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A Monte Carlo simulation approach to the texture formation during electrodeposition—I. The simulation model
Electrochimica Acta, Vol. 42, No. 1, pp. 3745, 1997 Copyright 0 1996 Elsevier Science Ltd. Printed in Great Britain. All rights reserved
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A Monte Carlo simulation approach to the texture formation during electrodepositionI. The simulation model D. Y. Li* and J. A. Szpunar Department
of Metallurgical
Engineering,
McGill University, 3450 University Street, Montreal, PQ H3A 2A1, Canada
(Received 27 September 1995; in revised form 18 March 1996) Abstract-A Monte Carlo model was developed to simulate the texture development during electrodeposition. In the model, a two-dimensional triangle lattice was used to map the microstructure of the deposit. As the deposit grew, previously empty lattice sites were occupied layer by layer. Each occupied lattice site was assigned an integer number to represent the orientation of the grain to which the site belonged. The authors proposed a minimum-energy texturing mechanism, which states that the texture evolution results from the minimization of the system’s free energy. Two energy components were considered responsible for the texture formation: the surface energy and the magnetic energy (when the deposition is conducted in external magnetic fields). In this paper, the simulation model of texture evolution has been described, and various parameters of the model have been tested and discussed. Copyright 0 1996 Elsevier Science Ltd Key words: texture, electrodeposition,
simulation,
surface energy, magnetocrystalline
INTRODUCTION
anisotropy.
There is a lack of a satisfactory explanation of the texture formation during electrodeposition, although a large body of experimental results is available. This retards controlling the electrodeposit’s texture during processing to improve the deposit’s properties. Several models were proposed to explain the texture formation during electrodeposition [3,4, 10-121. These models basically fall into two categories. One is called the two-dimensional nucleation theory [4]. The main idea behind this explanation is an assumption that different planes require different overpotentials to form two-dimensional nuclei, and this may result in different nucleation rates on these planes. As a result, the advance rates of different planes might be different, leading to the texture evolution. However, it is known that the overpotentials for generating two-dimensional nuclei on different planes are small (eg, a few mV for Cu [ 131) in comparison with the operating overpotential, which is usually around lo* mV. Under such a relatively high operating overpotential, nuclei could be generated on different planes in similar probabilities. The texture, therefore, should not be attributed to the difference in nucleation rate. Various researchers [6, 17-191 have already reported disagreement between the model and experimental
Since 1924 when Glocker and Kaupp [l] first demonstrated by means of X-ray diffraction that electrodeposited copper, silver, nickel, chromium and iron had fibre textures, the texture of electrodeposits has attracted the interest of researchers [l-14]. It is recognized that there are possible benefits from the texture, which is responsible for anisotropic properties of electrodeposits and electrocoatings. For example, the wear resistance is different for different crystallographic planes [15]. By introducing an appropriate texture, the wear resistance of protective coatings could be enhanced, and their service lifetime consequently prolonged. The intergranular corrosion is a frequent mode of corrosion, and the grain boundary is of particular importance to the intergranular corrosion. It has been demonstrated that the texture influences the grain boundary structure and, in turn, affects the corrosion resistance of materials [16]. If an appropriate texture is introduced in protective coatings, their resistances to environmental attacks could be improved. *Present address: Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, U.S.A. 37
D. Y. Li and J. A. Szpunar
38
observations. Another theory, the so-called geometrical selection theory [3], suggests that the formation of texture is attributed to the difference in growth rates of different planes and also the deposit’s surface morphology. According to the theory there are two modes for deposit growth, one is an outward growth mode and the other the lateral growth mode. In the outward growth mode, grains bounded by slow-growing faces, which are perpendicular to the substrate, may grow preferentially. Whereas, in the lateral growth mode, the grains bounded by slow growing faces, which are parallel to the substrate, may grow preferentially. This theory, however, cannot answer the question what determines the crystal growth mode. Disagreement between this theory and experimental observation has also been found. For example, one may explain the (110) fibre texture of iron deposits using the geometrical selection theory with an assumption that the iron deposit grows in the lateral growth mode. Iron deposits, however, usually have bumped surfaces, and the observed facets are neither perpendicular nor parallel to the deposit surface, no matter what kind of textures the deposits have. In this paper we propose a Monte Carlo approach towards explanation of the texture evolution during electrodeposition. It is assumed that the deposit growth takes a “path” leading to a decrease in the system’s free energy, and the deposit will be textured as a result. An attempt has been made to simulate the effect of minimizing the system’s free energy on texture formation. The Monte Carlo simulation technique was proposed by Anderson et al. [20]. The model relies on the representation of the microstructure by a two-dimensional discrete lattice and defines the interaction between lattice sites. In the Monte Carlo model, the lattice site is randomly selected to test its energy and orientation. A new state of this site is calculated according to the Metropolis algorithm [20]. This new state (eg, the change in grain’s orientation) can be accepted only if the energy of the site decreases. As a result, the whole system will reach a minimum-energy state. The Monte Carlo technique has been used to simulate the nucleation and grain growth during recrystallization, solidification and vapour deposition processes [21-231. Since the deposit growth during electrodeposition is a nucleation and grain growth process, it is also possible to apply the Monte Carlo technique to electrodeposition. Assuming that the deposit growth minimizes the system’s free energy, one may expect that the texture development could be -modelled by taking into account the factors
magnetization energy may also affect the texture development, if the deposited substance is ferromagnetic and the deposition is carried out in a magnetic field.
DESCRIF’TION OF THE MONTE SIMULATION MODEL
CARLO
Electrodeposition is a process in which mass transfer is accompanied by charge transfer [24-281. During the process, metal atoms are ionized at the anode, then hydrated and move to the cathode through the electrolyte under the influence of an imposed electrical field. When reaching the cathode surface, the ions are dehydrated and discharged (they are now called “ad-atoms”). These ad-atoms are not immediately incorporated into the metal lattice, and instead, they diffuse on the cathode surface until reaching low-energy sites, such as micro-steps and kinks, where they will be accommodated. This process could be simulated by adding atoms to random locations on the cathode surface and allowing them to diffuse until they reach the low-energy sites. However, direct modelling of this process is intractable because it needs to take into account a large number of atoms, eg lo’atoms, to simulate a surface as small as 1 pm’. Thus, the present approach is not an atomistic model but a quasicontinuum model that accounts for the statistical nature of the deposit growth.
#44k444 444444 44b4444 44444)
0000000000 00000000000 0000000000 00000000000 0000000000 00000000000 0000000000 00000000000 0000000000 00000000000 0000000000 00000000000
0000000000 00000000000 0000000000 00000000000 0000000000 00000000000 0000000000 00000000000 0000000000 00000000000 0000000000 00000000000
0000000000 00000000000 0000000000
00000000000 00000000000
00000000000 0000000000 00000000000
0000000000 00000000000
and
Fig. 1. The microstructure at a cross-section of the deposit is mapped on to a two-dimensional triangle lattice. Each
related to the free energy of the system. For example, the anisotropic surface energy could play an important role in the texture formation. Anisotropic
occupied site is assigned a non-zero integer number to represent the orientation of the grain to which the site belongs. The site having zero number is unoccupied.
