Journal of Cluster Science

https://doi.org/10.1007/s10876-022-02309-3

ORIGINAL PAPER

Temporal Growth and Aging of ZnO Nanoparticles in Colloidal

Solution: Phase Field Model
Priyanka Sharma1 · Sanjiv Kumar Tiwari1 · Partha Bir Barman1

Received: 6 January 2022 / Accepted: 19 June 2022

© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022

Abstract
The temporal growth and aging of ZnO nanoparticles (NP's) in colloidal solution were investigated both experimentally and
theoretically. UV–Vis spectroscopy revealed that the nucleation and growth of NP’s in solution occurs in less than 2 min.
Transmission electron microscopy images depict the morphology of aggregated NP’s. In atomically balanced reaction (for
sample S1), first growth takes place and then aging were observed. However, in the case of the atomically unbalanced reac-
tion (for sample S2), decoupling of nucleation from growth was seen after 20 min. This result was confirmed by the slopes
of ­dEg/dt ­(Eg = band gap) and dαmax/dt (αmax = absorption maximum) with time, which remains constant for sample S1 but
shows abrupt decrease for sample S2 after 20 min. Thereafter, growth was found to be controlled by the diffusion and reac-
tion parameters. The growth of NP’s was modelled using the phase-field model. The result from the current work reveals
that the nucleation, growth and aging of NP’s occur in the atomically balanced reaction whereas decoupling of nucleation
from growth happens in atomically unbalanced reaction.

Keywords PFM · ZnO nanoparticles · Temporal evolution · Nucleation and growth · Cahn–Hilliard equation

Introduction properties, therefore a fundamental understanding of the

numerous mechanisms that contribute to particle formation
Nanomaterial’s finds applications in multiple fields because is essential [14]. The three fundamental stages in chemical
of their large surface-to-volume ratio and other properties route synthesis are nucleation, growth, and aging [15]. How-
[1–7]. Recently, Zinc Oxide (ZnO) has emerged as a promis- ever, compared to nucleation and growth, the aging process
ing candidate in the field of bio nanotechnology due to its is relatively slow. Many authors have mainly studied the
wide bandgap (~ 3.27 eV) and large binding exciton energy aging process in detail and there exist very few reports on
(~ 60 meV). ZnO has potential applications in a variety of nucleation and growth. For real-time or in situ monitoring
optoelectronic and sensing devices, including UV lasing, of nucleation and growth, two main experiments have been
transparent conducting electrodes, gas sensors, antimicrobial proposed i.e. optical hyper Rayleigh scattering and small-
agents and drug delivery systems [8–12]. The solution-phase angle X-ray scattering (SAXS) and wide-angle X-ray scat-
approach, or chemical route, has become a very widespread tering (WAXS). These experiments have a time resolution
and cost-effective technology for the synthesis of nanopar- of a few hundred to a thousand milliseconds [16, 17], but
ticles (NP’s) and nano-architectures. To produce ZnO NP’s their limitation is that they were more suitable for metal
or their nano-architectures, either the hydrolysis of zinc NP’s such as gold (Au). Lianhua et.al has reported the in situ
salt in an organic solvent or a non-hydrolysis technique is observation of nucleation and growth of CdSe NP’s with a
used [13]. Furthermore, because the shape and size of NP’s fast integration time of the order of 2–4 ms. But, the experi-
have a substantial impact on their optical and other physical ment was carried out using a special high-temperature setup
of immersing transmission-reflection dip probe in reaction
solution [18]. Maria et al. explains the nucleation and growth
* Priyanka Sharma
sharmaps94@yahoo.in kinetics of NP’s by using real-small-angle X-ray scattering
(SAXS) and wide-angle X-ray scattering (WAXS) coupled
1
Department of Physics and Materials Science, Jaypee with the UV–Vis spectroscopy technique. They concluded
University of Information Technology, Waknaghat, Solan, that the early growth stage of particle radii from 1.1 nm to
Himachal Pradesh 173234, India

