Influence of Surfactant on Crystallization Kinetics of Stearic Acid

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Thermodynamics, Transport, and Fluid Mechanics

Influence of Surfactant on Crystallization Kinetics of Stearic Acid Hiya Goswami, and Jyoti R. Seth Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b05954 • Publication Date (Web): 03 Feb 2019 Downloaded from http://pubs.acs.org on February 4, 2019

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Industrial & Engineering Chemistry Research

Influence of Surfactant on Crystallization Kinetics of Stearic Acid Hiya Goswami and Jyoti R. Seth* Department of Chemical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai400076, India.

ABSTRACT

Crystallization of organic crystals from solutions is used in many industries. Depending on the application, control over size and shape of crystals is often desired and is achieved by tailoring process parameters such as rate of cooling and/or shear rate, or via the use of certain additives. In this study, the effect of Span 80 on crystallization of stearic acid from its supersaturated solution in mineral oil is presented. The rate of growth of the C-form crystals has been determined to increase almost quadratically with relative supersaturation. In the presence of Span 80, growth was much slower, which led to a gradual change in shape and aspect ratio of the crystals. A model based on competitive adsorption between surfactant and solute at the growing interface has been used to explain the role of surfactant. Additionally, the rate of nucleation decreased in the presence of Span 80, indicative of a higher interfacial energy penalty during nucleation.

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1. INTRODUCTION Crystallization is a widely used process for either purification and/or polymorph selection in the pharmaceutical, food, paint, pigment and other industries.1,2 In these systems, control over kinetics of nucleation and growth of crystals is crucial for successful design and implementation of the process. For crystallization from a supersaturated solution, key process parameters include degree of supersaturation, addition of seed crystals, rate of cooling and shear rate. Other controlling factors include choice of solvent, use of additives, etc..3,4 These influence the choice of polymorph crystallized and also the kinetics of nucleation and growth and hence the size and shape of the crystals. Typically, a minute amount of a surface active additive has a dramatic effect on crystal habit. It must be noted that final shape of the crystals thus formed is often vital for product performance. In chocolate, for example, aspect ratio of fat crystals is important for rheology modification and for pigments, shape and size affect their color and intensity.5,6 Several mechanisms have been proposed to understand how additives influence crystal growth. These include (i) by altering solution properties like solubility,7 (ii) by changing solid-liquid interfacial energy,8 (iii) by adsorbing at a growing crystal face, typically leading to inhibition of growth,9–11 (iv) by promoting aggregation of solute molecules in solution and/or micelle formation,12 and (v) by incorporation into the crystal i.e., co-crystallization.13 Of these, effect of additive on surface energy and adsorption of additive during growth are the most common ones.14– 18

Whereas the classical models of Cabrera-Vermilyea19 and Kubota-Mullin15 focus on adsorption

of additive at the advancing steps of the growing crystal, the model by Bliznakov20 focuses on adsorption of additive at the kink sites and subsequent effect on growth rate. Often, surface blockage in the presence of additives is related to the additive concentration by a multi-component, competitive-adsorption model, similar to the Langmuir isotherm.21 This has recently been used to


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explain the growth of sucrose crystals from aqueous solutions in the presence of raffinose.22 Here, we report a case where a surfactant influences the kinetics of crystallization of a long-chain carboxylic acid, stearic acid [see Figure 1(A)], in a manner consistent with the competitiveadsorption model22,23.

(A)

(B)



Figure 1. (A) Chemical structure of Stearic Acid (C17H35COOH); (B) Chemical structure of Span 80 (Sorbitan Monooleate, C64H124O26).

Long-chain carboxylic acids are regularly employed in soaps, shampoos, creams, ointments and other pharmaceutical drugs and are used as surfactants, rheology modifiers, fillers or agents for drug delivery. When crystallized from solution, carboxylic acids exhibit multiple polymorphic forms (each with a certain kinetics of nucleation and growth). Stearic acid (SA) has four main polymorphs A-, B-, C- and E-forms. While the A-form adopts a triclinic subcell and is needle shaped, the other polymorphic forms (B, C and E) have monoclinic lattices with plate-like crystal morphologies [see Figure 2(A)]. The B- and E-forms have almost the same structure with slightly different lamellar arrangements.24–28 Choice of polymorph crystallized is a function of the type of solvent, temperature of nucleation, use of additives, rate of cooling, shear rate, etc.. Some of these aspects are very well documented in the works of Sato and others.29–36 These observations, largely reported for SA, extend at least qualitatively, to other even-numbered, long-chain carboxylic acids as well. Despite abundant literature on polymorph selection, there is little quantitative information


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about the dynamics of crystallization, which is very sensitive to additives such as surfactants or polymers that may interact with the growing crystal face. Even in minute concentrations, these additives can induce a significant change in shape, number density and size-distribution of crystals. SA, being the most commonly used carboxylic acid, was the model solute for this study, with a focus on the role of additive on the kinetics of crystallization of C-form, generally its most desired polymorph [see Figure 2(B-D) for subcell arrangements of SA molecules in the C-form].

