crystals Article
Determination of the Nucleation and Growth Kinetics for Aqueous L-glycine Solutions from the Turbidity Induction Time Data Lie-Ding Shiau 1,2 1 2
Department of Chemical and Materials Engineering, Chang Gung University, Taoyuan 333, Taiwan;
[email protected] Department of Urology, Chang Gung Memorial Hospital Linkou, Taoyuan 333, Taiwan
Received: 22 September 2018; Accepted: 23 October 2018; Published: 24 October 2018
Abstract: As the turbidity induction time measurements are influenced by the size distribution of the nuclei at the detection point, these data should provide important information on both nucleation and growth. A model is developed in this work to determine the nucleation and growth kinetics of aqueous L-glycine solutions using the turbidity induction time data for various supersaturations from 293.15 K to 313.15 K. The photomicroscopic growth experiments of aqueous L-glycine solutions are also conducted to determine the growth kinetics of nuclei under the same conditions for comparison. The results indicate that the interfacial energy obtained from this model is consistent with that obtained based on the traditional method by assuming ti −1 ∝ J. The growth kinetics, including the growth activation energy and the kinetic growth parameter, obtained from this model using the induction time data are close to those obtained from the photomicroscopic growth experiments performed in this work. Keywords: crystallization; nucleation; induction time; interfacial energy
1. Introduction According to classical nucleation theory (CNT), only nuclei greater than a critical nucleus size are thermodynamically stable and can continue to grow to a detectable size [1–3]. The formation of critical nuclei is closely related to the interfacial energy of the crystallized substance, which is usually calculated from the induction time data in the literature [4–10]. The induction time is defined as the elapsed time between the creation of the supersaturation and the appearance of detectable nuclei at a constant temperature. Although the induction time can be detected by visual observation of the crystal’s appearance [7,11], turbidity measurements have been commonly adopted in recent years to determine the induction time by detecting the change in the intensity of transmitted light in solution at the onset of nucleation [12–17]. Traditionally, determination of the interfacial energy from the induction time data is often simplified by assuming ti −1 ∝ J [1,4–10]. Thus, it is implicitly assumed that at the detection of the nucleation point, only the number of the nuclei is accounted for regarding the change in the intensity of transmitted light in solution. The detection of nucleation point based on turbidity measurements should be influenced by both the number and the size of the nuclei [18] as the change in the intensity of transmitted light in solution is proportional to the size distribution of the nuclei instead of the number of the nuclei. To incorporate the effect of the nuclei size distribution on the detection of nucleation, Shiau and coworkers [18,19] have developed a model to examine the turbidity induction time data of aqueous L-glutamic acid solutions using the L-glutamic acid growth kinetics reported by Scholl et al. [20]. It is found that the obtained interfacial energy and growth activation energy of L-glutamic acid [19] are consistent with
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the literature data. L-glycine is the simplest amino acid and is often used as a model compound in the study of solution nucleation [21–25]. The objective of this work is to develop a model to study the nucleation and growth of aqueous L-glycine solutions based on the turbidity induction time data. To validate the obtained L-glycine growth kinetics from this model, the photomicroscopic growth experiments of aqueous L-glycine solutions are also conducted to determine the growth kinetics of nuclei at the same conditions for comparison. 2. Theory The nucleation rate based on CNT [1–3] is expressed as: J = A J exp(−
16πv2 γ3 3k B 3 T 3 ln2 S
),
(1)
where v = ρMNW and S = CCeq . For simplicity, the nucleation event is assumed to correspond to a point C A at which the total number density of the nuclei has reached a fixed (but unknown) value, fN [26,27]. One obtains at the induction time ti using: f N = J ti ,
(2)
Thus, it is implicitly assumed that the detection of the nucleation point is related to the number of nuclei. Substituting Equation (1) into Equation (2) yields: ln
AJ 1 16πv2 γ3 , = ln − ti fN 3k B 3 T 3 ln2 S
(3)
