Pattern Formation in the BromateâSulfiteâFerrocyanide Reaction - The

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Pattern Formation in the Bromate-Sulfite-Ferrocyanide Reaction István Molnár, and Istvan Szalai J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.5b06545 • Publication Date (Web): 15 Sep 2015 Downloaded from http://pubs.acs.org on September 18, 2015

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The Journal of Physical Chemistry

Pattern Formation in the Bromate-Sulte-Ferrocyanide Reaction István Molnár

†Institute ‡School

†, ‡

and István Szalai

∗, †

of Chemistry, Eötvös University, Budapest, Hungary

of Ph.D. Studies, Semmelweis University, Budapest, Hungary

E-mail: [email protected]

1

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Abstract Mixed Landolt type pH oscillators are versatile systems, which allow the experimental study a wide range of nonlinear phenomena including multistability, oscillations and spatio-temporal patterns. We report on the dynamics of the bromate-sulteferrocyanide reaction operated in a open one-side-fed reactor, where spatial bistability, spatio-temporal oscillations, front and Turing type patterns have been observed. The role of dierent experimental parameters, like the input ow concentrations of the hydrogen and the ferrocyanide ions, the temperature and the thickness of the gel medium (which aects the rate of the diusive feed) have been investigated. We point out that all these parameters can be eciently used to control the spatio-temporal dynamics. We show that the increase of ionic strength stabilizes the uniform states in expense of the patterned one. Some general aspects of the spatio-temporal dynamics of mixed Landolt type systems, which are based on the oxidation of sulte ions by strong oxidants, are emphasized.

Introduction Reaction-diusion systems, as general distributed models involving transport and nonlinear interactions, arise in dierent contexts. The patterns evolve in the diverse systems, despite the dierences in the local reaction terms, have universal features. The typical phenomena which arise are travelling fronts, periodic waves (spirals and targets) and stationary patterns in form of hexagons, stripes, labyrinths or localized structures.

13

Chemical reaction-diusion

systems are convenient to study pattern formation in well-dened experiments.

4,5

By using

proper reactors, that allow to maintain the reaction-diusion system far from equilibrium, one can explore and even control these complex physico-chemical phenomena.

Multistep

reactions which possess nonlinear kinetics and include positive and negative feedbacks are not peculiar in inorganic and biochemical reactions. Representative examples can be found among aqueous phase redox reactions, radical chain reactions and enzymatic or genetic

2


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networks.

4,6

On the base of their common dynamical features, well established practical

methods have been developed to nd conditions at which these reactions produce complex behaviors, such as oscillations in continuous-ow stirred tank reactors (CSTR) temporal patterns in open spatial reactors.

10,11

4,79

or spatio-

These methods are based on the control of

the time and length scale separation between the positive and negative feedback processes. Importantly, the use of these design algorithms does not require a detailed knowledge about the kinetics and mechanism of the applied reactions. Among the known aqueous phase inorganic oscillatory reactions, besides the classical Belousov-Zhabotinsky

12,13

and chlorite-iodide-malonic acid reaction families,

lators are preferably used to study reaction-diusion phenomena.

11,1624

14,15

pH oscil-

pH oscillators are

driven by a hydrogen or hydroxide ion autocatalytic process and ready to show large amplitude pH cycles.

25

The bromate-sulte-ferrocyanide (BSF) reaction

26

belongs to the category

of mixed Landolt type pH oscillators, which involve an autocatalytic oxidation of a weak acid by a strong oxidant and a suitable negative feedback. According to the works of Rábai and coworkers, the mechanism of the BSF reaction can be described by the following set of stoichiometric equations:

27,28

− + SO2− 3 + H HSO3

(R1)

+ HSO− 3 + H H2 SO3

(R2)

− 2− − + BrO− 3 + 3HSO3 → 3H + 3SO4 + Br

(R3)

2− − + BrO− 3 + 3H2 SO3 → 6H + 3SO4 + Br

(R4)

4− BrO− + 6H+ → Br− + 6[Fe(CN)6 ]3− + 3H2 O 3 + 6[Fe(CN)6 ] − − 2− BrO− 3 + 6HSO3 → 3S2 O6 + Br + 3H2 O

(R5) (R6)

Reactions (R1)-(R4) represent the protonation and the oxidation of sulte ions and provide the hydrogen ion autocatalytic subset.