(B)

(A) (i)

(ii)

(iii) (h11)

75o

55o

(100)

(C)

(D)

Figure 2. (A) Optical micrographs indicating typical crystal morphologies of various SA polymorphs observed in this study (i) A-form, (ii) B-form, and (iii) C-form; (B) Schematic showing the lattice arrangement within C-form of SA24; (C) Representation of the terminal methyl groups on the (100) face of the C-form crystal of a typical long-chain, carboxylic acid25; (D) Representation of the exposed carboxylic acid groups on the (311) face of the C-form crystal of a typical long-chain, carboxylic acid.25


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A non-ionic surfactant, sorbitan monooleate (Span 80) has been used as the crystal modifying additive for SA with mineral oil (MO) as the solvent [see Figure 1(B) for its chemical structure]. With a Hydrophilic-Lipophilic Balance (HLB) value of 4.3, it is readily soluble in the solvent, MO. Crystallization was carried out under quiescent conditions at room temperature. Relative supersaturations were kept small in order to achieve low crystal number densities. In this way, growth of individual crystals could be tracked. Lateral growth of the rhombic-shaped, C-form crystals was tracked and the initial growth rate was found to increase almost quadratically with relative supersaturation. Using this, crystal growth over long times could be modelled as a function of the local supersaturation. Further, with the addition of Span 80, a reduction in growth rates was observed. This was modelled as an effect of adsorbing surfactant molecules at the growing solidliquid interface. Reduced growth rates result in a change in the aspect ratio of the crystals and manifestation of additional crystal faces at long times.


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2. EXPERIMENTAL SECTION In order to observe crystallization of SA from its solution in MO, a tiny drop of the heated solution was pressed between two glass slides maintained at 25oC. The solutions were thus quickly quenched to a pre-determined supersaturation level. Crystal growth was monitored using an optical microscope. The number and size of crystals were tracked for various supersaturations and surfactant concentrations. In the following, materials and methods used, the experimental setup for image capture and analysis are described. 2.1. Materials The solute, stearic acid (SA), was procured from Merck Chemicals at 97% purity and was further purified by recrystallization from acetone (Emparta; analytical grade; > 98% purity; Merck Chemicals). Light mineral oil (MO) with density, 𝜌𝑙 = 845 kg/m3, from Sigma Aldrich, was the solvent. MO was passed through a 200 nm pore-size syringe-filter before use. Sorbitan monooleate (Span 80), an oil soluble, non-ionic surfactant from Loba Chemie, was used as the additive. 2.2. Determination of Solubility Solubility of SA in MO was estimated at different temperatures. Known quantities of SA and MO were added to a round-bottom flask and the solution was stirred at 70oC for at least an hour. The clear solution was then cooled to the desired temperature and kept overnight for crystallization. With crystals settled at the bottom, aliquots of the saturated supernatant were diluted with isopropanol and then titrated against 0.1 M NaOH solution to determine the concentration of SA at saturation. The solubility graph is included as Figure S1.


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2.3. Imaging Setup for Observation of Crystal Growth Solutions with desired concentrations of SA in MO, with and without Span 80, were prepared. The mass fraction of SA (𝐶SA ) was varied from 0.6% to 1.4%, and that of surfactant (𝐶surf ) was varied between 0.015% and 0.08% (i.e., between 0.3 mM and 1.57 mM). These concentrations 0 0 correspond to relative supersaturations, 𝜎0 = (𝐶SA /𝐶SA − 1), of 0.5 to 2.5. 𝐶SA = 0.4% is the

solubility of SA in MO at 25oC and does not change appreciably with up to 0.08% Span 80 (Figure S1). Prior to the optical experiments, glass slides and coverslips were cleaned with isopropanol and then kept at 25oC in an incubator for at least an hour before use. Ambient temperature was maintained at 25  2oC. The solutions were maintained at 60oC with constant stirring for an hour or more to obtain clear solutions. A glass slide was placed on the microscope stage on which a 10 𝜇l drop of the heated, clear solution was deposited and immediately, a coverslip was gently placed over it. Care was taken so that the solution would spread across the entire area under the coverslip. Based on the volume used and area of the coverslip (22x22 mm), the solution layer thickness is estimated to be 20 𝜇m. Given the volume of sample used and the expected amount of SA to be crystallized, the estimated rise in temperature of the glass slide is 0.01oC, i.e., it remains practically unaffected. The thermal diffusivity of mineral oil under these conditions is 0.8  10-7 m2/s. The entire solution would therefore quench to the glass temperature in just a few milliseconds. This method of quick cooling allowed us to easily capture and quantify the kinetics of crystal formation and growth. Images were recorded, from the time of pouring, in bright-field mode, at a rate of 5 fps or higher, using a Nikon LV100ND microscope equipped with a QICAM Fast 1394 camera. All experiments were repeated twice or more for repeatability.