which
are
crytallographically
anisotropic
39
Texture development during electrodeposition1. Representation of the deposit’s microstructure and texture In this simulation, the continuum microstructure at a cross-section of the deposit is represented using a two-dimensional triangle lattice as Fig. 1 shows. Using the triangle lattice is convenient because the triangle lattice has a relatively more isotropic character than other lattices, such as the square lattice [20]. Each site in this lattice is assigned an integer number, P, to represent the orientation of the grain to which the site belongs. The lattice site has a certain volume, but this volume is small enough, so that the change in the site’s state can provide the information about the microstructural evolution. Initially, all sites are not occupied and their assigned P values are zero. When the deposit grows, the sites are occupied layer by layer. The occupation of a site is indicated by a change of its P value from zero to a positive integer. The overall process is modelled in the following way. Nuclei are initially generated at the cathode surface, and the lattice is then scanned layer by layer. In the scanning process, the site is randomly selected and its P value is tested. If P = 0, the site will be occupied and its P value will change to a positive integer. The value of this positive integer is dependent on the change in the local free energy. If P > 0, the site is already occupied and it is, therefore, skipped. The major task of the simulation is to model the textural evolution. The texture component is represented by its volume fraction, defined as the ratio of the lattice sites which contribute to this texture component to overall lattice sites. For example, for a fibre texture whose axis is normal to {hkl} plane, its volume fraction is the ratio of the sites which belong to the grains having their {hk/j planes parallel to the deposit surface to overall lattice sites. 2. Nucleation and grain growth The nucleation during electrodeposition is similar to the nucleation from a supersaturated solution or condensation from a supersaturated vapour [24]. When an ion enters the diffusion layer, it is dehydrated, discharged, and then diffuses and joins other ad-atoms to form a nucleus. The more ad-atoms moving on the cathode surface, the higher is the frequency of forming nuclei. Since the density of ad-atoms is dependent on the flux of ions carried by the deposition current, the frequency of nucleation is therefore affected by the current density. The relation between the nucleation frequency and the current density is expressed as [24,25]: Fnccexp[-a/(RTiS)]
(1)
where it, T and R are the current density, temperature, and gas constant, respectively, and o is a parameter proportional to the surface energy of the nucleus. It is seen that high current densities result in
high nucleation rates. During the deposit growth, the nucleation may also occur on growing crystal’s surface. However, compared to the grain growth, this type of nucleation is relatively unfavoured because it requires an activation energy to overcome a nucleation barrier, which depends on the surface energy of the critical nucleus. Whereas the grain growth, ie direct incorporation of ad-atoms into the crystal lattice, is favoured because there is no such energy barrier during the grain growth process. Once a nucleus is formed, it starts to grow. Usually, an ad-atom does not get directly incorporated into the crystal lattice as it reaches the deposit surface. Instead, it diffuses on the deposit surface until reaching a place where the accommodation of the ad-atom leads to a decrease in the local free energy. In this simulation, the driving force for the deposit growth is, accordingly, assumed to be the minimization of the system’s free energy. A grain can grow, if by doing so the system’s free energy decreases. When the deposit grows, previously empty lattice sites are occupied. The decision on what is the site orientation is made after calculating the free energy of the site. The free energy of a site includes the interaction between the site and its neighbours. The energy of a lattice site consists of two terms and is expressed as: H = -J~(&,P,
- 1) +
Hhk/
(2)
nn
where J is one half of the free energy of the bond connecting two neighbour sites, Pi represents the P value of site i, and 8pipi has the following value: 8pip,= 1, if Pi = Pj; or SP,P,= 0, if Pi # P,. The first term in equation (2) represents the interaction between the testing site and its neighbours. If a pair of sites have the same P values, there is no interaction between these two sites because there is no interface between them. The interaction exists only if the sites have different P values. In equation (2) the bond energy term W has an average value of the interaction between a pair of neighbour sites which does not take into account the crystallographic anisotropy, so it only accounts for the “isotropic” grain growth. An additional term must be added to modify the energies of bonds between differently oriented sites to simulate the formation of texture. More details about this modification term will be given in the following sections. 3. Factors aIkcting textural development When an ion arrives at the cathode surface, it loses its charge and diffuses until reaching a low-energy location, where this ad-atom will be accommodated. The low-energy location, ie the site where there is a decrease in the system’s free energy as the ad-atom sits in, depends on the grain orientation; therefore, the minimization of the energy could be used to predict the texture development. A part of the free
40
D. Y. Li and J. A. Szpunar
energy is the surface energy which is crystallographically anisotropic, and hence may play a role in the texture formation. For example, when arriving at a junction of two grains, an ad-atom will tend to be accommodated at the crystallographic plane which has a lower surface energy, leading to a decrease in the overall surface energy. As the deposition proceeds, the interface will move towards the grain which has a higher surface energy, and another grain would therefore grow preferentially. Another factor considered here is the magnetocrystalline anisotropy. During electrodeposition of ferromagnetic substances, if a magnetic field is applied the grain having lower magnetization energy may grow preferentially, resulting in texture evolution. In our simulation, the effects of these two anisotropy factors on the deposit’s texture have been taken into account by adding the second term in equation (2). 4. The process controlling parameter: current density In this Monte Carlo model, the current density is used as a parameter controlling the process. Although the driving force for electrodeposition is the over-potential, and a plater may control the process by means of the potential, the current density is often used to control the deposition. This is because the current density directly relates to the deposition rate, or the weight of the deposited substance, as Faraday’s law described [26]: dh h x=7=-
iCoE,, 69
where h is the thickness of the deposit, and Eel is the electrochemical equivalent of the deposited metal, t, p and i, are the deposition time, density of the deposit, and the current density, respectively, and o is the current efficiency. According to this formula, the deposition rate, or the weight of deposited substance, is proportional to the current density. Another reason for using the current density to control the process is that the current density influences the deposit’s microstructure, because the nucleation frequency is affected by current density as equation (1) indicates. So, the size and number of grains are influenced by the current density. Moreover, the current density may influence the deposit’s texture. It is known that at a high current density deposits may grow without any texture [24], although the texture can be quite strong when current is low. It will be demonstrated in a following paper [271that a high current density makes the deposit less textured, whereas a lower current density makes the deposit strongly textured. It is difficult, however, to use the current density directly as an input in the simulation. In order to convert the current density into a Monte Carlo parameter, we define the operation of scanning one lattice layer with 200 iterations as one Monte Carlo step (MCS). Because each iteration takes a certain time, one MCS therefore corresponds to a certain
time interval. The deposition rate can thus be modelled by using different numbers (n) of MCSs in scanning each lattice layer. This is to say that the time, At, necessary for an increment of the deposit thickness, Ah, could be replaced by n(MCS): Ah/Al ccAh/n(MCS).
(4)
The deposition rate is lower if the number of MCSs, n, is higher. Since the deposition rate is proportional to the current density (see equation (3)), the current density can thus be expressed by n, that is i,ccAh/AtccAh/n(MCS)cC
l/n.
(5)
The effect of current density on the deposit’s texture and microstructure can, therefore, be modelled, using the number (n) of MCSs involved in the scanning of one layer. 5. Calculation of the surface-energy anisotropy and the magnetocrystalline anisotropy As discussed earlier, the surface-energy anisotropy and the magnetocrystalline anisotropy may play important roles in the texture formation during electrodeposition. These two anisotropic energies will be used as inputs for the Monte Carlo simulation. Before doing that, these two anisotropic energy contributions need to be calculated. (i) The surface-energy anisotropy. The Monte Carlo model will be applied, in a following paper [27], to iron electrodeposition. In order to focus our attention on the mechanism of texture formation and to avoid complex effects of alloying elements, electrodeposition of pure iron will be simulated. Surface energies of different crystallographic planes of iron were calculated using the LennardJones [28] potential. The surface energy of a crystallographic plane (hkl) is defined as the difference in free energy between the crystal’s surface layer and its middle layer, which are parallel to the (h/cl) plane. To calculate the surface energy, a cubic cell having dimensions of 1.43 x 1.43 x 1.43 nm3 and containing 2000 iron atoms was constructed. The surface layer of the cell was then relaxed. The surface relaxation is necessary because when a crystal is split into two parts and forms two fresh surfaces, the atoms on the surfaces cannot be in their original positions and must shift to new positions, otherwise the surfaces will not have a thermal-equilibrium configuration. In the relaxation process, each atom in the surface region was allowed to move randomly by a short distance. If the old configuration energy is defined as El and the new configuration energy due to the shift is defined as E2, each move is accepted with a probability P = exp[-(Er - E,)/KT]; where, K is the Boltzmann’s constant and T is temperature. Note that if Ez < El, the probability of such a move is always larger than unity, and such a move is then always accepted. After the relaxation, the energy of a surface layer (3 A thick) and that of a central layer with the same thickness were calculated.
41
Texture development during electrodeposition-
LO OS
03
3
0.1
f
E *
Fig.
