under the influence of two main driving forces namely, the Brownian motion of the ions and the radial ionic motion due to the applied electric field. The growth consists of two components, the random motion and the directional radial movement which leads to chain-like DLA growth with dense branches which protrude in the electric field direction towards the cathode. Thus, the transition occurs from DBM growth to chain-like growth as the voltage is varied. Growth halts near the cathode due to the presence of the acidic front. There is
7
no transition from the chain-like growth to the dendritic pattern. The experimental observations can be seen in Fig. 3. In the case of reverse polarity, there is no formation of DBM pattern since there is no accumulation of the anions near the anode. After application of the electric field, anions uniformly distribute around the circumference of the circle and the cations towards the cathode. So, an acidic zone is created near the center. Only chain-like growth pointing towards in the electric field direction is observed as seen in Fig. 4. (ii) Square geometry. After application of electric field (above threshold, (E = 23.0 = 0.666 V/cm)), there is separation of ions and the anions pile up at the anode near the two electrodes. Cations pile up at the cathode which leads to H2 bubbles as shown in Fig. 5(b). The experimental observation for growth under square geometry can be seen from Fig. 5(a)–(f). Due to aggregation of anions near the anode, Fig. 5(c), the electrode geometry effectively changes from planar to irregular. Hence, the DBM patterns in region ‘A’, Fig. 5(e), and chain-like growth start protruding towards the electric field direction, region ‘B’. Dendritic patterns grow near the cathode, region ‘C’, Fig. 5(f). Further studies on the optimization based on free-morphing parameters by means of recent optimization strategies such as Constructal theory [32–34] are underway.
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90
4. Conclusions
91 92
It has been seen that the fractal growth between two electrodes depends only on the distance separating the two electrodes, i.e., the electric field per unit length. For the present system the threshold has been found to be 0.30 V/cm. Below the specified threshold, there is no growth. Above the specified threshold, fractal growth kinetics is a combined effect of many competing processes, i.e., Brownian motion due to thermal agitation (high humid environment), electro-convective motion and very high local electric field near the tips of the aggregate. Thus, the combined effect of diffusion, electro-convection and migration make the aggregates undergo morphological transition from DBM to chain-like growth to dendritic pattern. The ion diffusion phenomenon that leads to large size fractal growth after the application of electric field has been analyzed on the basis of Nernst–Planck equation. Voltages greater than the threshold value have been chosen to simulate the growth using circular electrode geometry and square electrode geometry. This is a step ahead for a better understanding of morphogenesis of fractal growth kinetics in ion conducting polymer electrolyte without the use of complicated equations and theories.
93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113
Acknowledgements
114 115
The financial support by Delhi University’s R&D grant is gratefully acknowledged. One of the authors (A.D.) thanks UGC (University Grants Commission) for grant of Junior Research Fellowship.
116 117 118 119
References
120 121
[1] B.B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco, 1982. [2] T. Vicsek, Fractal Growth Phenomenon, 2nd ed., World Scientific, Singapore, 1992. [3] M. Ben Amar, P. Pelcé, P. Tabeling (Eds.), Growth and Form: Nonlinear Aspects, NATO ASI Ser., Ser. B: Phys., vol. 276, Plenum, New York, 1991. [4] L.P. Kadanoff, J. Stat. Phys. 39 (1985) 267. [5] T.A. Witten, L.M. Sander, Phys. Rev. Lett. 47 (1981) 1400. [6] S.C. Hill, J.I.D. Alexander, Phys. Rev. E 56 (1997) 4317. [7] M. Castro, R. Cuerno, A. Sanchez, Phys. Rev. E 62 (2000) 161. [8] l. Niemeyer, L. Pietronero, H.J. Wiesmann, Phys. Rev. Lett. 52 (1984) 1033. [9] J.S.S. Hele-Shaw, Nature 58 (1898) 34. [10] L. Paterson, J. Fluid Mech. 113 (1981) 513. [11] Z.Y. Zhang, M.G. Lagally, Science 276 (1997) 377. [12] V.A. Bogoyavlenskiv, N.A. Chernova, Phys. Rev. E 61 (2000) 1629.
122 123 124 125 126 127 128 129 130 131 132
JID:PLA
8
1 2 3 4 5 6
[13] [14] [15] [16] [17] [18] [19]
7 8 9 10 11
[20] [21] [22] [23] [24]
AID:22777 /SCO Doctopic: Nonlinear science
[m5G; v 1.134; Prn:26/08/2014; 9:26] P.8 (1-8)
A. Dawar, A. Chandra / Physics Letters A ••• (••••) •••–•••
J.S. Langer, Rev. Mod. Phys. 52 (1980) 1. R.F. Voss, Phys. Rev. B 30 (1984) 334. P. Meakin, Phys. Rev. B 28 (1983) 5221. R. Tao, M.A. Novotny, E. Kashi, Phys. Rev. A 38 (1988) 1019. R.F. Voss, J. Stat. Phys. 36 (1984) 861. M. Nazzarro, F. Nieto, A.J. Ramirez-Pastor, Surf. Sci. 497 (2002) 275. C. Huo, X.H. Reo, B.P. Liu, Y.R. Yang, S.X. Rong, J. Appl. Polym. Sci. 90 (2003) 1463. A. Chandra, S. Chandra, Phys. Rev. B 49 (1994) 633. A. Chandra, Solid State Ion. 86 (1996) 1437. S. Amir, N.S. Mohamed, S.A. Hashim Ali, Cent. Eur. J. Phys. 8 (2010) 150. S. Amir, N.S. Mohamed, S.A. Hashim Ali, Adv. Mater. Res. 93 (2010) 35. S. Amir, S.A. Hashim Ali, N.S. Mohamed, Ionics 17 (2011) 121.
[25] [26] [27] [28] [29] [30] [31] [32]
S. Amir, S.A. Hashim Ali, N.S. Mohamed, Phys. Scr. 84 (2011) 045802. A. Dawar, A. Chandra, Commun. Nonlinear Sci. Numer. Simul. 18 (2013) 959. A. Dawar, A. Chandra, Phys. Lett. A 376 (2012) 3604. V. Fluery, J. Kaufmann, B. Hibbert, Phys. Rev. E 48 (1993) 3831. F. Sagues, M.Q. Lopez-Salvans, J. Claret, Phys. Rep. 337 (2000) 97. Z.-J. Tan, X.-W. Zou, W.-B. Zhang, Z.-Z. Jin, Phys. Lett. A 268 (2000) 112. W. Huang, D. Brynn Hibbert, Physica A 233 (1996) 888. M.R. Hajmohammadi, S. Poozesh, S.S. Nourazar, Proc. Inst. Mech. Eng., Part D, J. Proc. Mech. Eng. 226 (2012) 324–336. [33] M.R. Hajmohammadi, S. Poozesh, A. Campo, S.S. Nourazar, Energy Conversation and Management 66 (2013) 33–40. [34] M.R. Hajmohammadi, M. Rahmani, A. Campo, O.J. Shariatzadeh, Energy 68 (2014) 609–616.
67 68 69 70 71 72 73 74 75 76 77
12
78
13
79
14
80
15
81
16
82
17
83
18
84
19
85
20
86
21
87
22
88
23
89
24
90
25
91
26
92
27
93
28
94
29
95
30
96
31
97
32
98
33
99
34
100
35
101
36
102
37
103
38
104
39
105
40
106
41
107
42
108
43
109
44
110
45
111
46
112
47
113
48
114
49
115
50
116
51
117
52
118
53
119
54
120
55
121
56
122
57
123
58
124
59
125
60
126
61
127
62
128
63
129
64
130
65
131
66
132