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Interplay between Kinetics and Thermodynamics on the Probability Nucleation Rate of Urea - Water Crystallization System Shijie Xu, Yifu Chen, Junbo Gong, and Jingkang Wang Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.7b01735 • Publication Date (Web): 26 Feb 2018 Downloaded from http://pubs.acs.org on February 27, 2018
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Interplay between Kinetics and Thermodynamics on the Probability Nucleation Rate of Urea - Water Crystallization System Shijie Xu†,‡, Yifu, Chen†,‡, Junbo Gong*,†,‡, Jingkang Wang†,‡ †
State Key Laboratory of Chemical Engineering, School of Chemical Engineering
and Technology, Tianjin University, Tianjin 300072, People’s Republic of China. ‡
Collaborative Innovation Center of Chemical Science and Chemical Engineering
( Tianjin ), Tianjin University, Tianjin 300072, People’s Republic of China.
ABSTRACT In this contribution, by employing the Possion distribution combined with the regular solution theory in the CNT framework, we construct a new model to uncover the relationship between induction time (tind) and supercooling (∆T) and saturation temperature (T0) at different specific probability. By choosing the urea aqueous solution as a benchmark system, we show that the value of ln (1/tind) follows a reasonable linear relationship with (T0/∆T)2/(T0-∆T) except the probability value tends to 0 or 1. Furthermore, we also shed new light into the role of chemical potential difference and nucleation temperature in determining the nucleation rate, namely, although the chemical potential difference is the driving force for the crystallization process, it does not always favor the nucleation process. We demonstrated that when the chemical potential difference increases as the nucleation temperature decreases (∆T gradually increases), in this case, the kinetic factor overwhelms the thermodynamic factor thus leading to a faster nucleation rate by employing the CNT theory. However, when the chemical potential difference decreases as the nucleation temperature increases, we found that increasing the nucleation temperature favor the nucleation process both in kinetic and thermodynamic aspects.
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1. Introduction Nucleation is a process of forming a new phase and the resulting creation of crystalline materials from liquid-phase precursors are central to the science and process engineering of materials in their broadest sense1. In general, in the context of crystal materials, the ability to precise control the process of molecular assembly from solution will be crucial for determining the product quality such as crystal structure and size distribution of the crystals 2, 3. Unfortunately, the fundamental understanding of crystal nucleation from solution still remains unclear due to the small length and time-scales involved 4,
and crystallization processes have to be developed based on
scaling-up of laboratory experiments. Thus, the key strategy to control the product quality is to understand and control the nucleation behavior in crystallization process. The formation of crystals (or other ordered solids) is a first-order phase transition 5, 6
. The thermodynamic of the classical nucleation theory (CNT), stemming from the
work of J.W Gibbs 7, and then the kinetic of the CNT was developed by Becker and Döring 8 and has been refined over the decades 9. One of the most basic assumptions of CNT is that the emerging nucleus possesses the same macroscopic structure of the bulk material. This simplified landscape has been repeatedly challenged
3, 10-16
. The
study of nucleation mechanism has recently focused on seeking a fundamental understanding of detailed molecular pathways of crystal nucleation from the perspective of solution chemistry6,
17-19
. Typically, through examining structural
evolution of solute molecules in solution, researchers are committed to finding the relationship between the structure of solute species in solution and the synthons in the macroscopic crystal phase. A direct structural resemblance suggests that the application of classical nucleation theory may be meaningful, but not always, while the lack of such a link indicates involvement of non-classical nucleation. Unfortunately, the non-classical nucleation mechanism still remains obscure, hence, the classical nucleation theory is still the main framework to study the nucleation kinetic from solution. One of the most important tools to probe the nucleation process is the induction time, which is the time elapsed from the instance of achievement of supersaturation to ACS Paragon Plus Environment
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create a nucleus of thermodynamically stable phase at a constant supersaturation 20. In the past decade, many researchers had reported many nucleation kinetics data based on the assumption that the induction time is both deterministic and volume independent
21-26
. However, as early as 1952, Turnbull has addressed the stochastic
nature of the nucleation process 27. Then, Toschev 28 further developed the probability nucleation theory, as well as the difference between steady-state and unsteady-state nucleation rate. Later, Alexander F. Izmailov and his co-authors
29
performed
nucleation experiments with the same solution under exactly the same conditions many times N (N ~ 150-300) based on simultaneous levitation of N (N ~ 150-300) identical microdroplets (1 ~ 20 µm in diameter) of supersaturated solutions in a solvent atmosphere by employing the linear quadrupole electrodynamic levitator trap (LQELT) which enable fast observation of nucleation. Recently, Jiang and ter Horst 30 applied the steady-state stochastic formulation of the nucleation theory to determine the nucleation rate of m-aminobenzoic acid and L-histidine. Although nucleation is stochastic in nature 29, 31-33, this aspect of nucleation appears only in small sample and relatively high supersaturation 34. Even for large samples, the stochastic aspect could be observed in theory, but not necessary. In some special cases, metastable zone widths and induction time may be treated as deterministic quantities 35-38. From the perspective of the classical nucleation theory, the nucleation rate is governed by the nucleation work and the pre-exponential factor, which explains the existence of a metastable state and indicates the difficulty of controlling nucleation process 39. In the induction time experiment, create chemical potential difference has two ways which involving fix the supercooling (change the saturation temperature T0 and crystallization temperature T1) or fix the saturation temperature (change the T1 to create different chemical potential difference). The latter has been validated that the nucleation rate increases as the chemical potential difference increases
21-24, 40-44
.