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2.4. Image Analysis The captured images were processed using the software ImageJ. For each experiment, induction time (𝑡ind ) was measured as the time from placing the drop until the very first speck of crystal was spotted. The major (𝐿1 ) and minor (𝐿2 ) axes of the rhombic shaped C-form crystals were marked and tracked over time (Figure S2). Growth rates were estimated from initial slopes of the plot of 𝐿1 and 𝐿2 versus time. About five crystals were analyzed from each set. For samples containing surfactant, there were notable changes in aspect ratio and morphology of the crystals. Even though the final crystals were not always rhombic, the major and minor axes were marked in a similar manner for ease of comparison with the control case i.e., without surfactant. Further, the entire sample area was scanned and crystals were counted over ten randomly chosen snapshots to estimate the total number of crystals (N).


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3. RESULTS AND DISCUSSION As SA crystals nucleated and grew in the thin layer of solution, their images were captured and analyzed. Without surfactant, predominantly C with some A and a negligible amount of B-form crystals were observed. This is as expected for SA and other long-chain carboxylic acids crystallized from hydrocarbon solvents like MO, hexane, etc.. In the presence of Span 80, A was suppressed and mainly C and some B-form were observed. These findings are consistent with the earlier studies of Garti and others.28–33 While these polymorphs are easily identifiable from their shapes, the same was also verified by measuring the X-ray powder Diffraction (XRD) patterns of the filtered crystals (Figure S3). Normally, the selection of polymorph is also influenced by other factors such as rate of cooling and shear rate, both of which were circumvented here. In the following sub-sections, observations for induction time, number and rate of growth of Cform crystals are presented as a function of relative supersaturation and concentration of surfactant. 3.1 Induction Time In Figures 3(A) and 3(B), induction time (𝑡ind ) is plotted with respect to relative supersaturation and surfactant concentration, respectively. Nucleation is expected to be faster at higher supersaturations and a rapid decrease in 𝑡ind with 𝜎0 is as expected from classical nucleation theory,21

𝑡ind

1 16𝜋𝛾 3 𝜗 2 1 𝐵 = 𝑒𝑥𝑝 ( ) = 𝑒𝑥𝑝 ( ) 𝐴 3(𝑘𝑇)3 (ln(1 + 𝜎0 ))2 𝐴 (ln(1 + 𝜎0 ))2

(1)

where 𝛾 is the interfacial energy between the nucleus and the solution and 𝜗 is the molecular volume. As seen in Figure 3(A), this expression fits the data well with A = 0.11 s-1 and B = 0.67.


9


Further, the number of crystals increased with relative supersaturation. Representative optical micrographs at different relative supersaturations are included in Figure S4.

(A)

(B) 103

103

107

1x105

0.8

0.6

102

N

105

tind (s)

102

N

106

tind (s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.4 10

4

10 3.0

3

0.2 10

1

0.0

0.5

1.0

1.5

2.0

2.5

10

1

0.00



0.02

0.04

0.06

Csurf (wt. %)

Figure 3. Plots for induction time, 𝑡ind (△), and total number of crystals, N (▽). (A) Induction time and the number of crystals are plotted for different relative supersaturations, 𝜎0 Dashed line is a fit for the induction time given by eq 1 (A = 0.11 s-1, B = 0.67). (B) Induction time and number of crystals for 𝜎0  plotted versus concentration of surfactant, 𝐶surf (wt. %). Error bars measure one standard deviation.

In the presence of Span 80, induction time progressively increased, as seen in Figure 3(B). Since the solubility, and therefore the relative supersaturation did not change appreciably with 𝐶surf , an effective crystal-solution interfacial energy (𝛾eff ) was estimated by fitting eq 1 to the data for 𝑡ind versus 𝜎0 for different 𝐶surf . The estimated (𝛾eff /𝛾0 )are plotted versus 𝐶surf in Figure 4. Several studies in the literature point out a multi-step nucleation mechanism.37–39 Here, however, the


10

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classical nucleation theory expression fits the data for all surfactant concentrations, thereby confirming that a single nucleus forms and subsequently grows. 1/3

It can be seen that the effective surface energy increased as 𝐶surf . This may be attributed to Span 80 being more polar than either SA or MO. The exact cause could not be inferred, however, a couple of scenarios may be considered: (i) Incorporation of Span 80 within the SA crystal. But there was no shift in XRD patterns in the presence of surfactant (Figure S3). This may be because Span 80 with a bent oleate chain cannot be easily incorporated alongside the linear stearate chains (Figure 1); (ii) Span 80 may be forming micelles or mixed micelles. However, measurements of viscosity as a function of 𝐶surf and Dynamic Light Scattering (DLS) experiments (Figure S5) indicate that micelles do not form under these conditions; (iii) The hydroxyl groups of Span 80 may associate with the exposed carboxyl groups of SA. This may lead to a layer of surfactant molecules covering the nuclei thus effectively increasing the interfacial energy. Similar cases have been reported for crystallization of sodium sulphate in the presence of phenol as an additive. There as well, induction time increased and rate of nucleation decreased with phenol concentration due to a higher Gibbs free-energy barrier in the presence of addtive.8 The presence of surfactant also affected the number of crystals. It was observed that fewer crystals were formed at higher surfactant concentrations [Figure 3(B)]. Consequently, since the amount of solute crystallized did not change appreciably with 𝐶surf , crystals grew to larger sizes as compared to the ones without surfactant (Figure S6).


