Modeling Thin Film CdS Development in a Chemical Bath Deposition

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Ind. Eng. Chem. Res. 2000, 39, 3272-3283

Modeling Thin Film CdS Development in a Chemical Bath Deposition Process M. Kostoglou, N. Andritsos, and A. J. Karabelas* Department of Chemical Engineering and Chemical Process Engineering Research Institute, Aristotle University of Thessaloniki, P.O. Box 1517, GR 540 06 Thessaloniki, Greece

The chemical bath deposition (CBD) process is currently favored for the preparation of CdS thin films of commercial interest. A chemical engineering type analysis of the CBD process is carried out in this paper to aid its design and optimization. Model equations are developed (based on a population balance formulation) for the temporal variation of reactant concentrations as well as of the solid phase, both in the bulk and on the substrate. A possible sequence of elementary mechanisms (i.e., nucleation, surface reaction, etc.) is suggested, and the resulting comprehensive model is solved numerically. Computational results show that the model is consistent with available experimental data on film thickness evolution. Furthermore, the influence of the process parameters on the results is studied theoretically and discussed extensively for several cases. The model may prove very useful for optimization of the CBD process with respect to its design variables (reactant concentrations, process time, etc.) as well as for efficient experimental determination of presently uncertain or missing parameter values. Introduction The significance of CdS as a wide energy band-gap semiconductor has greatly increased in the last 2 decades, mainly because of its application to photovoltaic solar cells, laser materials, and piezoelectric transducers. Among the available techniques for the preparation of CdS films, deposition from solution, and, more specifically, chemical bath deposition (CBD) appears to be a fairly simple and convenient method for large area coverage. Such a CBD process involves the slow release of sulfide ions via the controlled hydrolysis of thiourea in the presence of a cadmium salt and a chelating agent (commonly NH3) resulting in the precipitation of CdS on glass substrates.1,2 Despite its practical significance, the CBD process of CdS preparation has not been studied at all from an engineering point of view. The majority of relevant studies have apparently concentrated on the evolution of film characteristics during its growth, whereas the process mechanisms are still poorly understood.3 Until recently the common assumption was that the homogeneous precipitation of CdS was responsible for film growth. However, this is far from reality because homogeneous precipitation may actually prevent film growth through the consumption of cadmium and sulfide ions. A few years ago, Ortega-Borges and Lincot1 proposed a specific surface reaction mechanism based on the formation and adsorption of intermediates on the substrate surface, as a preliminary step to the formation of CdS, by the decomposition of these intermediates. Recently, Donã and Herrero2 improved this surface reaction model (at least in the range of concentrations they employed), taking advantage of their experimental results. During the past decade, considerable parallel activity has been evident (mainly in the chemical engineering literature) toward the quantitative description of pre* To whom correspondence should be addressed. Telephone: 30-31-996201. Fax: 30-31-996209. E-mail: karabaj@ alexandros.cperi.certh.gr.

cipitation processes. The main objective of this effort is to advance the treatment of precipitation processes beyond the level of pure empiricism and to model them using the extensive experience already gained from the study of chemical reactors (e.g., Marchal et al.4). Along these lines the CBD-CdS process is mathematically modeled here, using typical methods employed in the area of precipitation phenomena. However, a question may be raised as to whether there is really a need for a mathematical model of such a process. Obviously, a reliable model can help to better understand the system and the interaction between the input variables (e.g., reactant concentrations) on a quantitative basis. Beyond that, a rational model would be very useful in conjunction with experimental process optimization procedures, which are very constrained by the requirement to study the influence of one variable, while holding fixed all of the others (e.g., Oladeji and Chow5). Using the model, it is possible to examine the combined effect of the variables (reactant concentrations) and to identify an appropriate (relatively narrow) range of conditions for experimental optimization. For example, there are competing effects of ammonia in the film growth process, as will subsequently be discussed. On the one hand, ammonia prevents the undesirable homogeneous precipitation by forming complexes with Cd ions, but, on the other hand, it slows down the surface reaction. This implies that an optimum concentration of ammonia can be determined. The problem, however, is that such an optimum concentration is dependent on other reactant concentrations as well. This fact makes a full-scale experimental study of the system very tedious and the model invaluable for selecting reduced concentration ranges that are worth exploring experimentally. To be sure, quantitatively accurate predictions are not possible at present because of the uncertainty associated with some parameter values; for instance, even Ksp values for CdS differing by several orders of magnitude may be found in the literature. Nevertheless, the model can also be employed for the development of efficient experimental procedures in order to determine the uncertain or missing parameter values. Ultimately, if

10.1021/ie990472q CCC: $19.00 © 2000 American Chemical Society Published on Web 08/16/2000

Ind. Eng. Chem. Res., Vol. 39, No. 9, 2000 3273

reliable estimates of all parameters are obtained, the model can be used for global optimization of the process and even for model-based control. The approach pursued with the chemical system described in this work can also be followed for other systems of potential interest for thin film solar cells. The particular advantage of the chemical system used in the currently favored CBD process is that the driving force for film growth is different from that for bulk nucleation. Thus, there is an opportunity to adjust the relative magnitude of those driving forces in order to achieve (for a certain time period) film growth without appreciable bulk precipitation. Another feature of the chemical system employed here is that the supersaturation is brought about by a slow chemical reaction. The difference of this type of system compared to those in which the supersaturation is controlled by direct mixing of reactants (instantaneous nucleation) is well recognized. For example, the case of supersaturation induced by chemical reaction has been studied in the context of aerosol reactors,6 crystallization processes,7 and colloidal processes.8 Also, Okuyama et al.9 and Tandon and Rosner10 have studied the influence of bulk phenomena on deposit evolution in the context of chemical vapor deposition (CVD) processes. In the following section, a comprehensive mathematical model of the CBD process is developed and a method of solution is described. An oversimplified case of the complete model is solved first to show that it is capable of predicting qualitatively salient features of the system, already observed experimentally. The study of the complete model follows; it is first adjusted to the experimental film growth curves of Donã and Herrero,2 and model results for other problem variables are obtained and discussed. In particular, the influence of the ammonia concentration and pH on the process is examined. Furthermore, the relation between the film growth process and film quality is discussed. Model Development In a stirred vessel of active volume V (where a substrate of area A is immersed) specific quantities of CdSO4, (NH4)2SO4, NH4OH, and SC(NH2)2 are added; CdSO4 and SC(NH2)2 are used as sources of Cd and S ions, respectively. NH4OH is employed to control the ammonia concentration and (NH4)2SO4 as a pH buffer. Other salts may also be used instead of CdSO4, such as CdCl2, Cd(NO3)2, and Cd(CH3COO)2, but the analysis of the process remains the same. A schematic of the film growth process is shown in Figure 1, offering a simplified view of the contribution by the input variables (Cd, thiourea, NH3, and OH- concentration) to the elementary steps and of the main interactions. To model the system, one has to take into account the chemistry involved and the possible formation of a solid phase on the substrate surface as well as in the bulk of the solution. The chemistry is the basic building block of the proposed model, requiring a thorough quantification; a brief description follows. Bulk Chemistry.