This is consistent with the common method adopted in the literature to calculate γ from induction time data [1,4–10]. The turbidity induction time measurements are based on the change in intensity of transmitted or scattered light along the detector direction, which should be related to the size distribution of the nuclei instead of the number of the nuclei at the detection point [18,19]. As nuclei are progressively generated during the induction time period (t = 0–ti ), the nuclei born in the earlier stage will grow to a greater size than those born in the later stage at ti . To incorporate the effect of crystal growth at the nucleation point, Shiau and Lu [18] proposed a model to correlate the nucleation and growth with the turbidity induction time data using the predetermined growth kinetics. However, as it is often difficult to experimentally measure the growth kinetics of small nuclei, the application of this model is restricted. As the turbidity induction time measurements are influenced by the size distribution of the nuclei, these data should provide important information on both nucleation and growth. A model is developed in the following to investigate the nucleation and growth of nuclei based on the turbidity induction time data without the predetermined growth kinetics. In the derivation, a simple empirical power-law growth rate is proposed as: G = k G ( S − 1) g ,
(4)
where the value of g mostly falls between 1 and 2. Based on Burton–Cabrera–Frank (BCF) growth theory [28–30], the value of g is found close to 2 for low supersaturations [31]. Mohan and Myerson [32] indicated for aqueous L-glycine solutions at 293.15 K that Equation (4) with g = 2 is consistent with the BCF growth kinetics reported by Li and Rodriguez-Hornedo [33]. In the induction time study, nuclei born at any time t (0 < t < ti ) can grow from t to ti and their size at time ti is: (5) L ( t ) = G ( ti − t )
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The corresponding volume of nuclei with size L at time ti is V ( t ) = k V L ( t )3 = k V G 3 ( t i − t )3 ,
(6)
As nuclei are progressively generated from t = 0 to ti , the total volume of all the nuclei per unit solution volume at time ti is given by: fV =
Z t i 0
JV (t)dt,
(7)
Substituting Equation (6) into Equation (7) yields: f V = JkV G3
Z t i 0
(ti − t)3 dt =
JkV G3 ti 4 , 4
(8)
Note that J and G remain unchanged as S is kept at a particular supersaturation during each induction time experiment. Substituting Equations (1) and (4) into Equation (8) with g = 2 leads to: fV =
A J k V k G 3 ( S − 1)6 t i 4 16πv2 γ3 ), exp(− 4 3k B 3 T 3 ln2 S
(9)
Rearranging Equation (9) yields: " ln A plot of ln
4 k V t i 4 ( S −1)6
#
4 k V t i 4 ( S − 1)6
versus
1 ln2 S
A J kG 3 = ln fV
!
−
16πv2 γ3 3k B 3 T 3 ln2 S
,
(10)
at a given temperature should give a straight line, the slope A k
3
and intercept of which permit determination of γ and Jf G , respectively. V The temperature dependence of kG can be expressed in terms of the Arrhenius equation as: k G = AG exp(− Once
A J kG 3 fV
EG ), RT
(11)
is determined at different temperatures, substitution of Equation (11) yields: A J kG 3 ln fV
Thus, a plot of ln
A J kG 3 fV
versus
permit determination of EG and if fV is known.
A J AG fV
3
1 T
!
A J AG 3 = ln fV
!
−
3EG , RT
(12)
should give a straight line, the slope and intercept of which
, respectively. It should be noted that AJ AG 3 can be determined
3. Experimental 3.1. Induction Time Measurements The experimental apparatus of a 250 mL crystallizer was the same as that used by Shiau and Lu [18]. Deionized water and L-glycine (>99%, Alfa Aesar, Haverhill, MA, USA) were used to prepare the supersaturated solution. In each experiment, a 200 mL aqueous L-glycine solution with the desired supersaturation was loaded into the crystallizer. The solution was stirred with a magnetic stirrer at a constant stirring rate of 350 rpm. A turbidity probe with a Near-Infrared source (Crystal Eyes
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manufactured by HEL limited, Hertford, UK) was used to detect the nucleation event during the induction time study. The solution is held at 3 K above the saturated temperature for 5–10 min to ensure a complete dissolution at the beginning of the experiment, which was also confirmed using the turbidity measurement. As the cooling rate to reach a particular supersaturation influences the nucleation induction time [34], the solution was rapidly cooled at 25 ◦ C/min to the desired constant temperature. Thus, the lag time was usually less than 60 s, which was much smaller than the measured induction times listed in Table 1. Figure 1 shows the variation of measured turbidity with time for S = 1.15 and a temperature of 293.15 K. The percentage threshold for the turbidity data was defined as the change inx turbidity to determine whether a nucleation event had occurred [35]. A setting of4 20% Crystals 2018, 8, FOR PEER REVIEW of 14 for the threshold was employed for all the turbidity induction time data in this work. Table 1. The average induction times and the corresponding standard deviations (SD) for L-Glycine. Table 1. The average induction times and the corresponding standard deviations (SD) for L-Glycine.