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3.0 2.5 2.0

()3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1.5 1.0 0.5 0.0

0.00

0.02

0.04

0.06

Csurf (wt.%)

Figure 4. Estimated change in interfacial energy between SA nucleus and solution with respect to 𝐶surf (wt. %) for 𝜎0 . Dashed line is a linear fit with slope = 26.4. 𝛾eff and 𝛾0 are the interfacial energies with and without surfactant, respectively.


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3.2 Kinetics of Growth Crystal growth was tracked by measuring lengths of the major (𝐿1 ) and minor (𝐿2 ) axes over time. In Figure 5(A), increase in 𝐿1 and 𝐿2 over time is depicted for a typical case. The C-form crystal was rhombic, with an acute angle of 55o and its shape was preserved over time.28 The aspect ratio (𝐿1 /𝐿2 ) was close to the expected value of 1.96 (Figure S2). Crystals grew rapidly at first and slowly at later times. Deceleration in growth is attributed to the gradual depletion of solute from the surrounding medium. This was observed at all supersaturations and will be discussed further in the next sub-section. Since supersaturations are precisely known at short times, only the initial growth rates could be correlated to the initial relative supersaturation values. For this, growth rates of the major and minor axes over the first 10-20 s were estimated (

𝑑𝐿1

| ,

𝑑𝑡 0

𝑑𝐿2

| )

𝑑𝑡 0

[the slopes of the straight lines in Figure 5(A) and 5(B)]. Rates increased rapidly with 𝜎0 as a power-law with the exponent ≅ 1.8 (see Figure 6). The constant pre-factor obtained for the major axis was approximately twice of that obtained for the minor axis, thus preserving the aspect ratio. Parabolic kinetics are expected for growth governed by kink or step advancement. As per the Burton-Cabrera-Frank (BCF) model, growth rate is proportional to 𝜎 2 tanh(D/𝜎), where D is a temperature-dependent constant. As per this model, at lower 𝜎, growth rate increases as 𝜎 2 , whereas at higher 𝜎, growth rate increases linearly with 𝜎. The exponent of 1.8 might thus indicate growth at kinks and along edges as per the BCF model.21,40,41


13


(A) (i)

(ii)

30

20

2.2

L1/L2

L1,L2 (m)

40

2.0

10

1.8 0

0

0

10

20

25

t - tind (s)

30

40

50

50

t - tind (s)

(B) 80

(i)

70

(ii)

60 50 40

2.2

30

2.0

L1/L2

L1,L2 (m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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20

1.6 0

10 0

1.8

0

200

400

t - tind (s)

600

100 200 300 400 500 600 700

t - tind (s)

Figure 5. Typical growth of SA crystals without (A) and with surfactant (B). Lengths of the major [𝐿1 , ()] and minor [𝐿2 , (□)] axes of crystals are plotted versus time in A (i) and B (ii). Insets are plots of aspect ratio (𝐿1 /𝐿2 ) versus time. Optical micrographs taken at (𝑡 − 𝑡ind ) = 10s, 14s, 24s, and 34s for A (ii), and at 170s, 223s, 367s, and 597s for B (ii). Relative supersaturation, 𝜎0 = 2, for both, and surfactant concentration, 𝐶surf = 0.08%, for B. Dashed lines in A (i) are fits given by


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eq 6 and in B (i) by eq 10. Fitting parameters for A (i) are: 𝐿1,max = 41 𝜇m, 𝐿2,max = 23 𝜇m and 𝜏 = 11 s, and for B (i) are: 𝐿′1,𝑚𝑎𝑥 = 106 𝜇m and 𝜏′1 = 112 s; 𝐿′2,𝑚𝑎𝑥 = 62 𝜇m and 𝜏′2 = 135 s. Solid lines, fitted to the first few points, indicate initial rates of growth and their slopes for the major and minor axes are: A (i) 1.1 𝜇m/s and 0.7 𝜇m/s, and B (i) 0.7 𝜇m/s and 0.4 𝜇m/s, respectively.

1.8

(μm/s)

1.4

|

1.6

1.0

1.2

0.8

| ,

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.6 0.4 0.2 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0



Figure 6. Initial growth rates of the major [𝐿1 , ()] and minor [𝐿2 , (□)] axes of SA crystals as a function of relative supersaturation, 𝜎0 . Dashed lines are power-law fits with the exponent ≅ 1.8 and pre-factors, 0.3 𝜇m/s and 0.2 𝜇m/s, for major and minor axes, respectively. Error bars indicate one standard deviation.