0.4 03 02 0.1 01) 2.
100 211 310 311 111 110 Calculated surface energies of iron (normalized).
Lennard-Jones potential was used in the calculation. Six planes were calculated; they are 100, 2 11, 3 10, 622 = 3 I I, 222 = 111 and 1IO planes.
The difference between these two energies gives the surface energy of the specific plane. Choice of the layer of 3 8, thickness for the surface energy calculation is made because the interaction between a pair of atoms declines with an increase in the distance between these two atoms. In addition, the interaction between nearest neighbouring atoms contributes to about 90% of the system’s energy. For these two reasons, the layer of 3 8, thickness was chosen. In such a layer, the nearest and secondary nearest neighbours (the distances to the nearest and secondary nearest neighbours are 2.48 and 2.87 A respectively) are included. Surface energies of a number of low-index iron crystal planes were calculated. Most of these planes correspond to the fibre textures, which were reported in the literature [8, lo]. Figure 2 illustrates the result of the calculation, from which one can see that the (110) plane has the lowest surface energy. (ii) The effect of hydrogen adsorption on surface energy. Metal deposition is often accompanied by
simultaneous hydrogen co-deposition [9]. When the hydrogen co-deposition takes place, the deposit’s texture may change because adsorbed hydrogen may modify the metal’s surface energy. According to Pet&s work [29], iron’s surface energy can be dramatically lowered by the hydrogen adsorption. It is possible that the surface-energy anisotropy is also changed, because the change in surface energy may vary for different crystallographic planes. It was demonstrated, using a quantum-mechanical calculation, that the greater is the distance between adjacent metal atoms, the lower is the activation energy for hydrogen adsorption [30]. This means that loose packed planes should have higher hydrogenadsorption abilities than close-packed planes [3]. Allowing for the fact that the hydrogen discharge is dependent on the deposition condition [26, 311, and that different crystallographic planes could have different hydrogen-adsorption abilities, it is expected that the change in deposition condition may change the metal’s surface-energy anisotropy
and this, in turn, leads to a change in the deposit’s texture. In order to investigate the texture variation with hydrogen co-deposition, the surface-energy changes, affected by the hydrogen adsorption, were evaluated using the following approach. A crystal cell containing about 1000 iron atoms and having dimensions 2.86 x 2.86 x 1.43 nm3 was first constructed, and then a certain amount of hydrogen atoms were randomly placed into the vicinity of the iron cell surface which was under investigation. The hydrogen and iron atoms in the surface region were then relaxed. After the relaxation, the energy of the surface layer (3 /i thick) was calculated. The difference in free energy between the relaxed surface layer and a central layer of equal thickness represents the surface energy, as influenced by adsorbed hydrogen. Figure 3 illustrates the effect of hydrogen on surface energies of various crystallographic planes. One can see that the iron surface energy is lowered with an increase in the hydrogen concentration, and that the rate of the lowering for different planes is different. It is,therefore, expected that with an enhancement in the hydrogen co-deposition, the iron deposit’s texture would change from one type to another, due to the changes in the surface-energy anisotropy. (iii) The magnetocrystalline anisotropy. When
electrodeposition of ferromagnetic metals is carried out in a magnetic field, the metals are magnetized. The magnetization energy is also crystallographically anisotropic, and there exists an easy axis for the magnetization [32]. The easy axis in BCC iron is along the (100) crystal axis. When the easy axis of a crystal is parallel to the magnetic field, the energy needed to magnetize the crystal is the lowest. The difference in magnetization energy between differently oriented grains is called the magnetocrystalline anisotropy; for cubic crystals, it is expressed as [32]:
031
0
,
,
,
2
,
4
,
,
,
6
~ a
, 10
Hydropa Concentration (moles/l)
q 310
+ 211
Oux)
A622
x 222
VllO
0 10 22
Fig. 3. The lowering effect of hydrogen on iron surface energy.
42
D. Y. Li and J. A. Szpunar
where, (XI,cc2and cr3are direction cosines of the angles between the applied magnetic field and three axes of the crystal reference frame, and K, and K2 are the magnetocrystalline anisotropy constants. For iron, KI = 4.8 x 104J m3 and KZ = 1.5 x 104J m3 [32]. This magnetocrystalline anisotropy has been used to simulate the effect of magnetic field on texture development. 6. The simulation procedure The input of the simulation includes the number of MCSs, ie the process parameter, used to scan one lattice layer, and the anisotropy factors: the surface-energy anisotropy and magnetocrystalline anisotropy. As mentioned earlier, the number of MCSs used to scan one lattice layer is inversely proportional to the current density and can be used to control the deposition rate. The anisotropy factors determine the competition between different crystallographic planes, and are therefore responsible for texture development. Nuclei are initially generated at the cathode surface. The lattice is then scanned layer by layer. In scanning one layer, a site to be tested is selected randomly. If the site has P = 0, energy of the site will be tested to decide the way in which metal ions will be deposited on to this site (ie grain growth or nucleation). The energy of the selected site is then calculated using the following form of equation (2):
boundaries and low-energy boundaries, are normally formed in polycrystalline materials. The first type are the high-angle grain boundaries which are the boundaries between grains with misorientation more than 15 degrees, and this type of boundaries usually have a high energy. The high-angle boundary structures are complicated and irregular, and their energies are represented usually using an average value which can be determined experimentally. The low-energy boundaries include the coincident site lattice (CSL) boundaries and low-angle boundaries, which are regular and have relatively low energies. However, the low-angle boundaries and the CSL boundaries usually have a small percentage of grain boundaries. Therefore, in the simulation, only the high-angle boundaries are taken into consideration. In summary, a “grain boundary bound” has an energy equal to 2J, and a “surface bound” has an energy equal to 2J.(y&h)( 1 + Ii&). If the deposition is carried out in a magnetic field, the magnetization energy is taken into account. HE,. is the magnetization energy term only for occupied sites. The notation, uuw, represents the site’s orientation which is parallel to the magnetic field. This term counts the effect of the magnetic field which is strong enough to saturate iron. The magnetization energy for each occupied lattice site is expressed as x.., = IEu = ~[K,(c+Y: + c&x: + &Y:) + K&&&4) (8)
H = -JC @f,P,“”
1) - JJ$I$,, !“rn
+ fp,
(7)
where J represents half of the energy of the bond connecting two neighbouring sites. If the bond connects two occupied sites with different orientations (ie, a “grain boundary bond”), the bond energy corresponds to the grain boundary energy ygb. If the bond connects an occupied site and an unoccupied site, the interface between these two sites is a surface and this “surface bond” corresponds to the surface energy ys. The bond energy 2J, therefore, has different values for the “surface bond” and the “grain boundary bond”. If the value of the “grain boundary bond” is defined as 25, the “surface bond” will be automatically adjusted and has an energy eqd t0 2J.(y&b). ys and ygb are average SUrfaCe energy and average grain boundary energy, respectively. In equation (7), Phk, is a modification term which makes the “surface bond” energy crystallographically anisotropic. This anisotropy parameter is the difference between the surface energy of the {hkl} plane and the average surface energy ys. Sum Z is taken over all of the six nearest neighbour sites, while Z’ is taken over only those sites which are connected to the testing site by “surface bonds”. It should be noted that in this simulation only the average value of the grain boundary energy is taken into account, which simplifies the energy calculation. In fact, two types of grain boundaries, so called high-energy
where 1 is a normalization coefficient which is the ratio of the magnetization energy of a lattice site to its bonds’ energy. This formula gives the magnetization energy of a site when it is in the state of saturation. The initial energy state, HO, of the selected site is first calculated using equation (7), and the P value of the site is then changed from zero to the P value of its nearest neighbour and an energy state, Hn, is calculated using equation (7) again. P values of all nearest neighbours are used in the same way to obtain a number of trial H,, values. The minimum H, is chosen as the final-state energy of the selected site and the corresponding P integer is assigned to the site. If AH = H, - HO< 0, the new P value is retained. Otherwise this change in P value will not take place. AH 2 0 means that the grain growth is not energetically favoured at the testing site. In this case, we generate a randomly oriented nucleus in a probability obtained by incorporating equation (5) into equation (1): Faexp(a/RTi,2)
= exp(a’n2/RT)
(9)
where u’ is a constant. Output of the simulation will provide information about texture, grain size and grain shape of the deposit.