13
Vol.:(0123456789)
 P. Sharma et al.

2.2 nm occurs in 500 s (8.33 min) at a temperature 40 °C, Materials and Methods
whereas at the higher temperature of 50 °C, growth occurs
within 200 s (3.33 min). Their investigation concluded that Sample preparation involves the simple chemical synthe-
nucleation and growth were quite slow [16]. However, based sis where ­ZnCl2 (Loba Chemie, India) and NaOH (Merck,
on recent research, we infer that UV-spectroscopy was com- India) were used as precursors and ethanol (99.9% pure) as
monly utilized by researchers to analyse the nucleation and a solvent. All the chemicals were used as received. Separate
development of NP’s. Therefore, UV–visible absorption solutions of two different concentrations of ­ZnCl2 (1 mM,
spectroscopy has been demonstrated to be the most effective 1.25 mM) and NaOH (2 mM) were prepared in ethanol with
tool for real-time monitoring of nucleation and development a small quantity of water followed by continuous stirring at
of semiconducting NP’s. The average scan period of any room temperature. Then both of the solutions (­ ZnCl2 and
sample (spectral range of 350–600 nm) in a conventional NaOH) were mixed slowly to obtain the homogenous colloi-
UV–visible spectrograph is on the order of a few min, pre- dal suspension of ZnO NP’s [26]. Thereafter, two different
senting only the specifics of growth and aging. As a result, samples S1 (atomically balanced reaction, 1 mM of Z ­ nCl2
a suitable theoretical model is required to complement the and 2 mM of NaOH) and S2 (atomically unbalanced reac-
UV–visible spectroscopy results with a thorough grasp of tion, 1.25 mM of ­ZnCl2 and 2 mM of NaOH) were chosen
the growth process. for the study of the growth process. The chemical reactions
Some well-known theoretical methods are classical of the synthesis process are given below:
nucleation theory (CNT), Ostwald ripening and Lifshitz,
Slyozov and Wanger (LSW) theory [19–21]. However, each nZnCl2 + 2nNaOH + lH2 O
model has its own merits and demerits. For example, CNT → Zn2+ −
n (OH )m
(H2 O)l Cl−k
fails to predict nucleation rate and dynamical parameters (1)
+ (2n − m)OH − + 2nNa+ + (2n − k)Cl−
such as kinetic pre factor and interfacial free energy which
→ nZnO + (l + n)H2 O + 2nNaCl
were used as fitting parameters. Whereas, the LSW model
assumes that the transport between the growing particles where m,n, and k represent the number of moles and
is diffusion-limited only. The phase-field model (PFM), to 2n = m + k represents the zero charge precursor for homoge-
the best of our knowledge, is the best known microscopic neous nucleation. We choose S1 according to the zero charge
approach to understanding nucleation, growth, and morpho- precursor (Zn ligand) and S2 according to the 2n ≠ m + k
logical evolution because it provides us to monitor the evolu- (also n ∶ 2n ≠ ZnCl2 ∶ NaOH ) for non-zero charge precur-
tion of the microstructure in two or three dimensions at the sor and was chosen purposely because of the large concen-
time of phase transitions progresses. Yang et al. addressed tration of ­Zn2+ ions.
the nucleation and growth by grain evolution using PF mod- To study the temporal evolution of nucleation and growth
elling [22]. To explain the homogeneous and heterogene- of NP’s in sample S1 and S2, the in situ UV–Vis absorp-
ous nucleation of NP’s, Laszlo et al. combined the density tion spectra of suspended NP’s was recorded at different
functional theory with the PF theory [23]. The PF model was time intervals viz. 0, 6, 12, 18, 24, 30 and 36 min using a
also used to elucidate the mechanisms of silver NP sintering UV–visible-NIR spectrophotometer (Perkin Elmer Lambda
using conserved and non-conserved parameters [24]. Talha 750, USA) with a scan time of 2 min in the spectral range
et al. also looked at the advancement of using the PF model of 280–450 nm. Furthermore, for morphological evolution
to study the degradation of metallic materials under environ- Transmission Electron Microscope (TEM: Hitachi-H-7500,
mental attack, employing both the Cahn–Hilliard (C–H) and Japan) images were recorded at the same time interval. The
the Allen–Cahn (A–C) equations [25]. However, there are no growth of the NP’s was modelled using PFM. All the meas-
such reports employing PFM to study the growth and aging urements were performed at room temperature, after prop-
process of ZnO NP’s until now. The PF can thus be used erly stirring the solution.
to investigate the growth mechanism right before the com-
mencement of particle growth (when crystal order is absent).
The present study used in situ UV–visible absorption
spectroscopy to study the temporal evolution of ZnO NP’s Results and Discussions
formation. A detailed investigation of balanced and unbal-
anced atomic processes was carried out, with the difference Experimental Analysis
between the two studied analytically using the UV–Vis tech-
nique. In addition, the PFM was used to simulate the growth Figure 1 represents the graph of the time derivative of 𝛼max
process. Furthermore, the key novelty of the current study (𝛼max corresponds to the maximum absorbance value noted
work is the decoupling of nucleation from growth in atomi- from the absorbance spectrum shown in the inset of Fig. 1)
cally unbalanced reaction system.