-

NH4 + OH T NH3 + H2O CD

2+

+ 4NH3 T Cd(NH3)4

2+

Thiourea hydrolysis: SC(NH2)2 + OH- f HS- + H2O + CN2H2 HS- + OH- T S2- + H2O

kH (3)

K3e

(4)

CdS precipitation: Cd2+ + S2- T CdS(s)

Ksp

(5)

The subscript e designates equilibrium constants. Surface Chemistry (Mechanism Proposed by Don ˜ a and Herrero2).

Reversible adsorption of dihydroxodiamminocadmium complex: Cd(NH3)42+ + 2OH- + site T [Cd(OH)2(NH3)2]ads + 2NH3 (6) Adsorption of thiourea by formation of a metastable complex: [Cd(OH)2(NH3)2]ads + SC(NH2)2 f [Cd(OH)2(NH3)2SC(NH2)2]adsq (7) Formation of CdS and site regeneration by the metastable complex decomposition: [Cd(OH)2(NH3)2SC(NH2)2]adsq f CdS + CN3H5 + NH3 + 2H2O + site (8) which is the rate-limiting step in the mechanism. Donã and Herrero2 proposed the following film growth rate expression that corresponds to the above mechanism, using the stationary state approximation

r ) {k1k2Cs[Cd(NH3)42+][SC(NH2)2][OH-]2}/

Formation of the dominant tetraamminocadmium complex ion: +

Figure 1. Schematic of the CBD process. CdS film may grow by decomposition of a metastable complex [Cd(OH)2(NH3)2SC(NH2)2] on the surface (preferred mechanism) and by nuclei/particle deposition from the bulk.

{

k1[NH3]4 +k1[Cd(NH3)42+][OH-]2 + k2[SC(NH2)2] +

K1e

(1)

K2e

(2)

k1k2 [Cd(NH3)42+][SC(NH2)2][OH-]2 k3

}

(9)

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Ind. Eng. Chem. Res., Vol. 39, No. 9, 2000

where k1, k2, and k3 are the individual reaction constants and Cs is the concentration of sites on the substrate surface. These constants must be determined from experimental data, but the procedure is very difficult because of their involvement in the rate equation. Fortunately, for the range of concentrations used in CBD processes, the rate equation can be greatly simplified and the unknown constants may be reduced to one, which can be easily estimated from experimental data. Under typical CBD conditions, the concentration of ammonia is large with respect to the concentration of the reacting species; thus, it remains practically constant during the reaction. Also the OH- concentration remains constant. These two concentrations may be considered as input variables because they can be independently adjusted through the NH4OH and NH4(SO4)2 concentrations, respectively. Assuming that the fluid in the vessel is perfectly mixed, the film growth is surface reaction dominated. Therefore, using CCd to denote the total Cd species concentration in the system (in whatever form they exist) and CS for the total S concentration (except that corresponding to thiourea), one can derive the following set of equations describing the process. The symbols employed here are as follows: t ) time, F ) density of solid CdS, mw ) molecular weight of CdS, R ) volume of the nucleus (it is a function of supersaturation), S ) supersaturation ()[Cd2+][S2-]/Ksp)1/2), A ) substrate area, x ) particle volume, f(x,t) ) particle number density concentration, B(x,y) ) coagulation rate between two particles of volume x and y, D(x) ) deposition rate of particles of volume x, hr ) film thickness due to ionic addition, hp ) film thickness due to particle deposition, h ) total film thickness (hr + hp), r ) surface reaction rate, G(x) ) growth rate of particles of volume x, and H(S) ) nucleation rate as a function of supersaturation. Overall Cd Balance.

dCCd Ar F )dt V mw

∫0∞G(x) f(x,t) dx - mFRwH(S)

(10)

The terms in the right-hand side of eq 10 represent (from left to right) the Cd consumed for film growth, particle growth in the bulk, and generation of new nuclei. Thiourea Balance.

d[SC(NH2)2] Ar )- kH[SC(NH2)2][OH-] dt V F ∞ G(x) f(x,t) dx (11) (1 - δ) mw 0

∫

The terms in the right-hand side of this equation represent (left to right) consumption of thiourea for film growth, by hydrolysis, and for particle growth in the bulk. Overall S Balance (Excluding S in Thiourea).

dCS F ) kH[SC(NH2)2][OH-] - δ dt mw

∫0∞G(x) f(x,t) dx FR H(S) (12) mw

The terms in the right-hand side of eq 12 correspond to

(left to right) S produced by thiourea hydrolysis, S consumed for particle growth in the bulk, and S consumed for generation of new nuclei. Particle Population Balance.

∫0∞B(y,x-y) f(y,t) f(x-y,t) dy -

∂f(x,t) 1 ) ∂t 2 f(x,t)

∫0∞B(x,y) f(y,t) dy -

∂G(x) f(x,t) - D(x) f(x,t) + ∂x H(S) δ(x-R) (13)

where the first two terms of the right-hand side represent coagulation between particles, the third particle growth, the fourth particle loss (by deposition) and the fifth one-particle generation (nucleation). Film Growth Due to Ionic Addition.

dhr mwr ) dt F

(14)

Film Growth Due to Colloidal Particle Deposition.

dhp V ) dt A

∫0∞xD(x) f(x,t) dx

(15)

Chemical Equilibrium Relations.