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3.3 Growth of Crystals Over Time The dynamics of crystal growth can be explained via depletion of solute from the surrounding medium and can be calculated using a solute-balance between the growing crystal and its surrounding solution. Consider a rhombic crystal of thickness h, at any time t, with the major and minor axes, 𝐿1 (𝑡) and 𝐿2 (𝑡) ≅ 𝐿1 (𝑡)⁄2, respectively, and area, 𝑆(𝑡) = 𝐿21 (𝑡)⁄4 of the rhombic face. Volume of solution surrounding each crystal is V, and relative supersaturation is 𝜎(𝑡). During growth, as solute is depleted from the surrounding medium, 𝜎(𝑡) decreases while 𝐿1 and 𝐿2 increase. If 𝐶SA (𝑡) is the average mass fraction of solute in the surrounding medium, then one may write:

𝑑𝑆 ℎ𝐿21 𝑑𝐶SA 𝑑𝐶SA 𝑑𝜎 0 −𝜌𝑠 ( ℎ) = 𝜌𝑙 (𝑉 − ) ≅ 𝜌𝑙 𝑉 = 𝜌𝑙 𝑉𝐶SA , 𝑑𝑡 4 𝑑𝑡 𝑑𝑡 𝑑𝑡 (2) where 𝜌𝑙 and 𝜌𝑠 are the mass densities of solution and crystal, respectively. Average thickness of the crystals, h, was estimated to be much smaller than the sample gap, H = 20 𝜇m, i.e., (ℎ⁄𝐻 ) ≪ 1 (Table S1 and Figure S2). Thus growth perpendicular to the rhombic plane (dh/dt) would be much slower and is neglected here. Using the initial conditions, 𝑆(0) = 0 and 𝜎(0) = 𝜎0, we obtain, 0 𝐿21 𝐶SA 𝜌𝑙 𝑉 = (𝜎0 − 𝜎(𝑡)) . 4 𝜌𝑠 ℎ

(3)

As established from Figure 6, growth rate increases with 𝜎 as 𝑑𝐿1 /𝑑𝑡 = 𝑏𝜎 𝑛 (where, b = 0.3 𝜇m/s and n =1.8). By substituting this expression in eq 3, we obtain, 𝑛

𝑑𝐿1 𝐿21 𝜌𝑠 ℎ = 𝑏 (𝜎0 − 0 ) . 𝑑𝑡 4𝐶SA 𝜌𝑙 𝑉

(4)


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0 Non-dimensionalizing 𝐿1 using 𝐿1,max = (4𝜎0 𝐶SA 𝜌𝑙 𝑉/𝜌𝑠 ℎ)1/2 and 𝑡 using 𝜏 = 𝐿1,max /𝑏𝜎0𝑛 , eq 4

is expressed as, 𝑑𝐿1 𝑑𝑡

2 𝑛

2

= (1 − 𝐿1 ) ≅ (1 − √𝑛 𝐿1 ) ,

(5)

where overbars indicate non-dimensional quantities. The above approximation is valid for small values of 𝐿1 and at short times. Upon solving with the initial condition, 𝐿1 (0) = 0, we obtain

𝐿1 ≅ −

1 (1 − 𝑒 √𝑛 𝑡̅ ) √𝑛 (1 + 𝑒 √𝑛 𝑡̅ )

. (6)

In Figure 5(A), the data for L1 and L2 versus 𝑡 were fitted using the above expression. Equation 6 seems to capture the kinetics of growth rate up to sufficiently long times. The parameters, 𝐿1,𝑚𝑎𝑥 and 𝜏, signify the maximum length a crystal may attain for a given supersaturation and the time taken to grow to 𝐿1,𝑚𝑎𝑥 , respectively. For the case of 𝜎0 = 2, h was estimated to be 0.2 𝜇m (Table S1) and the solution volume surrounding each crystal is approximately estimated to be, V ≅ 1.6  104 𝜇m3 (total volume / total number of crystals). Using this, 𝐿1,𝑚𝑎𝑥 and 𝜏 are estimated to be 34 𝜇m and 15 s, respectively. These agree reasonably well with the fitted values of 41 𝜇m and 11 s. A comparison between the predicted and fitted values of 𝐿1,𝑚𝑎𝑥 and 𝜏 for all other supersaturations is presented in Figure S7.


17


3.4 Effect of Surfactant When surfactant is added to the crystallizing solution, growth rates are appreciably slower. For example, upon addition of 0.08 % surfactant at 𝜎0 = 2, initial growth rate of the major axis reduced from 1.1 𝜇m/s to 0.7 𝜇m/s [refer Figure 5(B)]. Such reduction was more significant at higher surfactant concentrations. In Figure 7(A), initial growth rates for 𝜎0 = 1 are plotted with respect to the surfactant concentration, 𝐶surf . A three-fold decrease was observed with only 0.06% surfactant. Similar reductions were also seen at 𝜎0  = 1.5 and 2.0 (Figure S8). Further, shape of the crystal was not preserved. Aspect ratio gradually decreased over time i.e., growth of the major axis slowed more than that of the minor axis [Figure 5(B)]. This change occurred sooner at higher surfactant concentrations.