13
Temporal Growth and Aging of ZnO Nanoparticles in Colloidal Solution: Phase Field Model

Fig. 1  Variation of absorption rate with time, inset shows typical Fig. 2  Variation of the rate of bandgap with time, inset shows the var-
absorbance spectra of the ZnO colloidal particles in the solution at iation of bandgap calculated from Tauc plot with time
different growth times for atomically balanced sample

with respect to the time. As in our case, we are monitoring which makes nucleates unstable. Physically speaking, the
the growth dynamics, thus in a dynamic or non-equilibrium presence of excess Z ­ n2+ ions causes large fluctuations in
state, absorption is given by the equations: surface energy. Resultantly, the nucleation reaction gets
decoupled from growth after 20 min of reaction time. For
dCZnO
𝛼(t) =
n1
= krea CZnCl
n
CH2 O∕ethanol (2) colloidal solution, absorption is proportional to the sum of
dt 2 2
the volume of all the particles which gets a redshift with
The time derivative of the above equation is given by: an increase in time. It is also evident from the figure that
half of the absorption maxima ­( Imax/2) remain constant
n1 n2
d𝛼(t) d2 CZnO dkrea n1 n2
d(CZnCl CH O∕ethanol )
2 2
with time (marked with a dotted arrow) indicating that
= = C C + k
dt dt2 dt ZnCl2 H 2 O∕ethanol rea
dt the particle size distribution remains constant throughout
(3) the process. In steady-state, allowed direct optical absorp-
where CZnO, CZnCl
n1 n
, CH2 O∕ethanolrepresents the concentration tion transition between two electronic states is given as
α(hυ) = A (hυ-Eg)1/2, where hυ and ­Eg represent the energy
2 2
of ZnO NP’s, ­ZnCl2, ­H2O/ethanol, krea is the reaction rate
and ­n1 and ­n2 are the reaction order respectively. Therefore, of the photon and optical bandgap respectively, A is con-
the rate of change of absorption provides more information stant inversely proportional to the index of refraction (μ).
about the temporal change of reaction rate and reactant con- However, for transient growth of NP’s, μ is not constant
centration, since k rea changes according to the rate of and varies with the progress of time unless the steady-state
increase/decrease of temperature given by Arrhenius law is reached. The average optical energy gap of colloidal
[27]. If dαmax(t)/dt remains constant; it implies that both tem- ZnO NP’s was calculated from the equation (αhυ) 2 = A
poral changes of reaction rate and reaction concentration (hυ-Eg) with a crucial assumption of the constant refrac-
balance each other or simply aging of the sample. tive index of nucleates or particles with time i.e. density
of the particles remains constant during the whole process.
Derivative of bandgap (­ Eg calculated using above men-
( )
dkrea E E dT
= A2 exp − A (4)
dt RT RT dt tioned Tauc equation) with respect to time was plotted in
Fig. 2.
where the symbols have their usual meaning. Particle size was calculated from Brus equation [28,
Figure 1 shows the variation of dα max(t)/dt with time 29] given by:
for sample S1 and sample S2. Sample S2 was chosen pur-
posely because it has a larger concentration of ­Zn2+ ions
( )
h 1 1 1 1.8e 1
E = Egbulk + + − (5)
as compared to O ­ H− ions. It is evident from Fig. 1 that, 8e mo me mo mh r2 4𝜋𝜀𝜀o r
for sample S1, dα max(t)/dt is almost flat/constant which
indicates constant aging of the sample, whereas for S2, where ­Egbulk is the bandgap of bulk ZnO, r is the radius of
dα(t)/dt first increases up to 20 min then decreases dras- the colloidal particle, m
­ e and m
­ h are the effective mass of the
tically. This might be due to excess ­Z n 2+ ions present electron (0.26) and holes (0.59), ɛ is the relative permittivity

