Modelling

A STABLE MAGNETIC LOCOMOTIVE

5.3 Modelling

In order to account for the pH dependent vertical motion of the polymer bead in H2O2, a model was developed following Weiss’ theory [89] on metal catalyzed decomposition of alkaline H2O2. Essentially, it is assumed that the motion of the bead is solely due to the buoyancy of the O2 gas generated on the surface of the beads, from the catalytic decomposition of H2O2, while O2 produced from solution decomposition, does not contribute to the motion in any way. In addition, it is assumed that O2 is continuously produced on the bead leading to its accelerated motion. However, as will be mentioned later, there are other factors, which may also need to be accounted for in order to have a complete understanding of the process. Finally, in the theoretical analyses, the rate constants (corresponding to the decomposition of H2O2) used was obtained from experimentally measured values. The details of the model are given below.

Let n_O2be the number of moles of O2 required just to balance the weight of a bead.

pH Specific Catalytic Motion

The minimum time required to produce such an amount of O2that would remain attached to the bead may be denoted as t_min. By balancing the weight of O2 with that of the bead (0.5 mg), one would get n_O2= 1.6×10^-5 mol. Now, 1.6×10^-5 mol O2 is produced from

3.1×10-5 mol of H2O2, following the reaction,

2H O_{2 2}2H O+O (1)_{2 2} Further, as the decomposition of H2O2follows first order kinetics, we have,



H O2 2

 

_t= H O2 2 0



e^{-k teff min} (2)

Or, H O



2 2

 

0- H O2 2



decomposed= H O



2 2



0e^{-k teff min} (3)



- ^{eff min}

 ^

²

_

2 decomposed

^ _

2 2 0

Or, 1 - e = H O

H O

k t (4)

Here, as defined earlier, t_minis the minimum time required for a bead to start moving upward and k is the effective H_eff 2O2 decomposition rate constant catalyzed by a single bead. Now the initial concentration of H2O2 = 8.75% = 2.57 M= 2.6×10 mol m^{3 -3}. Therefore, in 5 mL H2O2 solution, number of moles of H2O2 present is= 5.0×10 m ×2.6×10 mol m = 1.3×10 mol. Hence, from Eq. (4), we get,^{-6 -3 3 -3 -2}



^{1 - e-k teff min}



^{= 2.4×10-3} (5)

Or, k t_{eff min}= 0.00243 (6) In the present set of experiments, the decomposition constant was actually measured using 400 polymer beads keeping the volume of the solution and concentration of H2O2

the same as those used for velocity measurement with a single bead. In other words, the effective decomposition constant k was measured by observing the time-dependent _eff changes in the H2O2 concentration in presence of 400 polymer beads. This was required as the change in solution concentration was too small to be measured when the decomposition was catalyzed by a single bead. Decomposition constant k corresponding _eff to a single bead was then obtained by dividing k by 400. _eff

Further, relation (6) does not incorporate factors such as the rate of sticking of O2

bubbles to the surface of the polymer bead, catalytic surface area available for reaction - in

Chapter 5 presence of the O2 gas bubbles, time required for surface activation (i.e. induction period [100]) and the change in solution temperature during H2O2 decomposition. Also, even though for experimental necessity, 400 beads were used to measure the decomposition rate of H2O2, the situation might be quite different from that in case of decomposition by a single bead. For example, since the volume of solution was kept the same at 5 mL and concentration was also the same at 8.75%, the effective H2O2 molecules available to the surface of a single bead would be very different than to the surfaces of 400 beads, especially for decomposition at later times. This in turn, would make the time required for production of sufficient O2 gas for upward propulsion of one bead different from that estimated using observed k . Finally, it is found that _eff k values do not actually follow _eff linear relationship with the number of beads, which is evident from the results obtained using 0.1 g, 0.2 g and 0.3 g polymer beads (Figure 5.8).

Figure 5.8: Determination of decomposition constant of 8.75% H2O2 solution in presence of different amount of catalytic beads. The constants are obtained as a slope of the straight lines which show the variation of ln[H O ]_{2 2} witht (in min).

In order to incorporate the collective effects of the above factors, we propose that a dimensionless auxiliary parameter λ be introduced and the expression for minimum time is re-written as,

eff min= 0.00243λ

k t (7) It may be noted that λ takes care of the unaccounted factors including the variation in

pH Specific Catalytic Motion

sticking of bubbles over the polymer, surface roughness and the induction period involved in catalysis. At this stage, we must mention that although this is not the most elegant treatment absence of a more refined approach prompts us to proceed with Eq. (7).

Following the above equation and using the experimentally observed values of k and _eff tmin,the value of λ were calculated for different pH. The values are listed in Table 5.2.

Table 5.2: k and _eff t_minvalues for decomposition of H2O2by a single bead at different pH values, as obtained from experiments with 400 beads. Here k_catal,k_uncatal,keff=



kcatal-kuncatal



/400 refer to the decomposition constants measured with Au NP coated resin beads, uncoated resin beads and that corresponding to the effective catalysis.

