Computer modeling on the kinetics of dissolution of silver halide dispersion by complexing agents

Computer Modeling on the Kinetics of Dissolution of Silver Halide Dispersion by Complexing Agents DANIEL D. F. SHIAO Research Laboratories, E a s t m a n K o d a k Company, Rochester, N e w York 14650

Received October 3, 1979; accepted January 30, 1980 A mathematical model for the dissolution of aqueous silver halide (AgX) dispersions by complexing agents has been developed to examine the experimental data reported by Shiao, Fortmiller, and Herz [J. Phys. Chem. 79, 816 (1975)]. This model takes into account the formation of all possible Ag + complexes at the AgX/solution interface and the dynamics of transferring them to the bulk solution. Numerical methods were used to solve a set of coupled rate equations and to compute the amount of dissolved AgX as a function of time. There is good agreement between the computed results and the experimental data correlating the effects of halide-ion concentration and A g X grain size on the rate of dissolution. Based on the magnitude of the apparent mass-transfer coefficients for various interfacial Ag ÷ complexes derived from this study, plausible mechanisms leading to the transfer of these complexes to the bulk solution are discussed. The electrical potential at the AgX surface plays an important role in the formulation of this model; its effect on the rate of dissolution by charged ligand is illustrated. In addition, the influence of enthalpies of AgX dissociation and of Ag+-complex formation on the temperature dependence of dissolution rates is examined. INTRODUCTION

where a' is a parameter that depends on factors such as ionic strength, halide excess, and ionic charge of the ligand, A is the effective surface area of the silver halide microcrystals per mole of AgX, K~p is the solubility product of AgX, L0 is the total ligand concentration, and/32 is the stability constant of the Ag+-ligand complex. Based on this empirical relation, a plausible mechanism for AgX dissolution, which can be described by Eqs. [2]-[4], has been proposed:

Computer modeling of chemical dynamics is a useful tool for chemists in elucidating quantitative mechanisms of complicated chemical systems. In colloid and surface chemistry, the rate processes taking place at interfaces are often complex, and such a technique is particularly valuable. The application of computer modeling to the kinetics of dissolution of silver halide microcrystals is discussed in this communication. As we shall see later, the use of this computational analysis does lead to further understanding of the mechanism of this important photographic process. The rates of dissolution of AgX dispersions have been studied extensively by Shiao and co-workers (1). The initial dissolution rate, ri, in the presence of complexing ligands was found to fit the equation ri =

o~'(AfizL~Ksp),

AgXs K~p Age+ + X/-, Ag~- + 2L/32 (agLz) +, k (AgL~)~ ~ (AgL~)~ulk.

[2] [3] [4]

Here s refers to the solid state, f signifies the AgX/solution interface, and the ratedetermining process is assumed to involve

[1]

535 Journal of Colloid and Interface Science, Vol.77, No. 2, October1980

0021-9797/80/100535-08502.00/0 Copyright© 1980by AcademicPress, Inc. All rightsof reproductionin any formreserved.

536

DANIEL D. F. SHIAO

diffusion of complexed Ag + from the surface into the bulk of the solution. This step is controlled by the rate constant k. The overall expression for the initial rate of dissolution of AgX derived from this mechanism is identical with Eq. [1], with 1

a' = kE

lira ~ , t~0 [X?]

[5]

where E is a variable introduced to account for coulombic effects associated with ligand and silver halide surface charges. Although this mechanism explains quantitatively the dependence of q on A, /32, L0, and Ksp, it does not provide a physical model for describing the surface dissolution process and the related charge effects. This communication discusses the formulation of such a model and shows how it provides additional information concerning the dissolution process of AgX from the analysis of previously published data.

the dissolution reaction and the concentration of the ith species. Since the total number of moles n in the sphere is n = 4rrR3/3 v

and a = 1 -

(Ra/R~),

where v is the molar volume of AgX, R0 is the initial radius of the sphere, and a represents the fraction of reacted AgX, it can be shown that da

--

dt

3v

- --

(l - a) z/a ~'. kiGi.