[Cd2+][NH3]4 [Cd(NH3)4)]

) K2e

[S2-] [HS-][OH-]

) K3e

Using the above relations and species mass balances, one obtains

[Cd(NH3)4] )

CCd 1 + K2e[NH3]-4 [Cd2+] )

K2e[NH3]-4CCd 1 + K2e[NH3]-4 [S2-] )

CS 1 + [OH-]-1K3e-1

The assumption of chemical equilibrium without proper choice of the conserved quantities may be erroneous, as has been shown by Sorensen and Stewart11 for general chemical reaction systems and more specifically by Gandhi et al.12 in the context of precipitation. Here, the analysis described by Gandhi et al.12 for more complex systems is used implicitly in selecting appropriate quantities to be conserved. The particle growth rate G(x) may be, in general, a function of Cd(NH3)42+, SC(NH2)2, OH-, NH3, and S2- concentrations. Furthermore, if the particles grow directly by S2- addition, the parameter δ takes the value 1, whereas if they grow through a mechanism similar to that of film growth, the δ value is zero. Dimensionless quantities are introduced as follows:


xj )

x R0

t ts

τ)

F(x,t) )

R0f(x,t) N0

B h (x,y) ) tsN0B(x,y)

G h (x) )

tsG(x) R0

tsH(S) CCd D h (x) ) tsD(x) X) N0 [CdSO4]0 [SC(NH2)2] CS Y) Z) [SC(NH2)2]0 [SC(NH2)2]0 tsr R(S) S rj ) S h) R j) S0 R0 [CdSO4]0

H h (S) )

In these quantities the subscript 0 denotes the initial concentration, ts is the total time period of simulation, N0 is a reference total particle number concentration, S0 is a reference supersaturation, and R0 is the nucleus volume for the reference supersaturation. The model equations (10)-(13) are written as follows:

∂F(xj,τ) 1 ) ∂τ 2

∫0 B(yj,xj-yj) F(yj,τ) F(xj-yj,τ) dyj ∫0

∞

∂G h (xj) f(xj,τ) ∂xj D h (xj) F(xj,τ) + H h (S h ) δ(xj-R) (16)

B(xj,yj) F(yj,τ) dyj -

dX Arj )- C1 dτ V

∫0 Gh (xj) F(xj,τ) dxj - C1Hh (Sh ) Rj(Sh ) ∞

(17)

[CdSO4]0Arj

dY )dτ [SC(NH2)2]0V [CdSO4]0 C [SC(NH2)2]0 1

∫0∞Gh (xj) F(xj,τ) dxj - C2Y

[CdSO4]0 dZ ) C2 Y CH h (S h) R j (S h) dτ [SC(NH2)2]0 1

(18)

(19)

Deposited Mass.

dhr ) C3rj dτ

(20)

∫0∞ xjDh (xj) F(xj,τ) dxj

(21)

dhp V ) dτ A

where C1 ) N0R0F/mw[CdSO4]0, C2 ) kH[OH-]ts, and C3 ) mw[CdSO4]0/F. To proceed with the simulation task, appropriate models must be utilized to determine the rates for the individual phenomena involved, i.e., nucleation, particle growth, coagulation, and deposition. Rates of Physicochemical Processes. (a) Nucleation. The size of a stable nucleus (critical size), R, is obtained from the classical theory of nucleation (e.g., Dirksen and Ring13)

(

4Vmσ 1 R) π 6 nRT ln(S)

)

H ) Ac exp(-∆Gcrit/kBT)

(23a)

3 2 16π σ Vm 3 (k T ln S)2

(23b)

where

∆Gcrit )

B

xj

F(xj,τ)

supersaturation, the theory of homogeneous nucleation is valid only in the case where the time scale of supersaturation change is much larger than that of the molecular collision process (i.e., molecular cluster distribution is in a quasi-steady state). For large values of supersaturation, eq 13 gives a small nucleus size, i.e., a small number of molecules per nucleus. If that number is smaller than the number of molecules constituting an elementary CdS crystal (4 molecules; Weast14), then the latter is considered to represent the nucleus. The classical theory of homogeneous nucleation can be used to estimate the nucleation rate (the number of nuclei formed per unit of volume and time); i.e.,

3

(22)

where Vm is the molecular volume of CdS, R the ideal gas constant, and σ the interfacial tension. Unlike the majority of published works, this modeling effort takes into account the dependence of critical size on supersaturation (evolving critical size). For time-dependent

kB is the Boltzmann constant, and Ac is a preexponential factor. According to Nielsen,15 this factor is given by

Ac =

Dn Vm5/3

(23c)

where Dn is the diffusivity of the nucleus. At present it appears that there is no method independent of the nucleation process to determine the surface energy of solids. The only available independent measurement in the literature seems to be that for SrSO4.16 An extensive review of the experimentally determined values of σ is presented by So¨hnel and Garside,16 where it can be seen that the estimated surface energy of a salt differs significantly from one experimental study to another. Furthermore, Narsimhan and Ruckenstein17 suggest that the surface tension of the tiny nuclei is generally smaller than that observed macroscopically, i.e., that the surface tension is a function of the nucleus size. Obviously, the literature values of the surface energy cannot be trusted for use in the present model. Furthermore, under realistic conditions heterogeneous nucleation usually takes place. For heterogeneous nucleation, the solid surface energy may be replaced by the apparent one, σa, which is a function of the conditions of the solvent (including the presence of impurities). The two possible procedures for the experimental determination of σa, for a particular solution used in the CBD process, are the film growth rate and the measurement of solution light absorbance. The former involves fitting of the present model to the film growth rate data; the latter is described by So¨hnel and Garside16 and was employed by Kostoglou et al.18 for PbS precipitation. (b) Particle Growth. It is assumed that the mechanism controlling particle growth is the surface reaction. This is consistent with the experimental observation that the particles in the process do not grow to a large size. Typically, in similar studies it is assumed that the surface reaction mechanisms for particle growth and film growth are different. However, upon closer inspection one may claim that the intrinsic mechanism of film growth is a “homogeneous” one, i.e., that CdS grows on a CdS surface. The substrate material may influence