Reaction (R5), the oxidation of ferrocyanide by

bromate is the main inhibitory process, while the last step (R6) corresponds to an alternative

3



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hydrogen ion consuming pathway. The latter has considerable contribution only in case of large bromate ions excess over sulte ions. In a CSTR the core bromate-sulte (BS) reaction (R1-R4 and R6) shows bistability between two stationary states.

The pH of the CSTR

content is around 8-7 at the ow state characterized with low extent of the oxidation reactions (R3 and R4) and about 3-2 at the "thermodynamic state characterized by a high extent of that reactions.

26

Additionally, high amplitude pH oscillations can develop due to

the presence of (R6) at appropriate conditions.

28

The BS reaction is also capable to show

damped pH cycles in a semibatch or closed reactor.

29

Even more robust oscillations can be

induced by addition of ferrocyanide, when the resulting bromate-sulte-ferrocyanide (BSF) reaction is performed in a CSTR.

26

Due to the presence of reaction (R5) the pH of the

"thermodynamic state is shifted to the region of 4-3 in the BSF system. While, the temporal dynamics and the kinetics of the homogeneous BS and BSF systems are well known, this is not the case for the corresponding reaction-diusion systems. Front propagation under batch conditions have been reported in both reactions.

30,31

Compared

to the batch condition more versatile phenomena can be explored when a reaction-diusion system is operated in an open spatial reactor. A popular reactor conguration is called open one-side-fed reactor (OSFR), in which a thin sheet of gel is in contact by one face with the contents of a CSTR. Opposite to the feed face the gel is pressed against an impermeable

out

GEL

CSTR in

w

Figure 1: Sketch of the disk-shaped OSFR used in the experiments:

w

is the thickness of

the gel disk.

and transparent wall.

Chemicals react both in the CSTR and in the gel, but the CSTR

4


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content is kept on a low extent of reaction state, thus the gel is fed by an almost unreacted mixture of the reactants. The gel matrix avoids the uid motions that would disturb the formation of the reaction-diusion patterns.

The study of the BS reaction in an OSFR

resulted in the observation of spatial bistability and spatio-temporal oscillations. bistability

3335

32

Spatial

corresponds to the coexistence of two dierent spatial stationary states at

the same conditions.

At the so called F (ow) state, the extent of the reaction is low

everywhere in the gel and the concentration distributions are nearly uniform. Contrary, the other spatial state is inhomogeneous: at the gel/CSTR surface the extent of the reaction is low, while in the depth of the gel the extent of the reaction is high. Thus, the concentration distributions along the feeding axis are nonuniform. This is called the M (mixed) state of the gel content. This type of spatial bistability is typical for autocatalytic reactions performed in an OSFR.

33,36

Numerical simulations revealed that the spatio-temporal oscillations in the

BS reaction develop through a kinetic instability, that is originated in the interplay of the autocatalytic subset (R1)-(R4) and the inhibitory process (R6). Because, both the positive and the negative feedbacks are supplied by the reaction between sulte and bromate ions, the oscillations can only develop in a narrow range of the experimental parameters where the time scale separation between the antagonist feedback processes is appropriate. In a preceding communication we have already shown that spatio-temporal pH and calcium ion waves can develop when the BSF reaction is linked to the pH-sensitive complexation of Ca

2+

by ethylenediaminetetraacetate and operated in an OSFR.

24

In principle all pH os-

cillators can be used to induce periodic changes in a coupled pH-sensitive physico-chemical equilibrium. However, due to their advantageous properties, like reproducibility and tolerance to the presence of the components of the driven system, the bromate-based pH oscillators provide the best base to build such kind of dynamical systems. In this report we present a detailed experimental investigation of the BSF reaction-diusion system in an OSFR. Besides the eect of the chemical composition of the input ow, the role of the thickness of the gel, the temperature and the ionic strength of the mixture are also presented. In order to

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control the length scale separation between hydrogen ions and the other reactants, we used a weak polyelectrolyte, sodium-polyacrylate (NaPAA). This is a well established method to obtain long range inhibition, since the binding of hydrogen ions by low mobility carboxylate functions, slows down the apparent mobility of hydrogen ions.