Ro

To compute Cf~, we assume that the concentration of the ith complex in the interface is proportional to the calculated value based on stability constants determined from bulk solution; namely, Cfi = y(~: /~iLf iJ-1 [Ag~])

Dissolution of AgX is a heterogeneous process involving surface reactions at nonmetal/solution interfaces. The kinetics of this class of reactions have been recently reviewed (2). The model for AgX dissolution which we are formulating is based on the assumption that the reaction products are soluble in the bulk solution and do not inhibit further dissolution reaction. A spherical shape is assumed for a AgX grain, although the results are equally applicable to cubes, octahedra, or other isometric shapes (2). From the above assumptions and the proposed mechanism indicated by Eqs. [2]-[4], it can be shown that dn

--

-

4~-R2 ~] kiCf~,

[6]

dt

where n is the number of moles of AgX molecules in a grain, R is the radius of the sphere, and k~ and Cf~ represent respectively the mass-transfer coefficient of Journal of Colloid and Interface Science, Vol. 77, No. 2, October 1980

[8]

i

and [Ag+] = K~./[X?],

FORMULATION OF THE MODEL

[7]

i

[91

where fi~ is the j t h stability constant for the ith species, Lf~ represents the free ligand concentration of the ith ligand at the solid/solution interface, X~- is the halide concentration at the interface, and y is a proportionality constant. According to our assumption indicated in Eq. [8], the value of y should be greater than one. The distribution of ligands between the interface and the bulk solution is calculated according to the Boltzmann equation, Lbi - e +z'e*/kr,

[10]

Lfi where Lb~ and Lf~ represent the bulk and interfacial concentration of the ith ligand (including halide ion), respectively, Z~ is the ligand charge, and tO is the potential of the charged AgX surface. Finally, the consideration of mass conservation in the bulk yields two more equations: Xg = Xo + a(AgBr)o - (AgBr)b -

2(AgBr~-)b,

[11]

MODELING OF AgX DISSOLUTION RATES Lbi = L~0 - (AgL+)b - 2(AgL+)b,

[12]

where the subscript 0 represents the initial concentration. Only the 1:1 and 1:2 Ag + ligand complexes are considered in Eqs. [11] and [12]; with the ligands to be discussed in this report, this simplification is not a cause of serious error. The combination of Eq. [7] with Eqs. [8]-[12] yields a differential equation relating o~ as a function of time, dc~ 3v - - - - - (1 - c~)213 ~ k~F~(t),

dt

[13a]

Ro

k~ = k~y,

[13b]

where F~(t) is a composite function o f t , the value of which can in principle be calculated. This equation cannot be s o l v e d in a closed form. H o w e v e r , it can be evaluated by numerical integration. The computation procedures are summarized in the following section. METHODOLOGY The numerical solution of Eq. [13a] is considered as an initial value problem for an ordinary differential equation. The Euler integration procedure (13) is the most often used and simplest method for tackling such a problem. It starts by calculating the concentration increment from a given rate equation and a chosen time interval. With the Euler approximation,

C~(to + At) = Cj(to) + ACj,

[14]

where Cj(to) is the initial C~ value, Cj at (to + At) can be calculated. Thus, by knowing the initial Cj value, the entire time dependence of Cj can be computed. The key to obtain successful results with this method is to use very small At values. With the availability of modern computers, this requirement is certainly easy to achieve. The application of the Euler technique to the solution of Eq. [13a] is straightforward if all the input constants and parameters are known.

537

The fl~ values for various ligands and the pKso and R0 values for AgX at experimental conditions were either known or available from the literature (see Table I). The molar volumes, v, of AgX were computed from their density values. Constants such as Z~ and T were known for a given ligand at a given temperature. The electrical potential, 4, at the AgX/ solution interface is a complicated quantity. In the presence of gelatin, the ~ potential of AgX is principally determined by the adsorbed gelatin layer (3), and hence it is difficult to determine ~b directly under our experimental conditions. F r o m C-potential measurements by Barr and Dickerson (4), a qJ value of - 4 0 mV at pBr 3 was suggested. In the absence of other more reliable information, we adopted this value in our computational analyses. With the above considerations in mind, the solution of Eq. [13a] is at hand if the apparent mass-transfer coefficients (k~) for various Ag + complexes formed at the AgX/ solution interface are available. Thus, with a given set of k~ values, a computed curve relating ~ and t can be obtained for each experiment, and the initial rate, ri, can be graphically evaluated. For a given ligand, the magnitude of various apparent masstransfer coefficients can thus be estimated by comparing either the entire " ~ vs t " curve or the ri value between the computed and the experimental results. For the convenience of presenting data, we define the following term: log (relative rate) = log (ri/r~f), where r~f refers to the initial dissolution rate at a reference state. The choice of the reference state is arbitrary and should not alter the interpretation of the results. RESULTS AND DISCUSSION