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only the number of seeds that grow to form the film. As will be discussed in a subsequent section, the above mechanism for film growth seems to be consistent with the experimental observations regarding the film growth rate as well as film morphology. Therefore, the two surface processes (film growth and particle growth) are considered to be identical. This approach has the additional advantage that it reduces the degrees of freedom of the system under consideration. It will be noted that, unlike film growth rate, the growth rate of small particles is extremely difficult to be directly assessed experimentally. By considering that particle growth rate is equal to the easily determined film growth rate, one may avoid additional assumptions regarding the former (mode of growth, undetermined constants that must be assessed by indirect methods). Using the above approach and assuming a spherical shape, the following equation is obtained for the particle growth rate:

G(x) ) (36π)1/3x2/3

mwr F

(24)

(c) Coagulation. Colloidal particle coagulation in a flow field is promoted by Brownian motion (Brownian coagulation) and by the relative motion of particles due to flow (shear or turbulent coagulation).19 Because Brownian coagulation controls in cases with small particle sizes (as in the CBD process), the flow effects may be safely neglected. The Brownian coagulation rate is given as

B(x,y) )

(

)

2kBT x1/3 y1/3 2 + 1/3 + 1/3 3µWc y x

0.3Dp2/3 u∞1/2 Wdν1/6L1/2

Mi )

(25)

where µ is the viscosity of the liquid and W the stability ratio of the system. The latter accounts for the effect of interparticle interaction (double layer, van der Waals, hydrodynamic) on the coagulation rate. In general, this stability ratio is a very complicated function of particle sizes x and y. It also depends on other particle parameters such as the ζ potential and Hamaker constant (which must be determined experimentally) and on solution composition, through the double layer thickness (Elimelech et al.20). Furthermore, the consideration of the exact form of the stability ratio creates serious difficulties for the solution of population balance equations (Kostoglou and Karabelas21). Therefore, for the present level of sophistication, it is convenient to assume that W is constant for any (x, y) pair of values. (d) Particulate Deposition. Recently, Carbonell and Riley22 studied the particulate deposition on submerged surfaces in stirred vessels both experimentally and theoretically. The result of their approach using a mass-momentum transfer analogy and the boundary layer analysis of Blasius is

D(x) )

employed.23 As regards the deposition stability ratio, the same comments and simplification are applied as those in the case of the coagulation stability ratio. Solution Procedure. The complete system of equations (16)-(21) must be solved numerically. Such a numerical solution is not trivial because of the nonlinear partial integrodifferential form of the population balance equations combined with the “stiff” behavior that arises from the chemical system. There are accurate direct discretization methods for tackling the population balance, but they require significant computational effort.24 In cases where there is no need to determine the exact particle size distribution (PSD), but only an integral quantity (as in the present case where the effect of PSD on the film growth rate is of primary interest), the method of moments is a very attractive alternative because of its simplicity and reduced computational requirements.25 Until recently, the main drawback of the moments method was the restriction in the number of unknowns allowed (up to three for a log-normal distribution). Recently, Barret and Jheeta26 and McGraw27 developed two different hierarchies of moment methods in which the accuracy increases with the number of unknowns and without the need to specify a certain form of the PSD. The method of Barret and Jheeta26 can be used for rate functions which have some particular properties, whereas the McGraw27 method is quite general. In this work the Barret and Jheeta method is used because it is somewhat simpler. The dimensionless moments of the PSD are defined as

∫0∞xjiF(xj,τ) dxj

Writing the rate functions in the forms G(xj) ) Gcxjg, D(xj) h (xj,yj) ) Bc(2 + xj1/3yj-1/3 + xj-1/3yj1/3), ) Dcxjd, and B multiplying eq 16 with xi, and integrating for all particle sizes, the following system of differential equations for the moments of the distribution is obtained:

dMi dτ

) iGcMi+g-1 - DcMi+d + 1 i-1 i! (2Mi-jMj + Mi-j+1/3Mj-1/3 + Bc 2 j)1j!(i - j)!

∑

h (S h) R j i(S h) Mi-j-1/3Mj+1/3) + H for i ) 0, 1, 2, ..., n (28) The other equations remain unchanged except the integrals including the growth rate and the deposition rate that must be substituted by the expressions GcMg and DcMd, respectively. The fractional moments in the above equations can be related with the integer moments via a Lagrange interpolation procedure using the logarithm of the moment as the dependent variable and the moment index as the independent variable, i.e.,

(26)

n

n

where Dp is the diffusivity of particles of volume x, ν the kinematic viscosity of the solution, u a characteristic velocity of the fluid, L the characteristic length of the submerged surface, and Wd the stability ratio for particulate deposition. This deposition rate has exactly the same form as that resulting from turbulent eddy diffusivity theory where the wall friction velocity is

(27)

log(Mx) )

log(Mi) ∑ i)0

∏ (x - j)

j)0,i*j n

(29)

∏ (i - j)

j)0,i*j

Selecting the number of moments equations, n ) 2 corresponds to the well-known and very extensively


used method of the log-normal distribution (e.g., Pratsinis28). This method is a very good approximation for all of the phenomena considered in the present case, provided that the dispersivity of the distribution is not very large.21 Unfortunately, the hydrolysis reaction step (of the present reaction system) gives rise to continuous nucleation and to extremely broad size distributions. The inadequacy of the log-normal method to tackle problems where nucleation is driven by a chemical reaction has been pointed out in aerosol literature.6 In the present work an attempt is made to improve the approximation using four moments equations (n ) 3). Model Analysis Simplified Model. Initially an oversimplified version of the problem is solved in order to demonstrate the capabilities of the whole approach as well as the potential practical benefits. Subsequently, the complete model is examined. In general, modeling the solid-phase generation is done through the use of population balance equations, which renders the problem very complex; the gradual increase of supersaturation leads to continuous nucleation and growth, which are strongly interrelated. To avoid this complexity (in this particular example), it is convenient to make a “zero-order” approximation by substituting the continuous nucleation and growth phenomena with instantaneous ones occurring at some (phenomenological) supersaturation value Sc. This means that during the time period when S < Sc there are no particles in the system, whereas at the moment when S ) Sc, the film growth stops. The simplified model requires the solution of eqs 1012 and 14 with H(S) ) 0 and f(x,t) ) 0. Using simple species balances, one can observe that in typical CBD processes the amount of the reactants (Cd and S) consumed to form the film is very small (up to 2%) in comparison with their total mass in the system. Therefore, a further simplification can be made, i.e., that the film growth does not influence the total Cd and S concentrations in the system, which means that the first term of the right-hand side of eqs 10 and 11 can be omitted. Finally, for concentrations typical of CBD, the surface reaction rate is given in a simplified form by Donã and Herrero2 as

r ) k0

[CdSO4][SC(NH2)2][OH-] [NH3]2

(30)