10,11

In mixed Landolt type

systems this approach have been successfully applied to nd a large variety of stationary structures including front and Turing patterns.

11,16,2023

The presented results help to un-

cover some general aspects of pattern formation in mixed Landolt type pH oscillators and can be useful in the development of coupled systems driven by periodic pH-stimulus.

Experimental section The experiments were performed in a thermostatted disc-shape OSFR, where the gel was made of 2 w/w% agarose (Fluka 05077). The gel disc has an eective contact diameter of 25 mm and thickness of 0.75 mm. The residence time in the CSTR was xed to 500 s. The feed solutions of the major chemicals were stored in four separated tanks but enter premixed into the CSTR. Reactants were distributed in the feed tanks as follows: Tank 1:

NaBrO 3 (Sigma-Aldrich)+ sodium bromocresol green (Sigma-Aldrich); Tank 2:

Na2 SO3 (Sigma-Aldrich) + sodium bromocresol green (Sigma-Aldrich) and in given cases sodium polyacrylate (Aldrich, average Mw

∼15,000);

Tank 3: K 4 Fe(CN)6

· 3 H2 O

(Sigma-

Aldrich); Tank 4: H 2 SO4 (diluted from 1.0 mol/L standard solution (Sigma-Aldrich)). All solutions were prepared with ion exchanged water and chemicals were used without further purication. The following input feed concentrations were kept xed in all the experiments: [NaBrO3 ]0

= 65 mM,

[Na2 SO3 ]0

= 80 mM,

[sodium bromocresol green] 0

= 0.2 mM.

Here,

[X]0 denotes the concentration that species X would have after mixing in the total inlet ow and prior to any reaction. In the case of polyacrylate, [NaPAA] 0 stands for concentration of carboxylate functions. To visualize the pH patterns bromocresol green indicator ( pKa

= 4.8)

was selected and the reactor was illuminated through a narrow band-pass lter centered at

6


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λ = 590 ± 5nm.

The pictures were taken by using a AVT Stingray F-033B ( 656

bit) camera and recorded by the Streampix (Norpix) software.

× 492,

14

The image processing are

made by using ImageJ. The observed grayscale pictures were colorized to mimic the color change of the applied indicator. The pseudocolor images were made by the lookup table that is shown in Figure 2. The state of the CSTR composition was monitored by the recording the potential of a pH electrode.

Results Fronts and waves In the experiments we followed the step by step method which has been suggested to explore the pattern forming capacity of reaction-diusion systems in an OSFR. with nding spatial bistability.

11

This method start

We used [H 2 SO4 ]0 as a control parameter, since it allows

to cross the stability limits of the two stationary states. Let us start the description of the dynamics at [Fe(CN) 6

4

]0 =5 mM, where at [H 2 SO4 ]0 =4 mM the gel content is in the F state.

By increasing [H 2 SO4 ]0 this state is stable up to 5.6 mM, where the M state appears and

Figure 2:

Propagating front in the BSF reaction:

increases at the expense of the F state.

2

τ =500 s, T =30

,

Experimental conditions:

w=0.75 mm,

[Fe(CN) 6

4

[BrO 3 ]0 =65 mM,

]=5 mM, [H2 SO4 ]0 =5.4 mM. The pseudocolor images were derived from grayscale ones by mapping each intensity value

[SO3

]0 =80 mM,

the circular M state (yellow) domain

to a color according to the presented lookup table. High intensity (yellow) color indicates a pH below 4.8 in the depth of the gel, while low intensity (blue) corresponds to a pH above 4.8 in the gel.

starts to occupy the whole gel. The M state of the gel is stable even if [H 2 SO4 ]0 is deceased 7



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now.

Page 8 of 26

The back transition occurs at [H 2 SO4 ]0 =4.8 mM. In between [H 2 SO4 ]0 =4.8 mM and

5.6 mM the the F and the M states coexist, this is spatial bistability.