13
 P. Sharma et al.

of ZnO (8.5), ɛ0 is the permittivity of free space (8.85 × 10 Fig. 3 represent the function plot of d = ­d0 + ­ktp, for p = 1/3
−12
F/m) and ­m0 = 9.1 × 10 −31 kg. and 1/2 respectively. For sample S1, fitting parameters are
Taking the derivative of Eq. (5) w.r.t time: ­d0 = 1.61 ± 0.06 nm, p = 0.37 ± 0.06 and k = 0.11 ± 0.03 with
)( ) fitting accuracy of 98%, whereas for sample S2, parameters
dEg are ­d0 = 1.78 ± 0.05 nm, p = 9.99 ± 1.72 and k = 6.28 × ­10–16
( ( )
h𝜋 1 1 1 dr 1.8e 1 dr
=− + +
dt 3e mo me mo mh 4
𝜋r3 dt 𝜀𝜀o 4𝜋r2 dt respectively. The function plot for the samples S1 & S2
3
(6) reveals completely different behaviour in curvature which
was due to interfacial energy constant ‘k’, which further
The above equation implies that for zero slopes (dE g/
depends upon the local curvature of interfacial energy. In
dt = 0) or in steady-state, the surface to volume ratio of par-
sample S2, the local thermo-dynamical fluctuations induce
ticles remains constant. However, for finite dEg/dt values
large perturbation in local interfacial energy curvature which
the first term indicates volumetric dependence and the sec-
decouples nucleation rate and growth rate kinetics. There-
ond term indicates surface dependence. Figure 2, shows the
fore, because of thermo-dynamical fluctuations and excess
variation of dEg/dt with time. For sample S1, the constant
­Zn2+ ions, the sample S2 shows the unexpected value of k
value of dEg/dt indicates a constant surface-to-volume ratio
and further different behaviour of the nature of radius of
or the aging of the sample, which is in accordance with the
curvature in the function plot of Fig. 3 (inset).
absorption data (Fig. 1). For sample S2, dEg/dt remains ini-
To visualize the effect of interfacial energy curvature on
tially positive for 20 min and then turns negative, which
the morphology of colloidal particles of S2, nanoparticles
implies surface-mediated growth and gradual decoupling of
were deposited on the TEM grid after sonication. Figure 4
nucleation and growth parameters respectively. The inset of
shows the TEM image for different times. Although it seems
Fig. 2 shows the variation of bandgap with time, which is in
that a large number of particles are clustered together, clear
accordance with Brus Eq. (5). Figure 3, shows the variation
variation in morphology can be noticed in Fig. 4e and f. The
of particle size with time for S1 and S2 (inset). Experimental
inset of Fig. 4 shows log-normal particle size distribution.
data points were fitted to the equation:
Log-normal distribution becomes very narrow in Fig. 4f,
d = d0 + ktp (7) which is a visual signature of the decoupling of nucleation
rate and growth rate kinetics. Figure 5 shows the cumulative
where d,d0 and k represent the particle size, initial parti- log-normal distribution of particle size and the inset shows
cle size and proportionality constant (physically represents the variation of average particle size (calculated from the
interfacial energy) respectively. The parameter p = 1/3 indi- TEM image) with time. The inset figure indicates a decrease
cates purely diffusion-controlled growth for uniformly dis- in particle size after 30 min due to the decoupling of nuclea-
tributed colloidal particles. Blue and green dotted lines in tion and growth. However, the size calculated from TEM
is larger than the size calculated from the Brus equation
because of clustering (Figs. 3, 5 (inset)).