𝑻(𝐊)
293.15
303.15
313.15
𝑺(−) T(K)1.10 S(−) 1.12 1.10 1.13 1.12 293.15 1.16 1.13 1.16 1.07 1.08 1.07 1.08 1.10 303.15 1.10 1.12 1.12 1.06 1.06 1.07 1.07 313.15 1.08 1.08 1.10 1.10
ti (SD)(s) 2120 (471) 1090 (242) 936 (253) 563 (157) 2672 (588) 1390 (276) 831 (247) 442 (198) 2327 (534) 953 (273) 737 (201) 429 (175)
𝒕𝒊 (𝑺𝑫)(𝐬) 2120 (471) 1090 (242) 936 (253) 563 (157) 2672 (588) 1390 (276) 831 (247) 442 (198) 2327 (534) 953 (273) 737 (201) 429 (175)
Figure The variation variation of of measured measured turbidity 1.15 and and 293.15 293.15 K. Figure 1. 1. The turbidity with with time time for for SS = = 1.15 K.
3.2. Growth Rate Measurements 3.2. Growth Rate Measurements The photomicroscopic experiments shown are performed to investigate the growth rates of The photomicroscopic experiments shown are performed to investigate the growth rates of LL-glycine in water. This growth cell shown in Figure 2 [36] has a solution chamber of 20 mL in the glycine in water. This growth cell shown in Figure 2 [36] has a solution chamber of 20 mL in the upper upper part and a chamber for temperature-controlled water in the lower part. The growth rates of part and a chamber for temperature-controlled water in the lower part. The growth rates of aqueous L-glycine solutions for various supersaturations from 293.15 K to 313.15 K were studied isothermally in the upper stagnant solution.
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aqueous L-glycine solutions for various supersaturations from 293.15 K to 313.15 K were studied isothermally thePEER upper stagnant solution. Crystals 2018, 8, xinFOR REVIEW 5 of 14
Top view
Side view (a)
Top view
Side view (b)
The photomicroscopic photomicroscopic growth growth apparatus apparatus (a) (a) the the real real picture picture of growth cell; (b) schematic Figure 2. The diagram of growth cell with the features: (1) solution chamber; thermistor; (3) solution solution inlet and diagram chamber; (2) thermistor; outlet; (4) constant-temperature water chamber; (5) water inlet and outlet.
The growth growth of of crystals through aa microscope and analyzed The crystals was was monitored monitored photographically photographically through microscope and analyzed using an image analyzer (Imaging Software, NIS-Elements, Nikon, Japan) to determine the area of of using an image analyzer (Imaging Software, NIS-Elements, Nikon, Japan) to determine the area each crystal. For simplicity, the characteristic size of the small crystal is taken as the equivalent circular each crystal. For simplicity, the characteristic size of the small crystal is taken as the equivalent √ 2 /4, leading to L = diameter, i.e., A = πL 4A/π. The sizes against time for circular diameter, i.e., 𝐴 = 𝜋𝐿 /4, leading to 𝐿 = 4𝐴/𝜋. The were sizes then wereplotted then plotted against timeeach for crystal with the slope equal to the growth rate. In each run, 8–10 crystals were analyzed to calculate each crystal with the slope equal to the growth rate. In each run, 8–10 crystals were analyzedthe to mean growth rate among these condition. 3 shows the photograph of the the calculate the mean growth ratecrystals amongunder theseeach crystals under Figure each condition. Figure 3 shows needle-like α-form crystals in solution taken inin ansolution experimental forexperimental S = 1.12 at T run = 303.15 It was photograph of the needle-like α-form crystals taken run in an for S K. = 1.12 at found for various supersaturations from 293.15 K to 313.15 K that the needle-like α-form crystals were T = 303.15 K. It was found for various supersaturations from 293.15 K to 313.15 K that the needle-like nucleated in thewere photomicroscopic experiments. α-form crystals nucleated in the photomicroscopic experiments.