Taking into account all of the above simplifications (including that for the conditions under consideration 1 + K1e/[NH3]4 ≈ 1 and 1 + [H2O]/([OH-]K2e) ≈ [H2O]/ ([OH-]K2e), the reduced model can readily be solved to give the following equations for the evolution of film thickness by surface reaction and supersaturation:

h)

mwk0[CdSO4]0[SC(NH2)2]0[OH-] (1 - e-kH[OH ]t) kHF[NH3]2 (31)

S)

K1ekH[CdSO4]0[SC(NH2)2]0[OH-] (1 - e-kH[OH ]t) 4 Ksp[H2O][NH3] (32)

These two equations combined with the condition that the film growth stops at time tf when S ) Sc (at which

Figure 2. Final film thickness hf versus ammonia concentration (simplified model).

the film thickness reaches its final value hf) constitute the simplest possible model, which despite its simplicity can explain some salient features of the process. Indeed, the competing trends concerning the concentration of NH3, observed experimentally by Oladeji and Chow5 and shown in their Figure 1, can be explained with the above model. In Appendix I an explicit relationship is derived between the film thickness and NH3 concentration. It is clear that, on the one hand, a decrease in the ammonia concentration tends to enhance the film growth rate, but, on the other hand, it increases the supersaturation level (which promotes the undesirable bulk precipitation), leading to a reduction of the film growth. Thus, there must be an optimum ammonia concentration. The same holds true for the OH- concentration, which, however, has the opposite effect compared to that of ammonia. Furthermore, an increase of the OH- concentration tends to accelerate the process (reduction of the typical film formation time tf) through the hydrolysis reaction. Additionally, there is an upper limit in the concentration of OH- to avoid Cd(OH)2 precipitation. To demonstrate the capabilities of the above elementary model, numerical results are obtained for a particular case. All of the values of the variables and constants are taken from the literature except Sc (which is by definition a parameter to be determined) and kH. For these parameters, values are selected that give results of the same order as the corresponding experimental data. Here the variation of the film thickness will be examined with respect to NH3 and OH- concentrations, for a particular reactant concentration. According to the model, the final film thickness hf is independent of the OH- concentration, and it is proportional to the square of the NH3 concentration. The dependence of hf on the NH3 concentration is shown in Figure 2. The time tf required to reach this final film thickness is shown in Figure 3 as a function of the ammonia concentration for three typical pH values (OH- concentrations). These two figures could be used as a “nomogram” for design purposes; i.e., one could select directly from Figure 2 the appropriate ammonia concentration for a desirable film thickness. Then the required solution pH may be obtained from Figure 3 for a specified growth time and the already selected ammonia concentration. The optimal design of the process requires that the corresponding pH curve displays a rather sharp change of slope near the selected ammonia concentration. This choice warrants that the region of

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Figure 3. Duration tf of the film growth process versus ammonia concentration (simplified model).

Figure 4. Film thickness, h, at time t ) 1000 s versus ammonia concentration for three pH values (simplified model).

low reaction rate is avoided and that adequate use of thiourea is made during the process. In Figure 4, the film thickness at a specific time (t ) 1000 s) is displayed versus ammonia concentration, for several pH values. Obviously, the simple model can satisfactorily represent the characteristic dependence of the film thickness on ammonia concentration that has been found experimentally.5 In conclusion, the above “zero-order” model can describe the main features of the process qualitatively, and it is considered a significant step toward better understanding of the process. However, the complete process simulation (described in the next section) is deemed necessary in order to proceed with the practical (quantitative) application of the model. Comprehensive Model. In this case the system of equations (17)-(21) and (28) must be solved numerically. This system is very stiff because of the combination of chemical reactions with nucleation, necessitating an implicit Runge-Kutta integrator with a self-adjusted step and prespecified accuracy.29 Even so, for a relative accuracy of 10-5, several thousand steps are required. Preliminary calculations show that, for typical values of the characteristic fluid velocity and for conditions similar to those of the Donã and Herrero2 experiments, the resulting thickness of a particulate deposit (for deposition stability ratio Wd ) 1) is 2 orders of magnitude larger than the thickness of an “ionic”-type deposit. However, no appreciable particulate deposit has been observed experimentally. This may imply that the

Figure 5. Comparison between experimental (Donã and Herrero2) and theoretical film thickness evolution for two temperature values ([NH3] ) 1.68 M, pH ) 11.7).

deposition stability ratio Wd has a large value. The computations indicate that particulate deposition starts after a film of “ionic” deposit has developed. Therefore, the suspected large value of Wd may be due to the CdSCdS repulsion and not to the CdS-substrate interaction. Recalling previous arguments, one would also expect large Wc values. In view of these observations, for the present simulation, a value of Wc ) 1000 is considered (quite typical because the sensitivity around this value is small) and Wd ) ∞ (because no particulate deposit has been reported under the conditions of simulation). One may envision as follows the sequence of steps during the film growth and precipitation process. Initially the film grows linearly with time, under almost constant reactant concentration, while the supersaturation gradually increases until it reaches a value high enough for the onset of nucleation and crystal growth phenomena. At this instance the consumption of reactants due to bulk processes may cause a drastic decrease of the film growth rate, effectively arresting film thickness growth at a certain level. The film growth rate constant can be easily estimated from experimental data of the linear (initial) growth rate period. The time required for the termination of the film growth depends on the hydrolysis rate constant, which dominates the rate of supersaturation increase. In Figure 5 comparison is made of experimental data2 for CdS film thickness growth versus time (for two temperatures) with the simulation results. Actually, the only fitting parameter here is the surface energy σa, because the surface reaction rate constant k0 is given by Donã and Herrero2 based on the initial film growth rate. The surface energy value resulting from the fitting procedure (σa ) 492 N/m) is rather large because typical values seem to be smaller than 200 N/m (So¨hnel and Garside16). It must be pointed out, however, that σa appears in the model in combination with the solubility product Ksp, for the appropriate value of which there is much uncertainty. The value 10-25 suggested by Donã and Herrero2 is used here, while Oladeji and Chow5 suggest the value 10-28. If the true Ksp value is larger that 10-25, then the value of σ that would match the experimental results would be lower. The model results at the temperature of 75 °C are obtained under the assumption that the hydrolysis constant has the same temperature dependence (i.e., the same activation energy) as the surface reaction constant, which is accept-


Figure 6. Reactant concentration evolution at temperature T ) 65 °C ([NH3] ) 1.68 M, pH ) 11.7).