Texture development during electrodeposition-
(4
43
Figure 4(b) illustrates the case of iron deposition in which the surface-energy anisotropy has been taken into consideration. An inhomogeneous distribution of differently oriented grains indicates the existence of a texture.
a-((
The minimum number of iterations for the lattice scanning
10 0 100
211
310
311
111
110
Chklt.%iO~
Fig. 4. Volume fractions of differently oriented grains: (a) without taking into account any anisotropy factor; (b) with taking into account the surface-energy anisotropy of iron.
MODEL TESTING AND DISCUSSION The random-number generator The equal-frequency distribution of the random numbers is of particular importance to the simulation of texture development. Each time when a nucleus forms, its orientation is decided by the randomnumber generator which creates a random number that represents the orientation of the created nucleus. If the generated random numbers are not distributed with equal frequency, orientations of the created nuclei will not be homogeneously distributed. As a result, simulated deposits will be “textured” even without introducing any anisotropy factor. To ensure that no texture appears when the simulation does not take any anisotropy factor into account, the growth of iron deposit with and without inputting the anisotropy factors has been simulated using an adopted random-number generator. Figure 4(a) illustrates volume fractions of differently oriented grains simulated without introducing any anisotropy factor. It has been observed that the distribution of grain orientations is homogeneous, ie no texture appears, which convinces us that the random-number generator used in the simulation model is reliable.
Therefore, the deposit texture will be induced only by anisotropy factors used as input into our simulation.
In the Monte Carlo simulation, the number of iterations for scanning each lattice layer is required to be high enough to ensure each site be scanned in an equal probability. If the number of iterations is less than a minimum number, not all sites are scanned, and this will result in a “vacancy” in the deposit microstructure. Theoretically, there is no “vacancy” left when the number of MCSs (one MCS includes 200 iterations) used in the scanning of each lattice layer is infinite. However, in the simulation, the number of MCSs cannot be infinite, and the “vacancy”, therefore, cannot be avoided completely. The vacancies are caused by iterations which are not enough to cover all sites on the deposit surface. The vacancies can influence the energy calculation of the sites near the vacancies. So the number of vacancies should be reduced to their minimum. Theoretically, the higher the number of MCSs involved in the scanning of one lattice layer, the lower is the number of vacancies in the deposit. Figure 5 illustrates that the number of vacancies decreases as the number of MCSs increases. It is seen that when the number of MCSs exceeds n = 4, the total number of vacancies in the simulated deposit (containing 1.2 x lo4 sites) has a low value of about fifty, and further change in the number of vacancies with the increase in the number of MCSs is slight-that is to say that only about 0.4% of total lattice sites are skipped if n is higher than four. The effect of such a small amount of “vacancies” could be ignored. Therefore, n = 4 is chosen as the minimum number of the MCSs used to scan each lattice layer.
Fig. 5. Vacancies decrease as the number of MC.% is raised. When the number of MC% exceeds n = 4, the number of vacancies is about fifty, which takes only 0.4% of total site number of the simulated deposit (containing 1.2 x IO”sites).
44
D. Y. Li and J. A. Szpunar
The MCS clock and real time clock In the Monte Carlo simulation, the time scale is represented by the MCS, which corresponds to a certain period of real time, and therefore is chosen as a time unit for the simulation. The more MC% involved in the simulation, the longer the simulation takes. If the relationship between the MCS clock and the real time clock is known, the microstructure and texture development would be simulated quantitatively. This relationship could be rather complicated, and so far there is no satisfactory solution. It has been suggested that the conversion from the real time to the MCS time can be made by using atomic jump frequency which is related to an activation energy factor, exp(- WjkT), where W is the activation energy for each atomic jump [21]. However, in the Monte Carlo simulation model, which is a quasicontinuum model rather than an atomistic model, it is difficult to take real time into consideration. Therefore the present model is not a quantitative model but a qualitative one. Nevertheless, it is important that this Monte Carlo model is able to predict the tendency of the texture development during electrodeposition. The information provided by such a model is valuable to the deposition practice. Since a MCS corresponds to a certain time interval, the number of MC.%., n, used for scanning each lattice layer determines the deposit rate. The higher the number of MCSs for scanning one layer is, the longer it takes to form a deposit layer (ie the lower the deposition rate). At the same deposition rate, the more MCSs are involved in the overall deposition process, the thicker is the metal deposit. Although the corresponding real time interval is unknown, the linear relationship between the thickness of deposit and the deposition time can be described by using the number of MCSs. Since the deposition rate R = Ah/AtccAh/n(MCS) (see equation (4)), the overal thickness of a deposit may, therefore, be expressed as
Thickness = ZAh = R.ZAtaR.Z,n(MCS)
expected that crystallographic anisotropy factors which are related to the system’s free energy play an important role in the texture formation. Such anisotropy factors may include the anisotropies of surface energy, and the magnetization energy, if the deposition involves ferromagnetic metals and applied magnetic fields. The surface energy anisotropy and the magnetocrystalline anisotropy have been calculated and used as input in the simulation of texture evolution during iron electrodeposition, reported in a following paper [27]. The random-number generator, the minimum number of iterations, and the relation between the real time and the MCS time have been tested and discussed. REFERENCES R. Glocker and E. Kaupp, Z. P/t@. 24, 121 (1924). A. W. Hothersall, Trans. Faraday Sot. 31, 1242 (1935). A. K. N. Reddy, J. Elecrroanal. Chem. 6, 141 (1963). N. A. Pangarov, J. Electroanal. Chem. 9, 70 (1965). J. R. Park and D. N. Lee. J. Korean Inst. Metals 12.243 (1976). 6. I. Epelboin, M. Froment and G. Maui-in, Plating 56,
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1356 (1969).
7. A. N. Baraboshkin, Z. S. Martem’yanova, S. V. Plaksin and N. 0. Esina, Soviet Electrochemistry 13, 1558 (1977); 14, 6 (1978). 8. C. B. Barrett and T. B. Massalski. Structure of Metals. 3rd edition Chap. 19. Pergamon Press, Elm&d, New York (1980). 9. D. Y. Li and J. A. Szpunar, J. Electr. Mater. 22, 653 (1993).
10. N. A. Pangarov and S. D. Vitkova, Electrochem. Acta 11, 1719 (1966). 11. H. Wilman, Trans. Inst. Metal. Finishing 32,281 (1955). 12. A. N. Baraboshkin, Z. S. Martem’yanova, S. V. Plaksin and N. 0. Esina, Alektrohimiya 13, 1807 (1977); 14, 9 (1978).
13. D. Postl, G. Eichkom and H. Fischer, Z. Phys. Chem. 77, 138 (1972). 14. H. J. Pick, G. G. Storey and T. B. Vaughan, Electrochim. Acta 2, 165 (1960).
15. D. Kuhlmann-Wilsdorf, Fundamentals of Friction and Wear of Materials (Edited by D. A. Rigney) p. 119. American Society for Metals, Ohio (1981). 16. H. M. Kim and J. A. Szpunar, Materials Science Forum 157-162,
= R,N (9)
1997 (1994).
17. G. C. Ye and D. N. Lee, Plating and Surface Finishing, A, 60 (1981).
18. I. A. Menzies and C. X. Ng, Trans. Inst. Met. Fin. 47,
where Ah and At are the thickness of each lattice layer and the time for growing each layer of the deposit, respectively, R = Ah/At is the deposition rate, and N = Cn is the total number of MCSs for the deposit growth.
156 (1956).
19. J. R. Park and D. N. Lee., J. Korean Inst. Metals 14,359 (1976).
20. M. P. Anderson, D. J. Srolovitz, G. S. Grest and P. S. Sahni, Acta Metall. 32, 783, 793, 1429 (1984); 33, 509 (1985).