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The photograph photograph of of the the needle-like needle-like α-form α-form L-glycine L-glycine crystals crystals in in solution solution taken taken in the Figure 3. The photomicroscopic experiment at SS = 1.12 and 303.15 K. =
4. Results Results and and Discussion Discussion 4. The induction time data were measured The induction time data for for aqueous aqueous L-glycine L-glycine solutions solutions were measured for for various various supersaturations from 293.15 K to 313.15 K. Each run was carried out at least three times to determine supersaturations from 293.15 K to 313.15 K. Each run was carried out at least three times to determine the average average induction induction time time under under each each condition. condition. The The average average induction induction times times and and the the corresponding corresponding the standard deviations Table 1. α-form L-glycine L-glycine standard deviations (SD) (SD) are are listed listed in in Table 1. The The equilibrium equilibrium concentration concentration of of the the α-form −3 T 2 − 3.2022 × 10−1 T − 188.2 (C in kg/m3 , and T in in water was given by C (T) = 5.4397 × 10 eq eq in water was given by Ceq(T) = 5.4397 × 10−3T2 − 3.2022 × 10−1T − 188.2 (Ceq in kg/m3, and T in K) [37]. 3 , and ν = 7.757 × 10−29 m3 for L-glycine. K) [37]. Note MWkg/mol, = 0.075 ρkg/mol, ρC = 1607 kg/m 3, and Note that MWthat = 0.075 C = 1607 kg/m ν = 7.757 × 10−29 m3 for L-glycine. Although L-glycine can be crystallized in different polymorphs, Although L-glycine can be crystallized in different polymorphs, the the α-form α-form is is usually usually achieved achieved from nucleation of pure aqueous solutions [21–25]. To validate the polymorphm of the from nucleation of pure aqueous solutions [21–25]. To validate the polymorphm of the L-glycine L-glycine crystals, the thefinal finaldried driedcrystals crystals of induction the induction experiments were analyzed crystals, at at thethe endend of the time time experiments were analyzed using using both optical microscopy and Raman spectroscopy (P/N LSI-DP2-785 Dimension-P2 System, both optical microscopy and Raman spectroscopy (P/N LSI-DP2-785 Dimension-P2 System, 785 nm, 785 nm, manufactured by Lambda Solutions, INC., Seattle, WA, USA). The results all indicated manufactured by Lambda Solutions, INC., Seattle, WA, USA). The results all indicated that the that the needle-like obtained aqueous L-glycinesolutions solutionsfor for various needle-like α-form α-form crystalscrystals were were obtained fromfrom aqueous L-glycine various supersaturations from 293.15 K to 313.15 K. Figure 4 shows the Raman spectra of pure α-form crystals supersaturations from 293.15 K to 313.15 K. Figure 4 shows the Raman spectra of pure α-form crystals and product As compared compared with with the the Raman Raman spectra spectra of of and product crystals crystals obtained obtained at at various various supersaturations. supersaturations. As pure α-form crystals reported by Murli et al. [38], it was confirmed that α-form crystals were produced pure α-form crystals reported by Murli et al. [38], it was confirmed that α-form crystals were for variousfor supersaturations at 303.15 K. The section the Raman α-, β-,ofand γ-glycine produced various supersaturations at 303.15 K. Theofsection of thespectra Raman of spectra α-, β-, and γused for characterization are also depicted by Bouchard et al. [39]. glycine used for characterization are also depicted by Bouchard et al. [39]. Figure 55 shows induction time time data data from from 293.15 293.15 K K to to 313.15 K fitted Figure shows the the measured measured induction 313.15 K fitted to to Equation Equation (3) based on f . The fitted results are listed in Table 2, where γ was in the range 1.93–2.37 mJ/m2 N (3) based on f N. The fitted results are listed in Table 2, where γ was in the range 1.93–2.37 mJ/m2 and AJ andwas was the range × 10−×3 –6.42 10−3 s−1 . the Although the exact value only of AJbe could only be f N in theinrange 4.35 ×4.35 10−3–6.42 10−3 s−1× . Although exact value of AJ could determined determined with a known value of fN , γ was not influenced by the chosen value of fN . with a known value of fN, γ was not influenced by the chosen value of fN. Table 2. The fitted results of the induction time data to Equation (3) based on fN . Table 2. The fitted results of the induction time data to Equation (3) based on fN.