Modelling of Growth

Reaction–Diffusion Controlled Growth

The chemical reaction for the synthesis of the ZnO NP’s in the
solution phase is represented in Eq. 1. In the case of homoge-
neous nucleation, due to the concentration gradient of precur-
sors, NaOH will diffuse into the ­ZnCl2 solution. Qualitatively,
the first diffusion occurs and then a chemical reaction starts.
For a complete reaction to occur, ions must diffuse at least to
the order of the radius of the added droplet, hence
­rdroplet = (6Dt)1/2 (where r­ droplet = radius of the droplet, D = dif-
fusion constant and t = time). Typically 2 ml of NaOH droplet
has a radius of the order of 0.7 mm (4/3 π ­r3 droplet = 2 ml) since,
D ~ 1.3 × ­10–5 ­cm2 ­sec−1, hence t ~ 10 s. This simple calculation
Fig. 3  Variation of particle size with time for atomically balanced reveals that ions will take about 10 s to fully diffuse into the
sample S1, solid red line represents theoretical fit (d = ­d0 + ­ktp) to
­ZnCl2 solution. However, at the same time ­OH− ions present
experimental data points with p = 0.37, blue and green dotted lines
shows function fit with p = 1/3 and 1/2 respectively, inset shows the at the surface will react with Zn ions and subsequently form
variation of particle size with time for sample S2 Zn(OH)2 which acts as a nucleation center for further growth.

13
Temporal Growth and Aging of ZnO Nanoparticles in Colloidal Solution: Phase Field Model

Fig. 4  TEM images with the log-normal distribution of the ZnO nano-colloids for different times obtained after sonication for sample S2

Therefore, it is quite logical to assume that the firstly nuclea-

tion center forms and growth starts thereafter. Also, diffusion
of NaOH into ­ZnCl2 is slow compared to their chemical reac-
tion at the surface. Therefore, a local equilibrium is created
between free NaOH (in bulk) and immobilized component
Zn(OH)2 at the surface of the drops. In this case, the concen-
tration of Zn(OH)2 or nucleates must be correlated to the con-
centration of NaOH. We assume CZn(OH) = R ­Cn NaOH, where
2
R = constant and n = 1, 2, 3… and so on. Physically, R repre-
sents the nucleation rate in chemical diffusion reactions. If R
is very large then the nucleation rate or formation of Zn(OH)2
is also very large as compared to the temporal decay of initial
NaOH. For n = 1, S = R C (where CZn(OH) = S), the diffusion
2
equation is given by:

dC d2 C dS
= D 2 − , for linear equilibrium (8)
dt dx dt

Fig. 5  Temporal evolution of the log-normal distribution observed

from the TEM images. It represents the narrowing of the curve and dC d2 C dC
redshift in its peak value. The inset reveals the temporal evolution of
R =D 2 − (9)
dt dx dt
average particle size

13
 P. Sharma et al.

Fig. 6  Simulated images and log-normal distribution showing the temporal evolution of the nano-particles with different time steps having
parameters (C = 0.25, A = 2, M = 0.5, κ = 0.5. At Δt = 100, no nuclei were observed