21. D. J. Sorolovitz, G. S. Grest and M. P. Anderson, Acta
SUMMARY A Monte Carlo model was developed to simulate the texture formation during electrodeposition. In this model, a two-dimensional triangle lattice is used to map the microstructure at a cross-section of electrodeposit. During the deposition, sites of the lattice are occupied layer by layer. The texture development is assumed to be caused by the minimization of the system’s free energy. It is
Metal/. 34, 1833, 2115 (1986).
22. J. A. Spittle and S. G. R. Brown, J. Mater. Sci. 23, 1777 (1989).
23. D. J. Srolovitz, J. Vat. Sci. Technol. A4(6), 2925 (1986). 24. R. Winand, Fundamentals and Practice of Aqueous Electrometallurgy, p. 7. The Metallurgical Society of CIM, Montreal, PQ, Canada (1990). 25. J. O’M. Bockris and G. A. Razumney, Fun&mental Aspects of Electrocrystallization. Plenum Press, New York (1967). 26. E. Raub and K. Milller, Fundamentals of Metal Deposition. Elsevier Publishing Co., Amsterdam (1967).
Texture development during electrodeposition21. D. Y. Li and J. A. Szpunar, Electrochim. Acta. 42, 47 (1997)
28. T. Halicioglu and G. M. Pound, Phys. Stat. Sol. (a) 30, 619 (1975). 29. N. J. Petch, Phil. Mag. 1, 331 (1956).
30. G. Okamoto, J. Horiuti and K. Hirota, Sci. Papers Inst. Phys. Chem. Res. (Tokyo) 29, 223 (1936).
45
31. N. V. Parthasaradhy, Practical Electroplating Handbook. Prentice-Hall, Inc., Englewood Cliffs, NJ (1989). 32. C. W. Chen, Magnetism and Metallurgy of Soft Magnetic Materials, p. 61. North-Holland Publishing
Company, New York (1977).
Pergamon PII:
s00134686(%)00164-8
00134686/97$17.00+ 0.00
A Monte Carlo simulation approach to the texture formation during electrodepositionI. The simulation model D. Y. Li* and J. A. Szpunar Department
of Metallurgical
Engineering,
McGill University, 3450 University Street, Montreal, PQ H3A 2A1, Canada
(Received 27 September 1995; in revised form 18 March 1996) Abstract-A Monte Carlo model was developed to simulate the texture development during electrodeposition. In the model, a two-dimensional triangle lattice was used to map the microstructure of the deposit. As the deposit grew, previously empty lattice sites were occupied layer by layer. Each occupied lattice site was assigned an integer number to represent the orientation of the grain to which the site belonged. The authors proposed a minimum-energy texturing mechanism, which states that the texture evolution results from the minimization of the system’s free energy. Two energy components were considered responsible for the texture formation: the surface energy and the magnetic energy (when the deposition is conducted in external magnetic fields). In this paper, the simulation model of texture evolution has been described, and various parameters of the model have been tested and discussed. Copyright 0 1996 Elsevier Science Ltd Key words: texture, electrodeposition,
simulation,
surface energy, magnetocrystalline
INTRODUCTION
anisotropy.
There is a lack of a satisfactory explanation of the texture formation during electrodeposition, although a large body of experimental results is available. This retards controlling the electrodeposit’s texture during processing to improve the deposit’s properties. Several models were proposed to explain the texture formation during electrodeposition [3,4, 10-121. These models basically fall into two categories. One is called the two-dimensional nucleation theory [4]. The main idea behind this explanation is an assumption that different planes require different overpotentials to form two-dimensional nuclei, and this may result in different nucleation rates on these planes. As a result, the advance rates of different planes might be different, leading to the texture evolution. However, it is known that the overpotentials for generating two-dimensional nuclei on different planes are small (eg, a few mV for Cu [ 131) in comparison with the operating overpotential, which is usually around lo* mV. Under such a relatively high operating overpotential, nuclei could be generated on different planes in similar probabilities. The texture, therefore, should not be attributed to the difference in nucleation rate. Various researchers [6, 17-191 have already reported disagreement between the model and experimental
Since 1924 when Glocker and Kaupp [l] first demonstrated by means of X-ray diffraction that electrodeposited copper, silver, nickel, chromium and iron had fibre textures, the texture of electrodeposits has attracted the interest of researchers [l-14]. It is recognized that there are possible benefits from the texture, which is responsible for anisotropic properties of electrodeposits and electrocoatings. For example, the wear resistance is different for different crystallographic planes [15]. By introducing an appropriate texture, the wear resistance of protective coatings could be enhanced, and their service lifetime consequently prolonged. The intergranular corrosion is a frequent mode of corrosion, and the grain boundary is of particular importance to the intergranular corrosion. It has been demonstrated that the texture influences the grain boundary structure and, in turn, affects the corrosion resistance of materials [16]. If an appropriate texture is introduced in protective coatings, their resistances to environmental attacks could be improved. *Present address: Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, U.S.A. 37
D. Y. Li and J. A. Szpunar
38
observations. Another theory, the so-called geometrical selection theory [3], suggests that the formation of texture is attributed to the difference in growth rates of different planes and also the deposit’s surface morphology. According to the theory there are two modes for deposit growth, one is an outward growth mode and the other the lateral growth mode. In the outward growth mode, grains bounded by slow-growing faces, which are perpendicular to the substrate, may grow preferentially. Whereas, in the lateral growth mode, the grains bounded by slow growing faces, which are parallel to the substrate, may grow preferentially. This theory, however, cannot answer the question what determines the crystal growth mode. Disagreement between this theory and experimental observation has also been found. For example, one may explain the (110) fibre texture of iron deposits using the geometrical selection theory with an assumption that the iron deposit grows in the lateral growth mode. Iron deposits, however, usually have bumped surfaces, and the observed facets are neither perpendicular nor parallel to the deposit surface, no matter what kind of textures the deposits have. In this paper we propose a Monte Carlo approach towards explanation of the texture evolution during electrodeposition. It is assumed that the deposit growth takes a “path” leading to a decrease in the system’s free energy, and the deposit will be textured as a result. An attempt has been made to simulate the effect of minimizing the system’s free energy on texture formation. The Monte Carlo simulation technique was proposed by Anderson et al. [20]. The model relies on the representation of the microstructure by a two-dimensional discrete lattice and defines the interaction between lattice sites. In the Monte Carlo model, the lattice site is randomly selected to test its energy and orientation. A new state of this site is calculated according to the Metropolis algorithm [20]. This new state (eg, the change in grain’s orientation) can be accepted only if the energy of the site decreases. As a result, the whole system will reach a minimum-energy state. The Monte Carlo technique has been used to simulate the nucleation and grain growth during recrystallization, solidification and vapour deposition processes [21-231. Since the deposit growth during electrodeposition is a nucleation and grain growth process, it is also possible to apply the Monte Carlo technique to electrodeposition. Assuming that the deposit growth minimizes the system’s free energy, one may expect that the texture development could be -modelled by taking into account the factors
magnetization energy may also affect the texture development, if the deposited substance is ferromagnetic and the deposition is carried out in a magnetic field.
DESCRIF’TION OF THE MONTE SIMULATION MODEL
CARLO
Electrodeposition is a process in which mass transfer is accompanied by charge transfer [24-281. During the process, metal atoms are ionized at the anode, then hydrated and move to the cathode through the electrolyte under the influence of an imposed electrical field. When reaching the cathode surface, the ions are dehydrated and discharged (they are now called “ad-atoms”). These ad-atoms are not immediately incorporated into the metal lattice, and instead, they diffuse on the cathode surface until reaching low-energy sites, such as micro-steps and kinks, where they will be accommodated. This process could be simulated by adding atoms to random locations on the cathode surface and allowing them to diffuse until they reach the low-energy sites. However, direct modelling of this process is intractable because it needs to take into account a large number of atoms, eg lo’atoms, to simulate a surface as small as 1 pm’. Thus, the present approach is not an atomistic model but a quasicontinuum model that accounts for the statistical nature of the deposit growth.