In this range of

parameters a front can be initiated between these two stationary states as it is exemplied in Figure 2. The rate and the direction of propagation of this front depends on [H 2 SO4 ]0 and we did not observe morphological instability during its movement. As [Fe(CN)6

4

]0 increases, thus the rate of the negative feedback is increased, the pa-

rameter domain of spatial bistability decreases. Above a critical value of [Fe(CN) 6

4

]0 , that

is around 14 mM at the experimental conditions applied here, spatio-temporal oscillations develop in form of pH waves. The waves start from separated pacemakers, but this dynamics

Figure

3:

Spatio-temporal

oscillations

in

the

BSF

system

at

[Fe(CN) 6

4

]0 =40 mM,

[H2 SO4 ]0 =7.2 mM (a), the time interval between two successive snapshots is 300 s, and the [SO3

corresponding

2

]0 =80 mM,

time-space

plot

τ =500 s, T =30

,

(b).

Experimental

w=0.75 mm.

conditions:

[BrO 3 ]0 =65 mM,

The black vertical line on the rst snap-

shot indicates the position at which the time-space plot was constructed.

does not evolve in the entire gel. According to our experimental observations, the spatiotemporal dynamics of the BSF system is very sensitive to the thickness of the gel.

Even

small dierences in the thickness can stabilize or destabilize the actual dynamical state. As it is exemplied in Figure 3a, the oscillations develop in the large majority of the gel, except

8


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around the bottom left rim of it. As [H 2 SO4 ]0 =7.2 mM increased, the situation turns round, oscillations develop only around the the left rim of the gel (Figure 4a). The time-space plots

4

Figure 4:

Spatio-temporal oscillations in the BSF system at [Fe(CN) 6 ]0 =40 mM and [H2 SO4 ]0 =7.4 mM (a), the time interval between two successive snapshots is 300 s, and the

corresponding

[SO3

2

]0 =80 mM,

time-space

plot

τ =500 s, T =30

,

(b).

Experimental

w=0.75 mm.

conditions:

[BrO 3 ]0 =65 mM,

The black vertical line on the rst snap-

shot indicates the position at which the time-space plot was constructed.

(Figure 3b and Figure 4b) show, that these oscillations are characterized with sharp increase of hydrogen ion concentration (a switch from blue to yellow) that is followed by a smooth decrease and back switch (from yellow to blue), that is a avor of relaxation oscillations. The experimental observations are summarized in a nonequilibrium phase diagram along the [H2 SO4 ]0 -[Fe(CN)6

4

]0 plane (Figure 5). One can recognize its cross shaped topology,

the domain of oscillations develop above the critical [Fe(CN) 6 vanishes.

4

]0 at which spatial bistability

Within our experimental accuracy we did not observe bistability between the

oscillatory and the stationary states. We have performed experiments to uncover the eect of the thickness of the gel on the oscillatory dynamics. At the same conditions, except the [H 2 SO4 ]0 , the increase of the thickness promote the development of oscillations, while the decrease of the thickness can

9



Oscillations F state M state

40 [K4[Fe(CN)6]]0 / mM

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Osc.

30 20 F

10

M F/M 0 4

Figure

5:

Nonequilibrium

[H2 SO4 ]0 -[Fe(CN)6 [SO3

2

]0 =80 mM,

4

]0

phase

plane.

6 [H2SO4]0 / mM

diagram

The

τ =500 s, T =30

even quench it (Figure 6).

5

,

of

the

experimental

7

8

BSF

reaction

conditions

in

are:

OSFR

along

the

[BrO 3 ]0 =65 mM,

w=0.75 mm.

The phase diagram demonstrates, that

w

can be used as a

sensitive control parameter of the system. Around 0.75 mm, where most of the experiments were made, the full width of the parameter range of oscillations in

w,

is about 0.06 mm.

It explains the observations of dierent dynamics at dierent parts of the gel under same conditions.

It is dicult to avoid the development of a hundred of millimeter deviations

in the thickness of the soft hydrogel disc during its preparation. The reproducibility of the experiments made on the same gel is less than 0.1 mM in the [H 2 SO4 ]0 . In a thicker gel the parameter range of oscillations shits to lower values of [H 2 SO4 ]0 as the stability limits of both stationary states follows the same tendency. It is reasonable, since in a thicker gel the chemicals have longer time to react with each other, thus the uniform M state can appear at lower of [H2 SO4 ]0 . It was emphasized by Orbán, that the domain of CSTR oscillations increases while that of the bistability decreases with the rise of temperature.