(A) 0.40

(B)

0.30 0.25 0.20 0.15 0.10 0.05 0.00

1.2

0.02

0.04

0.06

16 12

1.0

1.2

1.6

2.0



0.8 0.6 0.4 0.2

0.00

20

K

Normalised Growth Rates

| (μm/s)

0.35

| ,

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 41

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Csurf (wt. %)

K*Csurf

Figure 7. Effect of surfactant on rate of crystal growth. (A) Initial rate of growth for the major 𝑑𝐿

𝑑𝐿

[ 𝑑𝑡1 | , ()] and minor [ 𝑑𝑡2 | , (□)] axes are plotted versus 𝐶surf for 𝜎0 . Dashed lines are fits 0

0


18

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given by eq 8 where K = 23 and growth rates without surfactant are

𝑑𝐿1

| = 0.3 𝜇m/s, 𝑑𝑡 0

𝑑𝐿2

| = 0.2

𝑑𝑡 0

𝜇m/s. (B) Growth rates versus 𝐶surf for different relative supersaturations [𝜎0 = 1(, ●), 1.5 (△, ▲) and 2 (□, ■)] are collapsed by normalizing the rates by that without surfactant and the surfactant concentration by K-1, a fitting parameter. Open symbols are for major and filled symbols are for minor axes. In the inset, fitted values of K are plotted with respect to 𝜎

Decrease in growth rates may be understood via attachment kinetics of surfactant or SA at the available growth sites. At any site, either a surfactant or an SA molecule may adsorb. Incorporation of an SA molecule would lead to crystal growth while adsorption of a surfactant molecule would block that site for SA and hence impede further crystal growth. Thus, while SA-SA intermolecular interactions would drive crystal growth, hydrogen bonding between hydroxyl groups of Span 80 and the exposed carboxyl groups of peripheral SA molecules of the crystal would drive adsorption of surfactant. The competition between surfactant and SA molecules over the available sites on the growing interface has been modelled as per the Langmuir adsorption isotherm. If at any time, 𝜃 is the fraction of sites on the growing interface that are blocked by the surfactant, then the fraction of sites available for growth would be (1 − 𝜃). Using the formalism of competitive adsorption,22,23 the effective number of sites must be,

1 − 𝜃 = (1 +

−1 𝐾surf 𝐶surf ) , 1 + 𝐾SA 𝜎

(7)

where, 𝐾𝑠𝑢𝑟𝑓 and 𝐾𝑆𝐴 are equilibrium constants for adsorption of surfactant and SA molecules, respectively. Rate of growth must be proportional to the number of available sites and therefore,


19


Page 20 of 41

the original rate of growth (eq 4) would be dampened by the fraction(1 − 𝜃). Thus, growth rate as a function of concentration of surfactant can be expressed as: −1 𝑑𝐿1 𝐾surf 𝑛 = 𝑏𝜎 (1 + 𝐶surf ) = 𝑏𝜎 𝑛 (1 + 𝐾𝐶surf )−1 . 𝑑𝑡 1 + 𝐾SA 𝜎

(8)

The above eq 8 is fitted to the data in Figure 7(A) with 𝐾 = 𝐾surf ⁄(1 + 𝐾SA 𝜎) as a fitting parameter. As expected, K decreases with 𝜎 [refer inset plot in Figure 7(B)]. Further, upon normalizing growth rates by those without surfactant, and 𝐶surf by the parameter K-1, data for the three supersaturations collapsed together [Figure 7(B)].


20

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3.5 Growth in the Presence of Surfactant Since growth rates in the presence of surfactant were lower as given by eq 8, eq 4 was modified and expressed in non-dimensional form as: 𝑑𝐿1 𝑑𝑡

2

2

≅ (1 − 𝑀𝐿1 ) (1 − 𝑛𝐿1 ), (9)

where 𝑀 = 𝐾0 𝐶surf /(1 − 𝐾0 𝐶surf );𝐾0 = 𝐾surf /(1 + 𝐾SA 𝜎0 ). The time scale becomes 𝜏 ′ = (𝑏𝜎0𝑛 (1 − 𝐾0 𝐶surf ))−1 in the presence of surfactant whereas 𝐿′1,𝑚𝑎𝑥 remains similar to 𝐿1,max defined in section 3.3. Using the initial condition 𝐿1 (0) = 0, one obtains √𝑛

1 + √𝑛 𝐿1 [ ] 1 − √𝑛 𝐿1

√𝑀

1 − √𝑀 𝐿1 [ ] 1 + √𝑀 𝐿1

= 𝑒 𝑡̅(𝑛−𝑀) . (10)