The initial motion of the bead at pH 9.10 and 9.45 originate from extraordinary slow production of O2, The observed values of t_minwere found to get changed as a function of pH, as listed in Table 5.2. Thus λ could be calculated from t_minobserved at these pH values, except at pH 9.1, where the rate of decomposition was found too slow and hence assumed zero. It is interesting to note that the values of λ did not change drastically with pH and were within the same order of magnitudes. Further, assuming the resin bead to be moving with an uniform acceleration a , the equation of motion can be written as [71]







3 3 3 2

w w

0

4 4 2 d

= π g + π g - 6π - π - 6 π - g

3 3 3 -

t t

ma r σ R σ ηRv R ρa R ηρ a t m

t t (8) The first two terms on the right hand side of Eq. (8) signify the forces of buoyancy on the polymer bead due to its own and the volume of the O2 bubble attached to it. The third term represents the steady state viscous drag acting on the polymer bead while the last two terms correspond to the drag forces due to the degree of irrotationality and the magnitude of initial acceleration involved in the motion.

pH k_catal(×10 )^-6 (s^-1) k_uncatal(×10 )^-6 (s^-1) k_eff(×10 )^-6 (s^-1) tmin (s) λ (×10 )^-6

9.10 0 0 0 746.43± 42.11 0

9.45 1.01±0.04 0.68± 0.04 0.0008 ± 0.0002 362.06± 23.24 1192 9.78 54.83 ± 2.01 25.83± 1.88 0.0725±0.0097 34.55± 4.07 10308 10.20 112.33± 6.70 48.83± 2.33 0.1588 ± 0.0226 28.44 ± 2.81 18585 10.55 770.17 ± 41.67 113.04±13.53 1.6429± 0.1380 4.28 ±1.41 28936 10.80 1136.03 ± 96.17 171.33±27.50 2.9117 ±0.3092 3.44± 0.16 41219

Chapter 5 Here, m and R are the mass



^{= 5×10 kg-7}



and radius



^{= 4×10 m-4}



of a single resin bead, ρ



= 1.9 10 kg m^{ 3 -3}



is the density of material. The parameters σw



= 1.0 10 kg m^{ 3 -3}



and η



= 1.0×10 N s m^{-3 -2}



represent the density and the viscosity of 8.75% H2O2 solution, which is nearly equal to those of pure water. The symbols



^-2



g = 9.8 m s ,^v



^{= 0.02 m s-1}



^{and t}

^

^{= 1 s}

^

signify the acceleration due to gravity, velocity of a polymer bead inside the liquid and the measured time taken by the polymer bead to reach the top of the liquid layer after it takes off from the bottom of the vessel. Also, the parameter r represents the time-dependent increase in the radius of the O_t 2bubble grown over the surface of the polymer bead. Using these values, we get,







3 3

w w

3 2

0

4 4

π g + π g - 6π - g

3 3

= + π2 +6 π d

3 -

t

t

r σ R σ ηRv m

a m R ρ R ηρ t

t t

(9)

3 3

w w

3 2

4 4

π g + π g - 6π - g

3 3

Or, =

+ π2 +12 π

3

r σt R σ ηRv m

a

m R ρ R ηρt

(10)

Or, = 7.6×10a ^{9 3}r_t - 0.45 (11) Further, differentiating [H O ] = [H O ] e_{2 2 t 2 2 0} ^{-k teff} with respect to time t and replacing the expression of decomposition rate of H2O2by the production rate of O2[89], we get,

d[O ]^{2 eff} _{2 2 0} ^{- eff}

= [H O ] e

dt 2

t k k t

(12)

d[ ^O2] ^eff _{H O2 2 0} ^{- eff}

Or, = [ ] e

dt 2

t k t

n k

n (13)

Here, [n_O2]tis the number of moles of O2 produced in time t .

The total volume of O2generated in time t is given by,

4 ³

= π3

t t

V r (14)

pH Specific Catalytic Motion

Here r is the radius of the oxygen bubble attached to the bead (assuming the bubble to be _t spherical) responsible for its movement.

Now,

 

^_ 2^_

-1

= 22.4 L mol O

t t

V n (15)

 ^O2 4 1 ^{-3 3 3}

Or, = π×

3 22.4×10 m ^t

n t r (16)

Differentiating with respect to t , we get,

 ^O2 2 -3 3

d 1 d

= 4π×

dt 22.4×10 m dt

t t

t

n r

r (17)

Using Eq. (13), we get,

4π× 1 _{-3 3} ²d = ^eff[ _{H O2 2}] e₀ ^{- eff}

22.4×10 m dt 2

t k t t

r k

r n (18)

eff 2 2

2 eff -3 3 -

H O 0

Or, d = ×22.4×10 m ×[ ] e dt 8π

k t

t t

r r k n (19)

Integrating we get,

³ 3 ^{-3 3 H O2 2 0} ^{- eff} 

= ×22.4×10 m ×[ ] 1 - e 8π

k t

rt n (20) Again, the initial concentration of H2O2 is [H O ] = 8.75% = 2.6×10 mol m_{2 2 0} ^{3 -3}. The number of moles of H2O2 present initially in 5 mL solution is,[n_{H O2 2}] = 1.3×10 mol₀ ^-2 , which results in r_t³= 3.5×10 1 - e^-5 ^{-k teff}  m³.

Therefore, the acceleration of the bead, using Eq. (11), is given by,

 

 ^{- eff} 

= 2.62×10 1 - e5 ^{k t} - 0.45

a (21) We define the time taken by one polymer bead to go from the bottom of the vessel to

Chapter 5