𝑻(𝐊)
T(K)
293.15 293.15 303.15 303.15 313.15
313.15
γ(mJ/m2 ) 𝟐
𝜸(𝐦𝐉/𝐦 )
2.37 2.372.07 2.071.93
1.93
AJ −1 ) fN𝑨(s 𝑱
(𝐬 −𝟏3)
AJ (m−3 s−1 )
𝑨𝑱 (𝐦 𝟑 𝐬 𝟏 )
4.35𝒇𝑵 × 10 −3 5.29 × 10 4.35 × 10−3 − 3 6.42 × 10 5.29 × 10−3
3.32 × 109 4.04 ×3.32 109 × 109 4.91 ×4.04 109 × 109
6.42 × 10−3
4.91 × 109
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Figure 4. 4. The Raman Raman spectra of of pure α-form α-form crystals and and product crystals crystals obtained in in the induction induction Figure Figure 4.The The Ramanspectra spectra ofpure pure α-formcrystals crystals andproduct product crystalsobtained obtained inthe the induction time experiments for various supersaturations at 303.15 K. time timeexperiments experimentsfor forvarious varioussupersaturations supersaturationsatat303.15 303.15K. K.
Figure The induction time data of L-glycine at 293.15–313.15 K fitted to Equation (3) based on Figure Figure5.5. 5.The Theinduction inductiontime timedata dataof ofL-glycine L-glycineat at293.15–313.15 293.15–313.15K Kfitted fittedto toEquation Equation(3) (3)based basedon on 𝑓f𝑓N...
As the the induction time time data are are measured by the intensity change of the transmitted light, Figure 6 As As theinduction induction timedata data aremeasured measuredby bythe theintensity intensitychange changeof ofthe thetransmitted transmittedlight, light,Figure Figure shows the measured induction time data from 293.15 K to 313.15 K fitted to Equation (10) based on 66shows showsthe themeasured measuredinduction inductiontime timedata datafrom from293.15 293.15KKto to313.15 313.15KKfitted fittedto toEquation Equation(10) (10)based basedon on Notethat thatggwas wasassumed assumedto tobe be22 due due to to low low supersaturations supersaturations (S (S == 1.4–2.4) 1.4–2.4) in in the the induction induction time time ffVfVV. ..Note Note that g was assumed to be 2 due to low supersaturations (S = 1.4–2.4) in the induction2 time experiments. The Thefitted fitted results are are listed in in Table 3, where γ was in the range 2.49–2.93 mJ/m and experiments. experiments. The fittedresults results arelisted listed inTable Table3,3,where whereγγwas wasin inthe therange range2.49–2.93 2.49–2.93mJ/m mJ/m2 2and and A J kG 3 − 5 − 3 − 4 was in the range 2.78 × 10 –2.58 × 10 s . It should be noted that γ was not influenced by f V was wasin inthe therange range2.78 2.78××10 10−5−5–2.58 –2.58××10 10−3−3ss−4−4. .ItItshould shouldbe benoted notedthat thatγγwas wasnot notinfluenced influencedby bythe the the chosen value of fV . chosen chosenvalue valueof offVfV. .
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Figure6.6.The Theinduction inductiontime timedata dataofofL-glycine L-glycineatat293.15–313.15 293.15–313.15KKfitted fittedto toEquation Equation(10) (10)based basedon on 𝑓f V. . Figure Table 3. The fitted results of the induction time data to Equation (10) based on f V . Table 3. The fitted results of the induction time data to Equation (10) based on 𝑓 .