Figure 7. Final film thickness hf versus ammonia concentration for three pH values (complete model).

able. In general, from Figure 5 one may conclude that the model has (at least) the capability to satisfactorily fitting the experimental film thickness evolution. The reactant concentration variation for the case with temperature T ) 65 °C is shown in Figure 6. There is an initial period (approximately 300 s) during which thiourea is consumed through hydrolysis to give HSand S2- ions (CS). In this period the CCd concentration remains practically constant because its consumption for the film growth is negligible. Thus, the film growth rate is linear with a small departure from linearity due to the decrease of the thiourea concentration. After this period, the CS concentration reaches the threshold value for the onset of nucleation. The consumption of thiourea and of Cd becomes significant because of the particle growth in the solution bulk. At this stage thiourea is consumed primarily in two ways: through hydrolysis and particle growth in the bulk. After a sufficient period of time, there is no thiourea left and the system becomes chemically inactive; film growth stops, and a final film thickness is attained. The final film thickness resulting from model calculations is shown (versus ammonia concentration) in Figure 7. This figure corresponds to Figure 2 obtained from the simplified model. One observes that the qualitative behavior of the simplified model is physically correct; i.e., the final film thickness increases with the ammonia concentration and has a slight dependence on pH. However, it does not perform quantitatively well because it predicts a rather strong dependence of the

Figure 8. Film thickness h at time t ) 1000 s versus ammonia concentration for three pH values (complete model).

Figure 9. Film thickness evolution for several values of ammonia concentration (pH ) 11.7).

final film thickness on the ammonia concentration as compared to a weaker one of the complete model. Figure 8 is the counterpart of Figure 4 for the complete model. The qualitative behavior is similar, but the complete model predicts a much smoother dependence of the film thickness on the NH3 concentration. The value of the optimum NH3 concentration is fairly close to that of the simplified model for the pH ) 11.5 case but much larger for the other pH values. For a specific pH value, the optimum ammonia concentration that the two models predict can be adjusted through the selection of a Sc value; however, the two models predict quantitatively different dependences of the optimum NH3 concentration on pH. Figure 9 depicts the film thickness evolution for several values of the ammonia concentration. An increase of the ammonia concentration leads to an increase of the initial growth rate and to a decrease of the effective termination time (i.e., time to approach the final film thickness) as expected. The evolution of the driving forces for the phenomena of nucleation (supersaturation) and of film and particle growth (Gr ) [CdSO4][SC(NH2)2]) is shown in Figure 10, in dimensionless form, for the case with T ) 65 °C. As expected, the driving force for the film growth decreases monotonically with time. The behavior of supersaturation is more interesting; after a steep initial increase, it reaches a maximum and then decreases slowly. This means that, in relation to simple models of nucleation, this behavior is closer to the constant nucleation rate model

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Figure 10. Evolution of the normalized supersaturation S/S0 and the normalized driving force for particle growth Gr/Gr0 ([NH3] ) 1.68 M, pH ) 11.7).

Figure 11. Particle number concentration evolution for several ammonia concentration values (pH ) 11.7).

(after an initial period, of course) than to the instantaneous nucleation mode. The consequences on the PSD will be examined next. The evolution of the particle number concentration for several values of the ammonia concentration is shown in Figure 11. For the case of low ammonia concentration, the supersaturation rises sharply, leading to the production of a large number of particles. The latter grow and consume the reactants, resulting in a significant reduction of the nucleation rate. The reduction in the particle number after the early nucleation burst is attributed to particle coagulation. In general, as the ammonia concentration is reduced, the situation tends to resemble the case of instantaneous nucleation. In the case of large ammonia concentration, the supersaturation rises slowly and the (relatively) smaller number of generated particles is not enough to lead to a rapid reactant depletion. This case resembles the constant nucleation rate model; the number of particles tends to remain fairly constant, after an initial period, because of the simultaneously occurring opposite processes of nucleation and coagulation. Figure 12, depicting the temporal evolution of the mean volume equivalent particle diameter, is in accord with the above observations. The apparent rate of increase of the mean particle size is smaller for the large ammonia concentration; indeed, the large particle numbers (see Figure 11 for large times) are accompanied by large coagulation rates, because of the continuous generation of nuclei that keep the mean size small. In

Figure 12. Volume equivalent mean particle diameter evolution for several ammonia concentration values (pH ) 11.7).

Figure 13. Evolution of the dispersivity of the PSD for several ammonia concentration values (pH ) 11.7).

this case the PSD has a tendency to spread at the lower side because of continuous nuclei generation and at the upper side because of creation of large particles by coagulation. This leads to very large particle dispersivities, as shown in Figure 13. The dispersivity of the distribution, defined as ln(M0M2/M12), is a measure of its nonuniformity. For comparison, it will be noted that the asymptotic dispersivity for Brownian coagulation (Lee30) is ln(2). The dispersivities shown in Figure 13 are indeed very large and can only be the result of a continuous nucleation process. Very recently, Sathyagal and McCormick31 have commented on such large dispersivities in a study of a simplified model with constant nucleation rate and Brownian coagulation. For a small ammonia concentration and large times, coagulation prevails over nucleation and the dispersivity of the distribution tends to decrease. On the contrary, for a large ammonia concentration, the dispersivity keeps increasing with time. The first part of the dispersivity evolution (for a large ammonia concentration), before the local maximum, is of limited interest because of the relatively small number of existing particles. At the local maximum of the dispersivity, the particle number is apparently sufficient for the onset of significant coagulation; this tends to reduce temporarily the dispersivity, but eventually nucleation prevails and the dispersivity increases again. Comments on Film Growth and Morphology. In the work of Oladeji and Chow,5 the concept of induction time is introduced. The experimental film growth curves


observations. For this reason, it is believed that the same approach should be extended to include more realistic models of transient nucleation and more sophisticated (than “grain in cell”) spatial interaction models between the grains. With the incorporation of the new approach for film growth, the general model is capable of explaining qualitatively the trends obtained experimentally by Oladeji and Chow5 (their Figures 2-4). For example, the model shows that with decreasing the initial reactants concentration the final film thickness tends to increase. Concluding Remarks