dC D d2 C constant D(R + 1) −1 and D(2R + 1) −1 for n = 1 and 2

= (10) respectively. It is important to note that, in this particular
dt R + 1 dx2 2
diffusion–reaction process dC dt
= D ddxC2 − R(C) , we did not
This means that in the overall process of diffusion, the consider local reaction or any nonlinear combination of
diffusion constant (D) will reduce by a factor of R + 1 if ­CNaOH. This is because, local reaction and diffusion–reac-
the reaction involves (Eq. (10)). Similarly, for n = 2, the tion are of the form, where R(C) accounts for all local
diffusion Eq. (10) with reaction will be modified to; reactions used in the field of pattern formation given by
the Fisher-Kolmogorov-Petrovsty-Piskunov equation [30].
dC D d2 C
= (11) Moreover, a different form of R(C) yields different equa-
dt 2R + 1 dx2
tions. For example for R(C) = C(1 − C), R(C) = C(1 − ­C2)
Also, D will be reduced by a factor of 2R + 1. This and R(C) = C(1 − C)(C − α) with 0 < α < 1 corresponds to
indicates that the ideal situation is to assume the existence Fisher’s equation, Newell-Whitehead Segel equation, and
of a large number of nucleation centers for the further Zeldovich equation respectively [31]. These equations
growth process. Therefore, the formation of ZnO nano- were used to describe the spreading biological popula-
colloidal in suspension is due to non-separable events tion, Rayleigh Benard convection and combustion theory
of diffusion and reaction both with effective diffusion respectively.

13
Temporal Growth and Aging of ZnO Nanoparticles in Colloidal Solution: Phase Field Model

Fig. 7  Simulated images and log-normal distribution showing the temporal evolution of the nano-particles with different time steps having
parameters (C = 0.30, A = 1, M = 1, κ = 0.5)

Phase‑Field Modelling (PFM) within the solution using PFM is represented by the order
parameter ( 𝜑 ) and 𝜑 = 0, 0.5 and 1 represent the disor-
Furthermore, we use a theoretical method to understand dered, interface and order phase respectively.
the nucleation and growth mechanisms of the NP’s because The temporal evolution of the solid phase is given by
UV–Vis spectroscopy results only reveal the growth after the C-H equation [34, 35].
2 min of scan time. Thus for the theoretical approach, we ( )
assume that all nucleation centers have been formed and 𝜕𝜑 𝛿F 𝛿F 𝜕𝜑
= ∇ ⋅ M∇ , and𝜇 = or = ∇ ⋅ M∇μ (12)
are available for the further growth process. Physically, 𝜕t 𝛿𝜑 𝛿𝜑 𝜕t
a solution with a nucleation center can be thought of or
where 𝜑, M, μ, and F represent phase, mobility of solute,
visualized as an ordered phase (nucleation centers), dis-
chemical potential and total free energy functional respec-
ordered phase (liquid) and interface of nucleates and liq-
tively. Here order parameter (𝜑 ) is a conserved quantity.
uid or interface of order and disorder phase respectively.
Total free energy functional is given as
A significant spatial variation of phase is assumed to be
larger than the width of the interfacial region. This is a
∫
1|
{ }
2
F(𝜑, T) = Wo ∇𝜑|| + f (𝜑(x), T(x)) d3 x (13)
very important assumption because the constant interface 2 |
will eventually evolve into spherical particles whereas; V

spatial variation of the interface will give rise to differ- where

ent morphology which is the main reason behind the dif-
ferent reported morphology such as nano-flower, nano- 𝜀 k T
f (𝜑, x) = 𝜑(1 − 𝜑) + B3 𝜑(x)ln𝜑 + (1 − 𝜑)ln(1 − 𝜑)
flakes, etc. [32, 33]. Since all the nuclei had already been 2a 3 a
formed, only growth is taken into account (there will not (14)
be any involvement of latent heat). For this, we prefer to Here, ‘a’ is the distance between two discrete points
choose the PFM, as PFM is based on the conserved field and we assume temperature is constant and no nearest-
variables and expresses the phase as a function of time. neighbour interaction. ­W 0 is a coefficient that includes
Moreover, growth is also responsible for continuous phase surface energy because it multiplies a gradient of order
transformation. Temporal evolution of the nano-structure