The data for growth in the presence of surfactant was fitted using eq 10. For the case in Figure 3(B), where 𝜎0 = 2 and 𝐶surf = 0.08%, 𝐿′1,𝑚𝑎𝑥 and 𝜏′ are predicted to be 50 𝜇m and 157 s, respectively. The corresponding fitted parameters were 110 𝜇m and 136 s, respectively. There is reasonable agreement between the predicted and fitted values of 𝜏′ but not of 𝐿′1,𝑚𝑎𝑥 . Table 1 lists the values of 𝐿′1,𝑚𝑎𝑥 , 𝐿′2,𝑚𝑎𝑥 and 𝜏′ for both major and minor axes of crystals grown at 𝜎0 =1 with varying 𝐶surf . It can be seen that the 𝜏′ values are similar for the major and minor axes and go on increasing with 𝐶surf . The predicted values of 𝐿′1,𝑚𝑎𝑥 and 𝐿′2,𝑚𝑎𝑥 do not agree with the fitted ones at higher 𝐶surf . This is expected, since growth rates for 𝐿1 and 𝐿2 were no longer proportional, which is when aspect ratio decreases.


21


Page 22 of 41

Table 1. Predicted and fitted values of 𝐿′1,𝑚𝑎𝑥 , 𝐿′2,𝑚𝑎𝑥 and 𝜏′ for 𝜎0 = 1 with varying 𝐶surf .

𝜏1

′𝑓

𝜏2

* 𝐿′𝑝 1,𝑚𝑎𝑥

𝜏 ′𝑝

𝜇m

s

s

𝜇m

s

57

31

45

50

56

87

0.03

124

68

279

280

88

277

0.04

147

85

280

280

92

289

0.05

136

89

300

300

95

299

0.06

210

110

294

300

95

301

′𝑓

′𝑓

𝐶surf

𝐿1,𝑚𝑎𝑥

𝐿2,𝑚𝑎𝑥 

wt.%

𝜇m

0

′𝑓

f - fitted; p - predicted * ′𝑝 𝐿2,𝑚𝑎𝑥 = 𝐿′𝑝 1,𝑚𝑎𝑥 /2


22

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3.6 Effect of Surfactant on Crystal Morphology The presence of surfactant caused a significant change in crystal morphology. Habit changed from rhombic to one with new faces emerging at ends of the major axis [refer images in Figure 5(B)]. For the C-form crystal, the top and bottom faces (100) have methyl terminal groups [as seen in Figure 2(C)], whereas the lateral faces (h11) expose the SA molecules stacked lengthwise [as seen in Figure 2(D)]. Attachment energies for (h11) face (Ea ≅ -100 kJ/mol) is higher than that for (100) face (Ea ≅ -9 kJ/mol). Thus, in accordance with the Gibbs-Wulf criterion, the (100) faces grow slowly and are larger in area and the (h11) faces grow much faster and have a smaller area.25,36 The orientation of SA molecules along the side faces might also be the reason why surfactant molecules reduce the edge growth rate. Driven by the association between exposed carboxyl groups of SA molecules and hydrophilic parts of Span 80, surfactant molecules may adsorb there (Figure S9). Adsorption of surfactant molecules results in a larger reduction in growth rate along the major axis than along the minor axis, which leads to a change in morphology. For C-form crystals, lattice spacing along the major axis is larger than that along the minor axis.25 Whether this facilitates preferential adsorption of surfactant molecules along the major axis direction needs to be evaluated. This, however, is beyond the scope of this present study. While crystals grew laterally, there was little evidence of surface nucleation, unlike other studies where very low supersaturations were maintained to avoid any secondary nucleation.42 Crosspolarized images taken in the absence of surfactant were almost completely and uniformly dark (Figure S10). At long times, when lateral growth slowed, surface nucleation was observed. As nuclei formed on the surface, bright-field images looked roughened and the cross-polarized images


23


Page 24 of 41

showed bright spots, indicating a mismatch in lattice arrangements between the parent and the secondary crystals (Figure S10). This was more prominent at higher surfactant concentrations. Since the top and bottom faces have very low attachment energies, the induction time for secondary nucleation would be large and was therefore observed only when lateral growth was suppressed in the presence of surfactant. The experiments capture 2D-growth of the crystals and therefore were inadequate for capturing growth via surface nucleation.


24

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4. CONCLUSIONS By observing crystallization of stearic acid under a quiescent and constant temperature setting, kinetics of nucleation and growth were measured as a function of supersaturation. The experiments focused on the C-form of stearic acid, which is generally the desired polymorph. Crystal growth rates increased with supersaturation raised to a power slightly less than 2, indicating that growth might be occurring along kinks and edges as per the BCF model. Growth rates inferred from shorttime data were, in turn, used to successfully predict evolution at later times, when the solution became less supersaturated. Thus the deduced power-law for growth is valid for lower supersaturations as well. Moreover, these are lateral growth rates, otherwise difficult to observe experimentally, since the growing face is only a few hundred nanometers thick. While these experiments were performed with mineral oil as the medium, a similar mechanism should be valid for other long-chain, carboxylic acids crystallized from either mineral oil or other non-polar hydrocarbon solvents. Influence of the oil-soluble surfactant, Span 80, on the kinetics of crystallization was quantified. With even a minute concentration of this surfactant, number of crystals reduced and growth rates decreased up to three-fold and induction times increased by an order of magnitude. Using the classical nucleation theory, an effective interfacial energy between crystal nucleus and solution was estimated, which increased with surfactant concentration to a power of one-third. The reduction in growth rates was modelled using a competitive-adsorption model where either stearic acid or surfactant may adsorb at the growing crystal face. As seen in the collapsed plot in Figure 7(B), this model fitted the data for different supersaturations as well as for different surfactant concentrations. Again, while these experiments were performed with Span 80, similar mechanisms may hold for other oil-soluble, low HLB surfactants as well. For other additives, parameters such