𝑻(𝐊)
γ(mJ/m2 )
T(K)
293.15 293.15303.15 303.15313.15
𝜸(𝐦𝐉/𝐦𝟐 ) 2.93 2.66 2.93 2.49 2.66
AJ kG 3 −4 (s 𝟑 ) f𝑨 V 𝑱 𝒌𝑮
(𝐬 𝟒 )
𝒇𝑽10− 5 2.78 × 4.152.78 × 10×−410−5 2.584.15 × 10×− 310−4
313.15 2.49 2.58 × 10−3 The turbidity induction time measurements in the current experiments were based on the intensity The induction time measurements current experiments were on the change ofturbidity the transmitted light, which is related toin fV .the Thus, compared to γ based on based fN , γ based on intensity of the transmitted light, related toenergy fV. Thus, compared Shiau to γ based on fN, γ fV shouldchange more accurately represent the which actual is interfacial of L-glycine. [40] reported 2 for accurately based on fV should more represent the actual interfacial of L-glycine. Shiau zone [40] γ = 1.35–2.02 mJ/m aqueous α-form L-glycine solutions usingenergy the turbidity metastable 2 for aqueous α-form L-glycine solutions using the turbidity metastable reported γ = 1.35–2.02 mJ/m width measurements at the saturation temperature between 308.15 K and 328.15 K. Using the visual zone width measurements at time the saturation between 308.15 K γand Using the observation of the induction data, Devitemperature and Srinivasan [25] reported = 5328.15 mJ/m2K.for aqueous 2 for visual observation of the induction time data, Devi and Srinivasan [25] reported γ = 5 mJ/m α-form L-glycine solutions at 303.15 K. A k 3 303.15 K. aqueous α-form L-glycine solutions 1 Figure 7 shows the plot of J Gat versus fitted to Equation (12). The fitted results are listed in T
fV
Figure 7 shows the plot of versus Afitted A 3 to Equation (12). The fitted results are listed in Table 4, which indicates EG = 58 kJ/mol and Jf G = 2.30 × 1026 s−4 . Although the exact value of V Table whichonly indicates EG = 58 kJ/mol = 2.30 s−4. Although the exact value of Avalue JAG3 AJ AG 34,could be determined with a and known value of fV×,10 EG26 was not influenced by the chosen of fV . only Because activation energy is usually 10–20 fornot diffusion andby 40–60 kJ/molvalue for surface V, EG was influenced the chosen of fV. could be determined with a known value of fkJ/mol integration [1], E = 58 kJ/mol obtained for the growth of L-glycine in the induction time experiments G energy is usually 10–20 kJ/mol for diffusion and 40–60 kJ/mol for surface Because activation should be integration controlled. integration [1], EG = 58 kJ/mol obtained for the growth of L-glycine in the induction time experiments should be integration controlled.
Table 4. The fitted results of EG and AG using Equation (12).
kJ EG ( mol )
Table 4. The fitted results of EG and AG using Equation (12). AG (m/s) AJ AG 3 −4 AJ AG 3 (s−4 ) fV (s 𝟑 ) 𝑨𝑱 𝑨𝑮 𝐤𝐉 293.15 K 𝑨𝑮 (𝐦/𝐬) 303.15 K 𝟑 𝟒 𝟒
𝑬𝑮 ( ) (𝐬 ) 𝐦𝐨𝐥 2.30 × 𝒇𝑽1026 58 58 2.30 × 1026
𝑨𝑱 𝑨𝑮 (𝐬 ) 9.18 ×
1022
9.18 × 1022
313.15 K
293.15 K 104303.152.83 K ×313.15 K 2.66 × 104 3.02 × 104 4 4 3.02 × 10 2.83 × 10 2.66 × 104
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Figure 7. 7. The Theplot plotof of Figure
A J kG 3 fV
versus versus
1 T
fitted to fitted to Equation Equation (12). (12).