13
 P. Sharma et al.

parameters that varies only at the interface. For simula- were subjected to time-dependent continuous nucleation
tions, we choose free energy function f (𝜑 ) given as: and growth studies. The findings from UV–Vis spectros-
copy revealed that the variation of dαmax(t)/dt with time for
f (𝜑) = A𝜑2 (1 − 𝜑)2 , (15) S1 was nearly constant which indicates constant aging of
where A is the free energy constant. the sample. However, for the sample S2, dαmax(t)/dt vari-
Using Eqs. (13) and (15), Eq. (12) governing the temporal ation first increased up to 20 min followed by a decrease
evolution of phase can be modified as thereafter. The reason for this variation is due to the excess
­Zn2+ ions present that make the nuclei unstable and results
𝜕𝜑 ) 𝜕𝜑 in the decoupling of growth from nucleation. Furthermore,
= ∇ ⋅ M∇ h − 2𝜅∇2 𝜑 , or = M∇2 (h − 2𝜅∇2 𝜑)
(
𝜕t 𝜕t the time-dependent variation of dEg/dt for S1 was also con-
(16) stant which signifies a constant surface to volume ratio over
Here, h = 𝜕f (𝜑)
= 2A𝜑(𝜑 − 1)(2𝜑 − 1), κ is the gradient the time according to Eq. (6). However, the dEg/dt for the
𝜕𝜑
energy coefficient which represents interfacial energy. S2 sample was initially positive till 20 min and then after
turned negative. The negative value of dEg/dt implies that it
is a surface-mediated growth and decoupling of nucleation
Numerical Implementations
and growth parameters. Also, Fig. 3 reveals positive and
negative radius of curvature for samples S1 and S2 respec-
Fourier transformation method is being used to simulate the
tively. The main reason for the negative radius of curvature
transformation Eqs. (16) to (17) given below:
behaviour of S2 was the thermo-dynamical fluctuations and
𝜕{𝜑}k the presence of an excess of Z ­ n2+ ions. Additionally, the
= −Mk2 {h}k + 2k2 {𝜑}k (17)
( )
𝜕t nucleation and growth process of the ZnO NP’s were also
investigated using the PFM model which explained the evo-
where {.}k indicates the spatial Fourier transform of the lution of ZnO NP’s with respect to increase in time. Moreo-
quantity{.} and k is the Fourier vector. The semi-implicit ver, the C-H equation from the PFM model also explained
discretization of Eq. (17) is given by: the temporal growth of the particles with the assumption of
𝜑(k, t + Δt) − 𝜑(k, t) a constant continuum term (ϕ). Thus, the findings from the
= −Mk2 {h}k − 2k4 M𝜅𝜑(k, t + Δt) current study advocated its importance in materials science
Δt
and device fabrication process.
𝜑(k, t) − Mk2 {h}k Δt Acknowledgements The authors would like to thank the Jaypee
𝜑(k, t + Δt) = (18)
1 + 2ΔtMk4 University of Information Technology, Waknaghat, Solan, Himachal
Pradesh, India for financial and infrastructural support for current work.
Here, ∆t represents the time step for numerical integra-
tion. For simulation we have used scaled parameters, grid Author Contribution All of the authors contributed to the writing of
size scaled to interfacial width (W), time step to (τ) and the manuscript. The final version of the manuscript has been approved
by all authors.
free energy functional to k­ BT. Scaled parameters make free
energy constant (A) and gradient energy dimensionless. So,
Declarations
we have mainly four scaled parameters used for computation
i.e. initial concentration of nucleates (C), free energy con- Conflict of interest The authors declare no competing interests.
stant (A), mobility (M) and interfacial energy (κ). However,
a very small noise term (0.002) was added to C to introduce
spatial fluctuation in C. Two different set of parameters i.e.
(C = 0.25, A = 2, M = 0.5, κ = 0.5 and C = 0.30, A = 1, M = 1, References
κ = 0.5) were used and size evolution were monitored at dif-
ferent time step ∆t as shown in Figs. 6 and 7. 1. J. Zhang, C. Li, Y. Zhang, M. Yang, D. Jia, G. Liu, Y. Hou, R.
Li, N. Zhang, Q. Wu, and H. Cao (2018). J. Clean. Prod. 193,
236–248. https://​doi.​org/​10.​1016/j.​jclep​ro.​2018.​05.​009.

Conclusions

Current study reports the synthesis of ZnO NP’s using ­ZnCl2

and NaOH at two different concentrations through atomi-
cally balanced reaction for sample S1 and atomically unbal-
anced reaction for sample S2 respectively. The ZnO NP’s

13
Temporal Growth and Aging of ZnO Nanoparticles in Colloidal Solution: Phase Field Model

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