25


Page 26 of 41

as effective interfacial energy (𝛾eff ) and equilibrium constant for adsorption (𝐾surf ) would be useful for quantifying the strength of their influence on nucleation and growth, respectively.


26

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Supporting Information Solubility of SA in MO, morphology of C-form crystal of SA, influence of surfactant on selection of polymorphs, representative images at different relative supersaturations, viscosity of MO with different surfactant concentrations, representative images at different surfactant concentrations, thickness of crystals calculated for different relative supersaturations and surfactant concentrations, comparison between predicted and fitted values of maximum crystal size and the time scale of growth for different relative supersaturations, effect of surfactant on growth rates, crystal growth mechanism, evidence of secondary nucleation. The Supporting Information is available free of charge on the ACS Publications website. AUTHOR INFORMATION Corresponding Author *Phone no: +91 (22) 2576 7226. *Fax: +91 (22) 2572 6895 * Email: [email protected] Notes The authors declare no competing financial interest ACKNOWLEDGMENT We acknowledge Mr. Aniket Talele and Mr. Nandan Singh Ruhela for their assistance in performing the experiments. The authors are grateful to Department of Science and Technology, India for their financial support (DST: SB/S3/CE/082/2013).


27


Page 28 of 41

ABBREVIATIONS SA, Stearic Acid; MO, Mineral Oil; HLB, Hydrophilic-Lipophilic Balance; XRD, X-ray powder Diffraction; DLS, Dynamic Light Scattering; Burton Cabrera Frank; BCF.

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(1)

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Wellner, E.; Garti, N.; Sarig, S. Solution-Mediated Polymorphic Transformation of Stearic


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For Table of Contents Only

Evolution of morphology of stearic acid crystals during growth in the presence of surfactant


34

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(B) Research Industrial & Engineering Chemistry





(B)

(A) 1 (i) 2 3 4 5 6 7 8 9 (C) 10 11 12 13 14 15 16 17 18 19 20

(ii)

(iii) (h11)

75o

55o

(100)

(D)


Page 36 of 41

(A) Page 37 of 41

1x105

0.8

106

0.6

102

N

105

tind (s)

tind (s)

1 2 3 4 5 2 6 10 7 8 9 10 11 12 13 101 0.0 14 15 16

N

10

(B) Industrial & Engineering Chemistry Research 7 103 10

3

0.4 10

4

0.2 0.5

1.0

1.5



2.0

101 103 0.00 2.5 3.0 ACS Paragon Plus Environment

0.02

0.04

Csurf (wt. %)

0.06

Industrial & Engineering ChemistryPage Research 38 of 41 3.0 2.5 2.0

()3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1.5 1.0 0.5 0.0

0.00

0.02

0.04

0.06

ACS Paragon Plus Environment Csurf (wt.%)

Page 39 of 41


(A) (i)

(ii)

30

20

2.2

L1/L2

L1,L2 (m)

40

2.0

10

1.8 0

0

0

10

20

25

t - tind (s)

30

40

50

50

t - tind (s)

(B) 80

(i)

70

(ii)

60 50 40

2.2

30

2.0

L1/L2

L1,L2 (m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

20

1.6 0

10 0

1.8

0

200

400

t - tind (s)

600

100 200 300 400 500 600 700 ACS Paragon Plus Environment

t - tind (s)

Industrial & Engineering ChemistryPage Research 40 of 41 1.8 1.6 1.4

𝒅𝑳𝟏 𝒅𝑳 , 𝟐 (μm/s) 𝒅𝒕 𝟎 𝒅𝒕 𝟎

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.5 1.0 1.5 2.0 2.5 ACS Paragon Plus Environment



3.0

0.35

Normalised Growth Rates

𝒅𝑳𝟏 𝒅𝑳 , 𝟐 (μm/s) 𝒅𝒕 𝟎 𝒅𝒕 𝟎

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Industrial & Engineering Chemistry Research (B)

0.30 0.25 0.20 0.15 0.10 0.05 0.00

0.00

0.02

1.2

20

K

Page of 41 (A)41 0.40

1.0

16 12 1.2

1.6

2.0



0.8 0.6 0.4

0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.04ACS Paragon 0.06 Plus Environment

Csurf (wt. %)

K*Csurf

Influence of Surfactant on Crystallization Kinetics of Stearic Acid

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