#44k444 444444 44b4444 44444)
0000000000 00000000000 0000000000 00000000000 0000000000 00000000000 0000000000 00000000000 0000000000 00000000000 0000000000 00000000000
0000000000 00000000000 0000000000 00000000000 0000000000 00000000000 0000000000 00000000000 0000000000 00000000000 0000000000 00000000000
0000000000 00000000000 0000000000
00000000000 00000000000
00000000000 0000000000 00000000000
0000000000 00000000000
and
Fig. 1. The microstructure at a cross-section of the deposit is mapped on to a two-dimensional triangle lattice. Each
related to the free energy of the system. For example, the anisotropic surface energy could play an important role in the texture formation. Anisotropic
occupied site is assigned a non-zero integer number to represent the orientation of the grain to which the site belongs. The site having zero number is unoccupied.
which
are
crytallographically
anisotropic
39
Texture development during electrodeposition1. Representation of the deposit’s microstructure and texture In this simulation, the continuum microstructure at a cross-section of the deposit is represented using a two-dimensional triangle lattice as Fig. 1 shows. Using the triangle lattice is convenient because the triangle lattice has a relatively more isotropic character than other lattices, such as the square lattice [20]. Each site in this lattice is assigned an integer number, P, to represent the orientation of the grain to which the site belongs. The lattice site has a certain volume, but this volume is small enough, so that the change in the site’s state can provide the information about the microstructural evolution. Initially, all sites are not occupied and their assigned P values are zero. When the deposit grows, the sites are occupied layer by layer. The occupation of a site is indicated by a change of its P value from zero to a positive integer. The overall process is modelled in the following way. Nuclei are initially generated at the cathode surface, and the lattice is then scanned layer by layer. In the scanning process, the site is randomly selected and its P value is tested. If P = 0, the site will be occupied and its P value will change to a positive integer. The value of this positive integer is dependent on the change in the local free energy. If P > 0, the site is already occupied and it is, therefore, skipped. The major task of the simulation is to model the textural evolution. The texture component is represented by its volume fraction, defined as the ratio of the lattice sites which contribute to this texture component to overall lattice sites. For example, for a fibre texture whose axis is normal to {hkl} plane, its volume fraction is the ratio of the sites which belong to the grains having their {hk/j planes parallel to the deposit surface to overall lattice sites. 2. Nucleation and grain growth The nucleation during electrodeposition is similar to the nucleation from a supersaturated solution or condensation from a supersaturated vapour [24]. When an ion enters the diffusion layer, it is dehydrated, discharged, and then diffuses and joins other ad-atoms to form a nucleus. The more ad-atoms moving on the cathode surface, the higher is the frequency of forming nuclei. Since the density of ad-atoms is dependent on the flux of ions carried by the deposition current, the frequency of nucleation is therefore affected by the current density. The relation between the nucleation frequency and the current density is expressed as [24,25]: Fnccexp[-a/(RTiS)]
(1)
where it, T and R are the current density, temperature, and gas constant, respectively, and o is a parameter proportional to the surface energy of the nucleus. It is seen that high current densities result in
high nucleation rates. During the deposit growth, the nucleation may also occur on growing crystal’s surface. However, compared to the grain growth, this type of nucleation is relatively unfavoured because it requires an activation energy to overcome a nucleation barrier, which depends on the surface energy of the critical nucleus. Whereas the grain growth, ie direct incorporation of ad-atoms into the crystal lattice, is favoured because there is no such energy barrier during the grain growth process. Once a nucleus is formed, it starts to grow. Usually, an ad-atom does not get directly incorporated into the crystal lattice as it reaches the deposit surface. Instead, it diffuses on the deposit surface until reaching a place where the accommodation of the ad-atom leads to a decrease in the local free energy. In this simulation, the driving force for the deposit growth is, accordingly, assumed to be the minimization of the system’s free energy. A grain can grow, if by doing so the system’s free energy decreases. When the deposit grows, previously empty lattice sites are occupied. The decision on what is the site orientation is made after calculating the free energy of the site. The free energy of a site includes the interaction between the site and its neighbours. The energy of a lattice site consists of two terms and is expressed as: H = -J~(&,P,
- 1) +
Hhk/
(2)
nn
where J is one half of the free energy of the bond connecting two neighbour sites, Pi represents the P value of site i, and 8pipi has the following value: 8pip,= 1, if Pi = Pj; or SP,P,= 0, if Pi # P,. The first term in equation (2) represents the interaction between the testing site and its neighbours. If a pair of sites have the same P values, there is no interaction between these two sites because there is no interface between them. The interaction exists only if the sites have different P values. In equation (2) the bond energy term W has an average value of the interaction between a pair of neighbour sites which does not take into account the crystallographic anisotropy, so it only accounts for the “isotropic” grain growth. An additional term must be added to modify the energies of bonds between differently oriented sites to simulate the formation of texture. More details about this modification term will be given in the following sections. 3. Factors aIkcting textural development When an ion arrives at the cathode surface, it loses its charge and diffuses until reaching a low-energy location, where this ad-atom will be accommodated. The low-energy location, ie the site where there is a decrease in the system’s free energy as the ad-atom sits in, depends on the grain orientation; therefore, the minimization of the energy could be used to predict the texture development. A part of the free
40
D. Y. Li and J. A. Szpunar
energy is the surface energy which is crystallographically anisotropic, and hence may play a role in the texture formation. For example, when arriving at a junction of two grains, an ad-atom will tend to be accommodated at the crystallographic plane which has a lower surface energy, leading to a decrease in the overall surface energy. As the deposition proceeds, the interface will move towards the grain which has a higher surface energy, and another grain would therefore grow preferentially. Another factor considered here is the magnetocrystalline anisotropy. During electrodeposition of ferromagnetic substances, if a magnetic field is applied the grain having lower magnetization energy may grow preferentially, resulting in texture evolution. In our simulation, the effects of these two anisotropy factors on the deposit’s texture have been taken into account by adding the second term in equation (2). 4. The process controlling parameter: current density In this Monte Carlo model, the current density is used as a parameter controlling the process. Although the driving force for electrodeposition is the over-potential, and a plater may control the process by means of the potential, the current density is often used to control the deposition. This is because the current density directly relates to the deposition rate, or the weight of the deposited substance, as Faraday’s law described [26]: dh h x=7=-
iCoE,, 69
where h is the thickness of the deposit, and Eel is the electrochemical equivalent of the deposited metal, t, p and i, are the deposition time, density of the deposit, and the current density, respectively, and o is the current efficiency. According to this formula, the deposition rate, or the weight of deposited substance, is proportional to the current density. Another reason for using the current density to control the process is that the current density influences the deposit’s microstructure, because the nucleation frequency is affected by current density as equation (1) indicates. So, the size and number of grains are influenced by the current density. Moreover, the current density may influence the deposit’s texture. It is known that at a high current density deposits may grow without any texture [24], although the texture can be quite strong when current is low. It will be demonstrated in a following paper [271that a high current density makes the deposit less textured, whereas a lower current density makes the deposit strongly textured. It is difficult, however, to use the current density directly as an input in the simulation. In order to convert the current density into a Monte Carlo parameter, we define the operation of scanning one lattice layer with 200 iterations as one Monte Carlo step (MCS). Because each iteration takes a certain time, one MCS therefore corresponds to a certain
time interval. The deposition rate can thus be modelled by using different numbers (n) of MCSs in scanning each lattice layer. This is to say that the time, At, necessary for an increment of the deposit thickness, Ah, could be replaced by n(MCS): Ah/Al ccAh/n(MCS).
(4)
The deposition rate is lower if the number of MCSs, n, is higher. Since the deposition rate is proportional to the current density (see equation (3)), the current density can thus be expressed by n, that is i,ccAh/AtccAh/n(MCS)cC
l/n.