Figure 14. Time evolution of the dimensionless film thickness resulting from the “grain-in-cell” model, for two values of the nuclei surface density (c is a constant), indicating induction period.

clearly show that there is a delay before the film starts growing linearly. The induction time in some of their experiments was as large as 50 min. One may surmise that the induction time increases with decreasing initial supersaturation (see Figures 2 and 4 of Oladeji and Chow5). In the Donã and Herrero2 experiments, the initial reactants concentration is much larger; thus, the induction time is too small to be observed. The classical theories of film growth predict only a linear film growth rate, which is a function of the substrate properties. The new theory proposed here, i.e., that the linear part of the film growth is determined only by the CdS properties (homogeneous growth), is compatible with the possible existence of induction time, as will be outlined using the following simplified arguments. As a first step in the process, a number of nuclei are created quite uniformly on the substrate. The surface number density of these nuclei is, in general, inversely proportional to the supersaturation and depends on the substrate properties. In the second step, each nucleus grows to form a grain in the homogeneous way. This means that the initial growth rate may be very small, restricted by the small surface area of available nuclei. However, later, when the immobilized grains grow and tend to “coalesce” with each other, total substrate coverage can be achieved with the concomitant linear growth rate. This mechanism may also influence the film morphology. For a time period, prior to the “coalescence” stage, the deposit consists of discrete grains (undesirable film quality); for longer times the film tends to become coherent and fairly uniform (good quality). This is in agreement with the experimental observations of Donã and Herrero2 that the film quality depends on the initial concentration of reactants. Using a simple “grain-in-cell” approach (Appendix II), one can determine the film growth rate that corresponds to the above simplified model. The computed film thickness evolution is depicted in Figure 14, where the linear growth rate and two cases of the new model (with a 1:4 ratio of densities of nuclei) are included. The time units are arbitrary because the rate is invariant with respect to time and nuclei density. In general, the induction time appears to be inversely proportional to the square root of the nuclei number density on the substrate. Although this new model for film growth is very simple, it has proved capable of predicting the existence of induction time and it is consistent with experimental

In this work an analysis is made of the CBD process for the preparation of thin CdS films. In the literature this process is usually treated as a “black box”, the only relevant reported studies involving an examination of instantaneous reactant concentrations and of the film growth rate. Here the principles of precipitation engineering are employed to describe the entire process, i.e., the relation between the film thickness evolution and the design parameters (initial species concentration and geometrical features). To this end, the recently proposed model by Donã and Herrero2 for the surface reaction is used, and a possible sequence is suggested for the elementary mechanisms taking place during the process. The resulting model seems capable of explaining the experimentally observed behavior of the system. The complete model is rather complicated; thus, a simplified version is also examined, which exhibits the same qualitative behavior as the complete one but requires additional (phenomenological-type) parameters. At present the model is validated only for a particular set of conditions (at specific temperature and Cd and thiourea concentrations) although it is very general in its derivation. To further test the model, additional experimental results are needed for several values of the above parameters. However, even at this stage, the model can be used for (at least partial) optimization of the process with respect to NH3 and OH- concentrations. As shown in Figure 8, the film grows linearly initially, but with time the growth rate is reduced, becoming asymptotically zero. The optimum design conditions apparently require that the process be stopped at a moment when the growth rate is no longer linear but still considerable (for example, a specified percentage of the initial growth rate); for smaller process time there may be poor utilization of reactants, whereas longer processing time may be undesirable because of economic penalties or off-specification film properties. Thus, for specified design parameters (e.g., final film thickness and process time), one must choose the NH3 concentration and pH in order to achieve near-optimum operation. The present model can be used for this choice. To the best of the authors’ knowledge, this work constitutes the first attempt to study comprehensively the CBD process for CdS film preparation from an engineering point of view. However, additional effort is required for further model development and validation as well as for exploiting this tool for parameter estimation. Acknowledgment Financial support by the European Commission under Contract JOR3-CT97-0124 is gratefully acknowledged.

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Appendix I. Results from the Simplified Model (Figures 2-4) Equations 31 and 32 are rewritten in the following simple form:

h ) c1(1 - e-c2t)

(A1)

S ) c3(1 - e-c2t)

(A2)

The constants c1, c2, and c3 can be determined by comparing eqs A1 and A2 with eqs 31 and 32. Figure 3 results easily by setting S ) Sc and t ) tf in eq A2 and solving for tf

tf ) -

( )

Sc 1 log 1 c2 c3

(A3)

Figure 2 is obtained by introducing the expression for tf in eq A1

hf )

c1 S c3 c

(A4)

The curves in Figure 4 consist of two parts: The film thickness is equal to the final film thickness and is computed from eq A4 whenever S > Sc. If S < Sc, the film thickness is computed from eq A1. Equation A1 shows a quadratic dependence on the ammonia concentration, whereas eq A2 shows an inverse quadratic dependence. Appendix II. Film Growth “Grain-in-Cell” Model Let N be the number density of nuclei on the substrate. According to the “grain-in-cell” model (similar to the “particle-in-cell” model used in the study of dense suspensions32), to each nucleus corresponds a circular area of radius a ) (1/πN)0.5. The entire substrate is assumed to be covered by such unit cells. Initially a grain is grown as a hemisphere, but after a certain period of time, it reaches a neighboring cell boundary and loses its hemispherical shape. From a quantitative point of view, the mean film thickness may be estimated by dividing the film volume in a cell by the cell substrate area. If k is the growth rate of the individual grain, the mean film thickness can be evaluated as follows:

h) h)

1 πa2

2(kt)3 3a2

for kt < a

(B1)

∫02π∫0ax(kt)2 - r2r dr dθ ) 2 [(kt)3 - ((kt)2 - a2)1.5] 3a2

for kt > a (B2)

The film evolution curve is continuous because the second equations tend to the first one as kt tends to a. By an expansion of eq B2 for kt . a, it can be seen that the film growth rate takes the expected linear form h ) kt. Literature Cited (1) Ortega-Borges, R.; Lincot, D. Mechanism of chemical bath deposition of cadmium sulfide thin films in the ammonia-thiourea system. J. Electrochem. Soc. 1993, 140, 3464.