A. Dissolution Kinetics As mentioned above, the numerical solution of Eq. [13a] allows one to compute the Journal of Colloid and Interface Science, Vol. 77, No. 2, October 1980

538

DANIEL D. F. SHIAO 1.0

versely proportional to its edge length, this conclusion is equivalent to the results shown in Fig. 2,

,c~ 0.8 *o O .~

0,6

e-

0,4

C. Silver Halide Surface Charge

.9 LI.

0,2

I 5

1 10

t,min

FIG. 1. Fraction of AgBr dissolved as a function of time from the dissolution of a cubic AgBr dispersion (0.45-txm edge length) at 25°C in water containing 0.2M Na2SO3, 0.13M Na2COz, 0.007M NaBr, and 0.5% gelatin. Solid line is the calculated curve. fraction of dissolved AgX, a, as a function of time. Assuming an equal value of k; for the 1:1 and 1:2 Ag+-SO~ - complexes (see Table I), the calculated results are shown in Fig. 1 as the solid line. The good agreement between the experimental and the computed data demonstrates the usefulness of our model. As mentioned in our previous publication (1), the dissolution kinetics followed an apparent first-order plot; such a plot certainly cannot be deduced from the functional form o f E q . [13a], Thus, from our viewpoint, the apparent first-order plot for the kinetics of AgX dissolution has no fundamental significance.

B. Silver Halide Grain Size

For a neutral ligand, since Z~ in Eq. [10] equals zero, the dissolution rate of AgX by this ligand is unaffected by the silver halide surface charge. On the other hand, for a charged ligand, the electrical potential on the AgX surface plays an important role in determining its dissolution kinetics. The quantitative explanation of this effect was stated in Eq. [10]. For a given ligand such as SO~-, the predicted change in the relative rate of dissolution as a function of AgX surface potential, ~b, is indicated in Fig. 3. The results clearly show that a decrease in negative potential of the AgX surface leads to an increase in dissolution rate. This conclusion is consistent with the observation that reversibly adsorbed additives that reduce the negative charge density at the AgX/X- interface, e.g., cationic surfactants (5), enhance AgBr dissolution rates in sulfite and thiosulfate.

o 0.5

~D

v

With a given ligand, Eq. [13a] predicts the dependence of dissolution rate of AgX -0.5 on its original grain size. The good agreement between the predicted and the experimental results as shown in Fig. 2 gives strong support to our model. In our pre-I.0 vious discussion (1), it was concluded that -3.5 -4.o -4.5 - .o Log (0.5 x edge -length in cm) the dissolution rate was directly proportional to the average surface area of the FIG. 2. Effect of grain size on the rate of disAgX grain. Since for a given assembly of solution of AgBr dispersions. Experimental condicubic AgX grains, the surface area is in- tions as in Fig. 1. Journal of Colloid and Interface Science, Vol. 77, No. 2, October 1980

MODELING OF AgX DISSOLUTION RATES 2.0

A

2

.~ i.o 4-o O _1

I

I

1

-40

-20

0

~,rnV FIG. 3. Effects of surface potential of AgBr on dissolution rate. See text for explanation.

D. Bromide-Ion Concentration The initial rate of AgX dissolution at initial p B r > 3.0 was difficult to determine because of the large change in B r - concentration during the initial phase of the experiment. To obtain reproducible data, the rt values were m e a s u r e d 1 rain after the beginning of each experiment. The results are shown as open circles in Fig. 4. In the computer-simulation experiments, similar p r o c e d u r e s were used to determine the initial rates. The a p p a r e n t mass-

539

transfer coefficient of B r - was a s s u m e d to be the same as that of the monoanionic H z N C H 2 C O O - (see Table I). The solid curve in Fig. 4 represents simulated data assuming log/32 for B r - to be 8.6, compared with the value 7.3 determined experimentally in aqueous solution (6). The difference in log/32 value m a y be attributed to the difference in e n v i r o n m e n t b e t w e e n the AgX/solution interface and the bulk solution as it is a well-documented fact that the/32 value, for B r - m a y vary o v e r several orders of magnitude, depending on the polarity of the solvent (7). Figure 4 shows good agreement b e t w e e n computer-simulated and experimental data and d e m o n s t r a t e s again that our model is consistent with empirical o b s e r v a t i o n s . H o w e v e r , since the true initial rates at time zero a b o v e p B r 3 are difficult to measure, the empirical information in this figure should be treated strictly as such; namely, the ri values represent the rates measured 1 min after the beginning of each experiment.