(5)
The effect of current density on the deposit’s texture and microstructure can, therefore, be modelled, using the number (n) of MCSs involved in the scanning of one layer. 5. Calculation of the surface-energy anisotropy and the magnetocrystalline anisotropy As discussed earlier, the surface-energy anisotropy and the magnetocrystalline anisotropy may play important roles in the texture formation during electrodeposition. These two anisotropic energies will be used as inputs for the Monte Carlo simulation. Before doing that, these two anisotropic energy contributions need to be calculated. (i) The surface-energy anisotropy. The Monte Carlo model will be applied, in a following paper [27], to iron electrodeposition. In order to focus our attention on the mechanism of texture formation and to avoid complex effects of alloying elements, electrodeposition of pure iron will be simulated. Surface energies of different crystallographic planes of iron were calculated using the LennardJones [28] potential. The surface energy of a crystallographic plane (hkl) is defined as the difference in free energy between the crystal’s surface layer and its middle layer, which are parallel to the (h/cl) plane. To calculate the surface energy, a cubic cell having dimensions of 1.43 x 1.43 x 1.43 nm3 and containing 2000 iron atoms was constructed. The surface layer of the cell was then relaxed. The surface relaxation is necessary because when a crystal is split into two parts and forms two fresh surfaces, the atoms on the surfaces cannot be in their original positions and must shift to new positions, otherwise the surfaces will not have a thermal-equilibrium configuration. In the relaxation process, each atom in the surface region was allowed to move randomly by a short distance. If the old configuration energy is defined as El and the new configuration energy due to the shift is defined as E2, each move is accepted with a probability P = exp[-(Er - E,)/KT]; where, K is the Boltzmann’s constant and T is temperature. Note that if Ez < El, the probability of such a move is always larger than unity, and such a move is then always accepted. After the relaxation, the energy of a surface layer (3 A thick) and that of a central layer with the same thickness were calculated.
41
Texture development during electrodeposition-
LO OS
03
3
0.1
f
E *
Fig.
0.4 03 02 0.1 01) 2.
100 211 310 311 111 110 Calculated surface energies of iron (normalized).
Lennard-Jones potential was used in the calculation. Six planes were calculated; they are 100, 2 11, 3 10, 622 = 3 I I, 222 = 111 and 1IO planes.
The difference between these two energies gives the surface energy of the specific plane. Choice of the layer of 3 8, thickness for the surface energy calculation is made because the interaction between a pair of atoms declines with an increase in the distance between these two atoms. In addition, the interaction between nearest neighbouring atoms contributes to about 90% of the system’s energy. For these two reasons, the layer of 3 8, thickness was chosen. In such a layer, the nearest and secondary nearest neighbours (the distances to the nearest and secondary nearest neighbours are 2.48 and 2.87 A respectively) are included. Surface energies of a number of low-index iron crystal planes were calculated. Most of these planes correspond to the fibre textures, which were reported in the literature [8, lo]. Figure 2 illustrates the result of the calculation, from which one can see that the (110) plane has the lowest surface energy. (ii) The effect of hydrogen adsorption on surface energy. Metal deposition is often accompanied by
simultaneous hydrogen co-deposition [9]. When the hydrogen co-deposition takes place, the deposit’s texture may change because adsorbed hydrogen may modify the metal’s surface energy. According to Pet&s work [29], iron’s surface energy can be dramatically lowered by the hydrogen adsorption. It is possible that the surface-energy anisotropy is also changed, because the change in surface energy may vary for different crystallographic planes. It was demonstrated, using a quantum-mechanical calculation, that the greater is the distance between adjacent metal atoms, the lower is the activation energy for hydrogen adsorption [30]. This means that loose packed planes should have higher hydrogenadsorption abilities than close-packed planes [3]. Allowing for the fact that the hydrogen discharge is dependent on the deposition condition [26, 311, and that different crystallographic planes could have different hydrogen-adsorption abilities, it is expected that the change in deposition condition may change the metal’s surface-energy anisotropy
and this, in turn, leads to a change in the deposit’s texture. In order to investigate the texture variation with hydrogen co-deposition, the surface-energy changes, affected by the hydrogen adsorption, were evaluated using the following approach. A crystal cell containing about 1000 iron atoms and having dimensions 2.86 x 2.86 x 1.43 nm3 was first constructed, and then a certain amount of hydrogen atoms were randomly placed into the vicinity of the iron cell surface which was under investigation. The hydrogen and iron atoms in the surface region were then relaxed. After the relaxation, the energy of the surface layer (3 /i thick) was calculated. The difference in free energy between the relaxed surface layer and a central layer of equal thickness represents the surface energy, as influenced by adsorbed hydrogen. Figure 3 illustrates the effect of hydrogen on surface energies of various crystallographic planes. One can see that the iron surface energy is lowered with an increase in the hydrogen concentration, and that the rate of the lowering for different planes is different. It is,therefore, expected that with an enhancement in the hydrogen co-deposition, the iron deposit’s texture would change from one type to another, due to the changes in the surface-energy anisotropy. (iii) The magnetocrystalline anisotropy. When
electrodeposition of ferromagnetic metals is carried out in a magnetic field, the metals are magnetized. The magnetization energy is also crystallographically anisotropic, and there exists an easy axis for the magnetization [32]. The easy axis in BCC iron is along the (100) crystal axis. When the easy axis of a crystal is parallel to the magnetic field, the energy needed to magnetize the crystal is the lowest. The difference in magnetization energy between differently oriented grains is called the magnetocrystalline anisotropy; for cubic crystals, it is expressed as [32]:
031
0
,
,
,
2
,
4
,
,
,
6
~ a
, 10
Hydropa Concentration (moles/l)
q 310
+ 211
Oux)
A622
x 222
VllO
0 10 22
Fig. 3. The lowering effect of hydrogen on iron surface energy.
42
D. Y. Li and J. A. Szpunar
where, (XI,cc2and cr3are direction cosines of the angles between the applied magnetic field and three axes of the crystal reference frame, and K, and K2 are the magnetocrystalline anisotropy constants. For iron, KI = 4.8 x 104J m3 and KZ = 1.5 x 104J m3 [32]. This magnetocrystalline anisotropy has been used to simulate the effect of magnetic field on texture development. 6. The simulation procedure The input of the simulation includes the number of MCSs, ie the process parameter, used to scan one lattice layer, and the anisotropy factors: the surface-energy anisotropy and magnetocrystalline anisotropy. As mentioned earlier, the number of MCSs used to scan one lattice layer is inversely proportional to the current density and can be used to control the deposition rate. The anisotropy factors determine the competition between different crystallographic planes, and are therefore responsible for texture development. Nuclei are initially generated at the cathode surface. The lattice is then scanned layer by layer. In scanning one layer, a site to be tested is selected randomly. If the site has P = 0, energy of the site will be tested to decide the way in which metal ions will be deposited on to this site (ie grain growth or nucleation). The energy of the selected site is then calculated using the following form of equation (2):
boundaries and low-energy boundaries, are normally formed in polycrystalline materials. The first type are the high-angle grain boundaries which are the boundaries between grains with misorientation more than 15 degrees, and this type of boundaries usually have a high energy. The high-angle boundary structures are complicated and irregular, and their energies are represented usually using an average value which can be determined experimentally. The low-energy boundaries include the coincident site lattice (CSL) boundaries and low-angle boundaries, which are regular and have relatively low energies. However, the low-angle boundaries and the CSL boundaries usually have a small percentage of grain boundaries. Therefore, in the simulation, only the high-angle boundaries are taken into consideration. In summary, a “grain boundary bound” has an energy equal to 2J, and a “surface bound” has an energy equal to 2J.(y&h)( 1 + Ii&). If the deposition is carried out in a magnetic field, the magnetization energy is taken into account. HE,. is the magnetization energy term only for occupied sites. The notation, uuw, represents the site’s orientation which is parallel to the magnetic field. This term counts the effect of the magnetic field which is strong enough to saturate iron. The magnetization energy for each occupied lattice site is expressed as x.., = IEu = ~[K,(c+Y: + c&x: + &Y:) + K&&&4) (8)
H = -JC @f,P,“”
1) - JJ$I$,, !“rn
+ fp,
(7)
where J represents half of the energy of the bond connecting two neighbouring sites. If the bond connects two occupied sites with different orientations (ie, a “grain boundary bond”), the bond energy corresponds to the grain boundary energy ygb. If the bond connects an occupied site and an unoccupied site, the interface between these two sites is a surface and this “surface bond” corresponds to the surface energy ys. The bond energy 2J, therefore, has different values for the “surface bond” and the “grain boundary bond”. If the value of the “grain boundary bond” is defined as 25, the “surface bond” will be automatically adjusted and has an energy eqd t0 2J.(y&b). ys and ygb are average SUrfaCe energy and average grain boundary energy, respectively. In equation (7), Phk, is a modification term which makes the “surface bond” energy crystallographically anisotropic. This anisotropy parameter is the difference between the surface energy of the {hkl} plane and the average surface energy ys. Sum Z is taken over all of the six nearest neighbour sites, while Z’ is taken over only those sites which are connected to the testing site by “surface bonds”. It should be noted that in this simulation only the average value of the grain boundary energy is taken into account, which simplifies the energy calculation. In fact, two types of grain boundaries, so called high-energy
where 1 is a normalization coefficient which is the ratio of the magnetization energy of a lattice site to its bonds’ energy. This formula gives the magnetization energy of a site when it is in the state of saturation. The initial energy state, HO, of the selected site is first calculated using equation (7), and the P value of the site is then changed from zero to the P value of its nearest neighbour and an energy state, Hn, is calculated using equation (7) again. P values of all nearest neighbours are used in the same way to obtain a number of trial H,, values. The minimum H, is chosen as the final-state energy of the selected site and the corresponding P integer is assigned to the site. If AH = H, - HO< 0, the new P value is retained. Otherwise this change in P value will not take place. AH 2 0 means that the grain growth is not energetically favoured at the testing site. In this case, we generate a randomly oriented nucleus in a probability obtained by incorporating equation (5) into equation (1): Faexp(a/RTi,2)
= exp(a’n2/RT)
(9)
where u’ is a constant. Output of the simulation will provide information about texture, grain size and grain shape of the deposit.