(2) Donã, J. M.; Herrero, J. Chemical bath deposition of CdS thin films: An approach to the chemical mechanism through study of the film microstructure. J. Electrochem. Soc. 1997, 144, 4081. (3) O’Brien, P.; McAleese, J. Developing an understanding of the processes controlling the chemical bath deposition of ZnS and CdS. J. Mater. Chem. 1998, 8, 2309. (4) Marchal, P.; David, R.; Klein, J. P.; Villermaux, J. Crystallization and precipitation engineeringsI. An efficient method for solving population balance in crystallization with agglomeration. Chem. Eng. Sci. 1988, 43, 59. (5) Oladeji, I. O.; Chow, L. Optimization of chemical bath deposited cadmium sulfide thin films. J. Electrochem. Soc. 1997, 144, 2342. (6) Landgrebe, J. D.; Pratsinis, S. E. Gas-phase manufacture of particulates: Interplay of chemical reaction and aerosol coagulation in the free-molecular regime. Ind. Eng. Chem. Res. 1989, 28, 1474. (7) Den Ouden, C. J. J.; Thompson, R. W. Analysis of the formation of monodisperse populations by homogeneous nucleation. J. Colloid Interface Sci. 1991, 143, 77. (8) Bogush, G. H.; Zukoski, C. F. Uniform silica particle precipitation: An aggregative growth model. J. Colloid Interface Sci. 1991, 142, 19. (9) Okuyama, K.; Huang, D.; Seinfeld, J. H.; Tani, N.; Kousaka, Y. Aerosol formation by rapid nucleation during the preparation of SiO2 thin films from SiCl4 and O2 gases by CVD process. Chem. Eng. Sci. 1991, 46, 1545. (10) Tandon, P.; Rosner, D. E. Co-deposition on Hot CVD surfaces: Particles dynamics and deposit roughness interactions. AIChE J. 1996, 42, 1673. (11) Sorensen, J. P.; Stewart, W. E. Structural analysis of multicomponent reaction models, Part II. Formulation of mass balances and thermodynamic constraints. AIChE J. 1980, 26, 104. (12) Gandhi, K. S.; Kumar, R.; Ramkrishna, D. Some basic aspects of reaction engineering of precipitation processes. Ind. Eng. Res. Chem. 1995, 34, 3223. (13) Dirksen, J. A.; Ring, T. A. Fundamentals of crystallization: Kinetic effects on particle size distributions and morphology. Chem. Eng. Sci. 1991, 46, 2389. (14) Weast, R. C., Ed. CRC Handbook of Chemistry and Physics; CRC Press: Cleveland, OH, 1974. (15) Nielsen, A. E. Diffusion controlled growth of a moving sphere. The kinetics of crystal growth in potassium perchlorate precipitation. J. Phys. Chem. 1960, 65, 46. (16) So¨hnel, O.; Garside, G. Precipitation. Basic Principles and Industrial Applications; Butterworth-Heinemann: Oxford, U.K., 1992. (17) Narsimhan, G.; Ruckenstein, E. A new approach for the prediction of the rate of nucleation in liquids. J. Colloid Interface Sci. 1989, 128, 549. (18) Kostoglou, M.; Andritsos, N.; Karabelas, A. J. Flow of supersaturated solutions in pipes. Modeling bulk precipitation and scale formation. Chem. Eng. Commun. 1995, 133, 107. (19) Friedlander, S. K. Smoke, Dust and Haze; Wiley-Interscience: New York, 1977. (20) Elimelech, M.; Gregory, J.; Jia, X.; Williams, R. Particle Deposition & Aggregation: Measurement, Modelling and Simulation; Butterworth-Heinemann: Oxford, U.K., 1995. (21) Kostoglou, M.; Karabelas, A. J. Comprehensive modeling of precipitation and fouling in turbulent pipe flow. Ind. Eng. Res. Chem. 1998, 37, 1536. (22) Carbonell, R. G.; Riley, D. T. Mechanisms of particle deposition from ultrapure chemicals onto semiconductor wafers: deposition from a thin film of drying rinse water. J. Colloid Interface Sci. 1993, 158, 274. (23) Williams, M. M. R.; Loyalka, S. K. Aerosol Science; Pergamon Press: New York, 1991. (24) Kumar, S.; Ramkrishna, D. On the solution of population balance equations by discretizationsIII. Nucleation, growth and aggregation of particles. Chem. Eng. Sci. 1997, 52, 4659. (25) McGraw, R.; Nemesure, S.; Schwartz, S. E. Properties and evolution of aerosol with size distributions having identical moments. J. Aerosol Sci. 1998, 29, 761. (26) Barret, J. C.; Jheeta, J. S. Improving the accuracy of the moments method for solving the aerosol general dynamic equation. J. Aerosol Sci. 1996, 27, 1135. (27) McGraw, R. Description of atmospheric aerosol dynamics by the quadrature method of moments. Aerosol Sci. Technol. 1997, 27, 255.

Ind. Eng. Chem. Res., Vol. 39, No. 9, 2000 3283 (28) Pratsinis, S. E. Simultaneous nucleation, condensation, and coagulation in aerosol reactors. J. Colloid Interface Sci. 1988, 124, 416. (29) Villadsen, J.; Michelsen, M. L. Solution of Differential Equation Models by Polynomial Approximation; Prentice Hall: New York, 1978. (30) Lee, K. W. Conservation of particle size distribution parameters during Brownian coagulation. J. Colloid Interface Sci. 1985, 108, 199.

(31) profile (32) Press:

Sathyagal, A. N.; McCormick, A. V. Effect of nucleation on particle size distribution. AIChE J. 1998, 44, 2312. Van de Ven, T. G. M. Colloidal Hydrodynamics; Academic London, 1989.

Received for review June 30, 1999 Accepted June 16, 2000 IE990472Q

Modeling Thin Film CdS Development in a Chemical Bath Deposition

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