E. Significance of the Apparent Mass-Transfer Coefficients As mentioned in the Methodology section, the apparent mass-transfer coefficients for various Ag + c o m p l e x e s can be estimated by matching the experimental results with

1,0

..........------%

-~ o

oj

o ._J

-I .0

I LO

I 2.0

I 3,0

I 4.0

I 5,0

pBr

Fro. 4. Effect of bromide-ion concentration on the rate of dissolution of a cubic AgBr dispersion (0.45-p~m edge length). Except for bromide, experimental conditions as in Fig. 1. Journal of Colloid and lnterj?tce Science, Vol. 77, No. 2, October 1980

540

DANIEL

D. F. SHIAO

TABLE

I

Thermodynamic Constants and Apparent Mass-Transfer Coefficients for Various Soluble Ag+ Complexes at 25°C k~ × 10~ (cm sec -~)

Ligand Br-

log fl~

log f12

AH~ (kcal/mole)

AH 2 (kcaYmole)

(0.7) a

5.0 °

(8.6) ~

-4.2 e

S O zz -

1.1

4.3 ~

7.4 b

(- 5.5) a

H2NCH2COO-

0.7

3.5 ~

6.4 b

--

--

H2NCH2CH2OH

1.6

3.4 b

6.8 o

--

---

HN(CH2CH2OH)2

1.1

2.5 b

5. I °

--

N(CH2CH2OH)~

1.6

1.6 °

3.3 °

--

$20~

0.008

7.6 a

12.5 a

-7.7 a

-8.4 e - 11.0a

-- 19.0a

Values in parentheses are assumed values. b d Data from the following references: bRef. (6); CRef. (9); dRef. (10). the computer-simulated data. The relevant constants used in computer simulation are summarized in Table I. The pKsp value for AgBr was taken to be 12.3 at 25°C. The apparent mass-transfer coefficients so obtained are also listed in Table I. In these calculations, the k~ value for the 1:1 and 1:2 Ag+-ligand complex were assumed to be equal. The transfer of material by diffusion from the bulk to suspended particles usually depends on the relative velocities of the fluid and particles. H o w e v e r , in the absence of convective diffusion, the solution of the Fickian equation for molecular diffusion through an infinite diffusion boundary layer leads to (11) k~ =

D~ R x 1000

[15]

where k~ and D~ are respectively the mass-transfer coefficient and diffusion coefficient of molecule i and R is the radius of the particle. The factor 1000 corrects the concentration unit from moles per cubic centimeter to moles per liter. For submicron AgX microcrystals, the justification of Eq. [15] was discussed by Wey and Strong (12). Thus, taking D~ = 2 x 10 -~ cm2/sec and R = 0.2 x 10-4 cm, the estimated masstransfer coefficient due to molecular diffusion is 1 x 10-3 cm/sec. In view of the Journal of Colloid and Interface Science, Vol. 77, No. 2, October 1980

k~ values listed in Table I and the fact that the true mass-transfer coefficient, k~, should be smaller than k~, it appears that the dissolution of AgX by complexing agents is too slow to be a diffusion-controlled process. Furthermore, if diffusion is assumed to be the rate-limiting step, there is no appropriate reason to explain the difference in k~ value between SO~- and $20~- as shown in Table I. Thus, we are led to consider other alternatives. Consider a (AgL+)f complex formed at the AgX/solution interface. If it does not diffuse to the bulk solution, some interaction, such as adsorption, must exist between the AgX surface and the complex. U n d e r such situations, two plausible mechanisms for mass transfer can be proposed: (i) mass transfer via desorption and (ii) mass transfer via ion exchange, namely, (AgL+)f + 2Lb ~ 2Lr + (AgL+)b. Both mechanisms can explain why the apparent mass-transfer coefficient for the Ag + complex of $20~- is much smaller than that of SO~-.