However, when the supercooling value is fixed and the chemical potential difference decreases with increasing the T1, then one may expect an unconventional relationship between chemical potential difference and nucleation rate since decreases the solid – liquid interfacial energy and increases the value of kinetic factor both facilitate the ACS Paragon Plus Environment
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nucleation process22,
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30, 45, 46
. In this contribution we question how the chemical
potential difference and crystallization temperature affect the possion probability distribution of the nucleation induction time. In particular we further explore the central question of how the chemical potential difference and crystallization temperature affect the nucleation rate from induction time experiments and associated possion distribution model in the CNT framework. As a vehicle for this journey we employ urea as our model since it has been the subject of significant previous study. Lee et al
47
had determined systematically the
solubility of urea in water, methanol, ethanol, isopropanol, and their mixtures. Then, the induction period of urea in methanol has been measured experimentally by the visual observation method
48
which has been used to estimate the nucleation
parameters. However, the authors only investigated the effect of chemical potential difference on the nucleation parameters but not point out the role of nucleation temperature in the nucleation process. Therefore, the aim of this work is 3-fold: (1) to construct the relationship between supercooling and saturation temperature and nucleation probability and answer the question that how the chemical potential difference and nucleation temperature affect the induction time Possion distribution; (2) to answer the question that does the chemical potential difference always facilitate the nucleation process? (3) to take further insight into the question that how the chemical potential difference and nucleation temperature affect the nucleation rate. 2. Theory 2.1. Nucleation rate from Possion distribution The formation of nuclei under fixed chemical potential difference can be seen as a series of random events occurring along the time axis in small working volume. If these events are independent of each other, then the probability of occurrence of m nuclei in the [0, t] period can be calculated using the Poisson distribution: 30 Pm =
Nm exp ( − N ) m!
(1)
Where N is the average number of nuclei formed within the same time frame. Then, the steady-state nucleation rate can be derived as Eq.(2) and the derived details can be ACS Paragon Plus Environment
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found elsewhere 30: t = 1 − exp −
(2)
where J is the steady-state nucleation rate and V is the volume of the solution, tind is the induction time as determined in the experiment. Therefore, by plotting 1/V·ln (1-p(t)) versus tind, a straight line should be yielded and the steady-state nucleation rate can be obtained. Behind the treatment leading to Eq. (2) is the assumption that the nucleation occurs at steady-state conditions with respect to the cluster distribution. Clusters constantly form and redissolve in each size class, but the distribution maintains a steady state in the solution 49. In the case of unsteady state conditions, the nucleation distribution will show a curvature at short induction times, as shown by Toschev
28
, and there is a time lag for the diffusional process to reach steady-state
conditions. 27 2.2. Classical Nucleation Theory Based on the classical nucleation theory, the steady state nucleation rate can be given by: = exp
γ
ln = ln −
!
γ
(3)
!
(4)
where the constant A is associated with the kinetics of nuclei formation in the growth medium lying between 1015 to 1042 m-3 s-1, and γ is the effective interfacial energy, Vs is the volume of a solute molecule (i.e., molecular volume), in this case, the molecular volume of urea is 7.47·10-29 m3, k is the Boltzmann constant equal to Rg/NA (NA is the Avogadro number), T1 is the nucleation temperature and S is the supersaturation. Thermodynamic parameter B is defined as
γ
. Furthermore,
based on the theory of regular solutions, one may write the relationship between the ratio of solute concentrations c0 and c1 corresponding to saturation temperature T0 and nucleation temperature T1, respectively, and the supercooling ∆T in the form: ln $ = ln
% %&
=
'() ' *+ &
(5)
Hence, the difference in chemical potential ∆µ between the solute in the solution µl
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and in the solid phase µs, may be written as Eq. (6): Δ- = - − -. = /012 $
(6)
Therefore, Eq. (3) can be rewritten as follow: J = Aexp [
6γ
*+ '()
8
7
& 98
& '
]
(7)
Then, combine Eq. (2) with Eq. (7), we can construct the relationship between nucleation probability and supercooling and saturation temperature from Eq. (8):