Based on the study of 28 inorganic systems, Mersmann and Bartosch [41] estimated fV = 10−4−4–10−−33 Based on the study of 28 inorganic systems, Mersmann and Bartosch [41] estimated fV = 10− –10 with a detectable size of 10 µm. In the calculations here, the intermediate value, fV = 4 × 10 −44 , for with a detectable size of 10 μm. In the calculations here, the intermediate value, f V = 4 × 10 , for −3 . Consequently, as indicated spherical nuclei with kV = π6 was assumed, leading to fN = 7.64 × 1011 m−3 11 leading to fN = 7.64 × 10 m . Consequently, as indicated spherical nuclei with kV = was assumed, in Table 2, AJ was in the range 3.32 × 109 –4.91 × 109 m−3 ·s−1 based on fN = 7.64 × 1011 m−3 . Table 4 9 9 − −1 11 m−3. Table 4 indicates J was 10 –4.91 fN = 7.64 ×if10 in Table 2,AAA 3 in the range 223.32 −4 ×based 4 . For on indicates on ×fV10 = 4m×3·s 10−based simplicity, AJ obtained based on fN J G = 9.18 × 10 s 22 −4 −4 3 A JAG = 9.18 × 10 s based on fV = 4 × 10 . For simplicity, if 10 AJ 4obtained based on fN was adopted to was adopted to find AG , one obtains AG = 2.66 × 104 –3.02 × m/s. find A G, one obtains AG = 2.66 × 104–3.02 × 104 m/s. The average growth rates and the corresponding standard deviations (SD) obtained from the The average and theare corresponding obtained photomicroscopic growth growth rates experiments listed in Tablestandard 5. Figuredeviations 8 displays (SD) the growth ratefrom datathe for photomicroscopic growth experiments are listed in Table 5. Figure 8 displays the growth rate data various supersaturation from 303.15 K to 323.15 K. The growth rate obtained here is consistent with for supersaturation from K toα-form 323.15L-glycine K. The growth rateatobtained is =consistent thatvarious reported by Han et al. [23] for303.15 aqueous solutions S = 1.35 here and T 303.15 K. with that reported by Han et al. [23] for aqueous α-form L-glycine solutions at S = 1.35 and T = 303.15 Substituting Equation (11) into Equation (4) for g = 2 yields: K. Substituting Equation (11) into Equation (4) for g = 2 yields: EG G= AG𝐴exp (− S − 11) )2 ,, 𝑒𝑥𝑝 (− )( )(𝑆 𝐺= (13) RT
Rearranging Rearranging Equation Equation (13) (13) leads leads to: to: ln[G( ) ] = ln𝐴 − E, (14) G ln [ ] = ln A − , (14) G 2 RT 1)versus ] should give a straight line, leading to E G = 57 As shown in Figure 9, a plot of ln[((S − ) 4 m/s. kJ/mol AG =in6.05 × 109, Asand shown Figure a plot of ln [
kJ/mol and AG = 6.05 × 104 m/s.
G ] ( S −1)2
versus
1 T
should give a straight line, leading to EG = 57
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Table 5. The average growth rates and the corresponding standard deviations (SD) for L-Glycine. T(K)
S(−)
G (SD) (×10−7 ) (m/s)
303.15
1.08 1.10 1.12 1.14
0.42 (0.17) 0.67 (0.29) 1.09 (0.36) 1.35 (0.55)
313.15
1.08 1.10 1.12 1.14
1.03 (0.42) 1.67 (0.85) 1.87 (0.64) 3.21 (1.37)
323.15
1.08 1.10 1.12 1.14
2.13 (0.73) 2.55 (0.82) 3.88 (1.03) 5.62 (2.14)
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Figure 8. The growth rate data of L-glycine for various supersaturations at 303.15–323.15 K. The lines Figure 8. The growth rate data of L-glycine for various supersaturations at 303.15–323.15 K. The lines represent the fitted values to Equation (13) using g = 2, EG = 57 kJ/mol and AG = 6.05 × 104 m/s. represent the fitted values to Equation (13) using g = 2, 𝐸 = 57 kJ/mol and 𝐴 = 6.05 × 10 m/s.
Figure 8 shows the growth rate data fitted well to Equation (14) using g = 2 and the fitted values Table 5. The average growth rates and the corresponding standard deviations (SD) for L-Glycine. of EG and AG . Thus, it was reasonable to adopt the power-law growth rate of Equation (4) with g = 2 in 𝑻(𝐊) 𝑺(−) (𝑺𝑫) (× 𝟏𝟎 𝟕 ) (𝐦/𝐬) derivation of Equation (10). It should be noted that the 𝑮 turbidity induction time data were measured −4 and f = 7.64 × 1011 m−3 ) in 1.08µm, as assumed here for fV = 40.42 (0.17) for nuclei of near-zero size (