F. Temperature Dependence Using the known values of enthalpy changes and activation energies for various equilibrium and rate steps, the overall dependence of the initial rate of silver

541

MODELING OF AgX DISSOLUTION RATES

halide dissolution on temperature can be computed according to our model. For thiosulfate dissolution, with the AH values given in Table I and assuming the activation energies for its apparent mass-transfer coefficients to be 6 kcal/mole, the calculated temperature dependence is shown in Fig. 5 as the solid line. The overall activation energy for AgBr dissolution by thiosulfate according to this analysis is 10.5 kcal/mole, which agrees well with the experimentally determined value. Owing to a computational error, the latter value was incorrectly reported to be 6.4 kcal/mole in our previous publication (1). To illustrate the point of view that the overall activation energy for AgX dissolution depends strongly on the thermodynamics of formation of Ag ÷ complexes according to our model, the dotted line in Fig. 5 shows the simulated temperature dependence of AgBr dissolution by sulfite assuming the activation energies for the mass-transfer process (the rate-limiting step) of thiosulfate and sulfite to be equal. An overall activation energy of 23.5 kcal/ mole for SO~- is obtained, which can be compared with 10.5 kcal/mole obtained for thiosulfate. This calculation shows that the apparent activation energy may not necessarily give information about the ratelimiting step in the dissolution process. Thus, in the absence of information concerning the mechanisms of Ag÷-complex formation, one should question the validity of using such a value to argue for or against any specific dissolution mechanism. In summary, we have demonstrated that the proposed mathematical model for silver halide dissolution can explain the various experimental observations such as the effects of halide-i0n concentration, silver halide surface charge, and grain size on dissolution rate. In addition, we have shown that computer modeling (simulation) is a powerful tool in elucidating quantitative mechanisms of chemical reactions at interfaces.

0,0

¢~

0,2

.5 v

0.4

2

0.6

o.o

I

I

3.0

3.2

(+)

3.4

x,o

FIG. 5. Effect of temperature on the rate of dissolution of a cubic AgBr dispersion (0.45-txm edge length) in water containing 0.01 M Na2SzO3, 0.001 M NaBr, and 0.5% gelatin at pH 10. Solid and dotted lines are computer-simulated results for Na2S~O3 and NazSO3, respectively. See text for explanation.

REFERENCES 1. Shiao, D. D. F., Fortmiller, L. J., and Herz, A. H., J. Phys. Chem. 79, 816 (1975). 2. Eyring, H. (Ed.), "Physical Chemistry--Volume VII: Reactions in Condensed Phases," p. 449. Academic Press, New York, 1975. 3. Padday, J. F., J. Photogr. Sci. 11, 334 (1963). 4. Barr, J., and Dickerson, H. 0., J. Photogr. Sci. 9, 222 (1961). 5. Weiss, G., Ericson, R., and Herz, A., J. Colloid Interface Sci. 23, 277 (1967). 6. Shiao, D. D. F., Photogr. Sci. Eng. 20, 179 (1976). 7. James, T. H. (Ed.), "The Theory of the Photographic Process," p. 7. Macmillan, New York, 1977. 8. (a) Owen, B. B., and Brinkley, S. R., J. Amer. Chem. Soc. 60, 2233 (1938); (b) Wagman, D. D., and Kilday, M. V., J. Res. Nat. Bur. Stand. Sect. A 77, 569 (1973). 9. Pouradier, J., Venet, M., and Chateau, H., J. Chim. Phys. 51, 375 (1954). 10. Sillen, L., and Martell, A. (Eds.), "Stability Journal of Colloid and Interface Science, Vol. 77, No. 2, October 1980

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DANIEL D. F. SHIAO

Constants of Metal Ion Complexes." The Chemical Society, London, 1964. 11. Satterfield, C. N., and Sherwood, T. K., "The Role of Diffusion in Catalysis," p. 47. AddisonWesley, Reading, Mass., 1963.

Journal of Colloid and Interface Science, Vol. 77, No. 2, October [980

12. Wey, J. S., and Strong, R. W., Photogr. Sci. Eng. 21, 14 (1977). 13. Lapidus, L., and Seinfeld, J. H., "Numerical S01utiofi Of Ordinary Differential Equations." Academic Press, New York, 1971.