26

At the actual conditions of our

experiments, the spatio-temporal oscillations disappear at 20

and below spatial bistability

can be observed (Figure 7). Above this temperature limit there is a slight increase in the width of the [H 2 SO4 ]0 range of the spatio-temporal oscillations.

Notably, the parameter

range of oscillations moves towards to smaller values of [H 2 SO4 ]0 . The change of temperature

10


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1.25

w / mm


Osc .

1

M

0.75 F

F/M

0.5

0.25 5

6

7 8 [H2SO4]0 / mM

9

10

Figure 6: Nonequilibrium phase diagram of the BSF reaction in OSFR along the [H 2 SO4 ]0 w plane. The experimental conditions are: [BrO 3 ]0 =65 mM, [SO32 ]0 =80 mM, τ =500 s,

T =30

, Fe(CN)6

4

]=30 mM.


Os

40

T / °C

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


c.

M

30

F/M

F 20

5

6

7 8 [H2SO4]0 / mM

9

10

Figure 7: Nonequilibrium phase diagram of the BSF reaction in OSFR along the H 2 SO4 ]0 T plane. The experimental conditions are: [BrO 3 ]0 =65 mM, [SO32 ]0 =80 mM, τ =500 s,

w=0.75 mm,

Fe(CN) 6

4

]=30 mM.

11



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Page 12 of 26

aects both the rate of the local reactions and the rate of the diusive exchange between the CSTR and the gel.

The increase of the rate of the autocatalytic process (R1)-(R4)

favors the development of M state. Contrary, faster exchange between the CSTR and the gel supports the F state, since the CSTR is kept on the low extent of reaction state. The experimentally observed tendency indicates that in between these two opposing eects in the BSF system the increase of the rate of the autocatalytic process with the rise of the temperature dominates. The variation of the period of oscillations with the temperature can be described by the temperature coecient, that is dened as the periods at temperature

T1

and

T2 ,

Q10 =

Tper2 Tper1

T 10 −T 2

1

, where

Tper1

and

Tper2

are

respectively. This temperature coecient has been

measured by Rábai and coworkers for several pH oscillators including the BSF system, where a value of 1.8-2.2 was obtained.

37

This is typical for a system driven by chemical reactions.

From the measured period of the spatio-temporal oscillations in OSFR we calculated a value of 1.9, that supports the kinetic origin of this reaction-diusion phenomena.

The period

of OSFR oscillations depends also on the time scale of the diusive exchange of chemicals between the gel and CSTR content.

Oscillations develop when this time scale is of the

order of the reaction rate, thus the connection between the period and the rate of the local reactions can be partly indirect.

Patterns in presence of sodium polyacrylate In reaction-diusion systems long range inhibition, e.g. slow diusion of the activator compared that of the inhibitor, can induce the formation of stationary patterns. NaPAA as a low mobility hydrogen ion binding molecule have already been eectively used to ensure long range inhibition in mixed Landolt systems with other oxidants.

11,16,2023

In the BSF reaction,

by using this approach dierent pattern formation scenarios could be detected. We present the dynamics of the BSF system in presence of NaPAA along the [H 2 SO4 ]0 -[Fe(CN)6

4

]0

parameter plane (Figure 8). At [NaPAA] 0 =9 mM all the interesting phenomena were found

12


Page 13 of 26

40

[K4[Fe(CN)6]]0 / mM

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


30

Pa

tte

Patterns F state 20 M state F

10

rn s

F/P

M

F/M 0 4.4

4.6

4.8 5 5.2 [H2SO4]0 / mM

5.4

5.6

Figure 8: Nonequilibrium phase diagram of the BSF reaction in OSFR along the [H 2 SO4 ]0 4 [Fe(CN)6 ]0 plane at [NaPAA] 0 =9 mM. The experimental conditions are: [BrO 3 ]0 =65 mM, [SO3

2

]0 =80 mM,

τ =500 s, T =35

between 4.4 mM

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