Texture development during electrodeposition-
(4
43
Figure 4(b) illustrates the case of iron deposition in which the surface-energy anisotropy has been taken into consideration. An inhomogeneous distribution of differently oriented grains indicates the existence of a texture.
a-((
The minimum number of iterations for the lattice scanning
10 0 100
211
310
311
111
110
Chklt.%iO~
Fig. 4. Volume fractions of differently oriented grains: (a) without taking into account any anisotropy factor; (b) with taking into account the surface-energy anisotropy of iron.
MODEL TESTING AND DISCUSSION The random-number generator The equal-frequency distribution of the random numbers is of particular importance to the simulation of texture development. Each time when a nucleus forms, its orientation is decided by the randomnumber generator which creates a random number that represents the orientation of the created nucleus. If the generated random numbers are not distributed with equal frequency, orientations of the created nuclei will not be homogeneously distributed. As a result, simulated deposits will be “textured” even without introducing any anisotropy factor. To ensure that no texture appears when the simulation does not take any anisotropy factor into account, the growth of iron deposit with and without inputting the anisotropy factors has been simulated using an adopted random-number generator. Figure 4(a) illustrates volume fractions of differently oriented grains simulated without introducing any anisotropy factor. It has been observed that the distribution of grain orientations is homogeneous, ie no texture appears, which convinces us that the random-number generator used in the simulation model is reliable.
Therefore, the deposit texture will be induced only by anisotropy factors used as input into our simulation.
In the Monte Carlo simulation, the number of iterations for scanning each lattice layer is required to be high enough to ensure each site be scanned in an equal probability. If the number of iterations is less than a minimum number, not all sites are scanned, and this will result in a “vacancy” in the deposit microstructure. Theoretically, there is no “vacancy” left when the number of MCSs (one MCS includes 200 iterations) used in the scanning of each lattice layer is infinite. However, in the simulation, the number of MCSs cannot be infinite, and the “vacancy”, therefore, cannot be avoided completely. The vacancies are caused by iterations which are not enough to cover all sites on the deposit surface. The vacancies can influence the energy calculation of the sites near the vacancies. So the number of vacancies should be reduced to their minimum. Theoretically, the higher the number of MCSs involved in the scanning of one lattice layer, the lower is the number of vacancies in the deposit. Figure 5 illustrates that the number of vacancies decreases as the number of MCSs increases. It is seen that when the number of MCSs exceeds n = 4, the total number of vacancies in the simulated deposit (containing 1.2 x lo4 sites) has a low value of about fifty, and further change in the number of vacancies with the increase in the number of MCSs is slight-that is to say that only about 0.4% of total lattice sites are skipped if n is higher than four. The effect of such a small amount of “vacancies” could be ignored. Therefore, n = 4 is chosen as the minimum number of the MCSs used to scan each lattice layer.
Fig. 5. Vacancies decrease as the number of MC.% is raised. When the number of MC% exceeds n = 4, the number of vacancies is about fifty, which takes only 0.4% of total site number of the simulated deposit (containing 1.2 x IO”sites).
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D. Y. Li and J. A. Szpunar
The MCS clock and real time clock In the Monte Carlo simulation, the time scale is represented by the MCS, which corresponds to a certain period of real time, and therefore is chosen as a time unit for the simulation. The more MC% involved in the simulation, the longer the simulation takes. If the relationship between the MCS clock and the real time clock is known, the microstructure and texture development would be simulated quantitatively. This relationship could be rather complicated, and so far there is no satisfactory solution. It has been suggested that the conversion from the real time to the MCS time can be made by using atomic jump frequency which is related to an activation energy factor, exp(- WjkT), where W is the activation energy for each atomic jump [21]. However, in the Monte Carlo simulation model, which is a quasicontinuum model rather than an atomistic model, it is difficult to take real time into consideration. Therefore the present model is not a quantitative model but a qualitative one. Nevertheless, it is important that this Monte Carlo model is able to predict the tendency of the texture development during electrodeposition. The information provided by such a model is valuable to the deposition practice. Since a MCS corresponds to a certain time interval, the number of MC.%., n, used for scanning each lattice layer determines the deposit rate. The higher the number of MCSs for scanning one layer is, the longer it takes to form a deposit layer (ie the lower the deposition rate). At the same deposition rate, the more MCSs are involved in the overall deposition process, the thicker is the metal deposit. Although the corresponding real time interval is unknown, the linear relationship between the thickness of deposit and the deposition time can be described by using the number of MCSs. Since the deposition rate R = Ah/AtccAh/n(MCS) (see equation (4)), the overal thickness of a deposit may, therefore, be expressed as
Thickness = ZAh = R.ZAtaR.Z,n(MCS)
expected that crystallographic anisotropy factors which are related to the system’s free energy play an important role in the texture formation. Such anisotropy factors may include the anisotropies of surface energy, and the magnetization energy, if the deposition involves ferromagnetic metals and applied magnetic fields. The surface energy anisotropy and the magnetocrystalline anisotropy have been calculated and used as input in the simulation of texture evolution during iron electrodeposition, reported in a following paper [27]. The random-number generator, the minimum number of iterations, and the relation between the real time and the MCS time have been tested and discussed. REFERENCES R. Glocker and E. Kaupp, Z. P/t@. 24, 121 (1924). A. W. Hothersall, Trans. Faraday Sot. 31, 1242 (1935). A. K. N. Reddy, J. Elecrroanal. Chem. 6, 141 (1963). N. A. Pangarov, J. Electroanal. Chem. 9, 70 (1965). J. R. Park and D. N. Lee. J. Korean Inst. Metals 12.243 (1976). 6. I. Epelboin, M. Froment and G. Maui-in, Plating 56,
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SUMMARY A Monte Carlo model was developed to simulate the texture formation during electrodeposition. In this model, a two-dimensional triangle lattice is used to map the microstructure at a cross-section of electrodeposit. During the deposition, sites of the lattice are occupied layer by layer. The texture development is assumed to be caused by the minimization of the system’s free energy. It is
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