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A simple method for estimating the size of nuclei on fractal surfaces
Journal of Crystal Growth 475 (2017) 49–54
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Journal of Crystal Growth journal homepage: www.elsevier.com/locate/crys
A simple method for estimating the size of nuclei on fractal surfaces Qiang Zeng Institute of Advanced Engineering Structures and Materials, College of Civil Engineering and Architecture, Yuhangtang Rd 866, Zhejiang University, 310058 Hangzhou, PR China
a r t i c l e
i n f o
Article history: Received 2 November 2016 Received in revised form 23 May 2017 Accepted 31 May 2017 Available online 3 June 2017 Communicated by T.F. Kuech Keywords: Nucleation Size Fractal Scale
a b s t r a c t Determining the size of nuclei on complex surfaces remains a big challenge in aspects of biological, material and chemical engineering. Here the author reported a simple method to estimate the size of the nuclei in contact with complex (fractal) surfaces. The established approach was based on the assumptions of contact area proportionality for determining nucleation density and the scaling congruence between nuclei and surfaces for identifying contact regimes. It showed three different regimes governing the equations for estimating the nucleation site density. Nuclei in the size large enough could eliminate the effect of fractal structure. Nuclei in the size small enough could lead to the independence of nucleation site density on fractal parameters. Only when nuclei match the fractal scales, the nucleation site density is associated with the fractal parameters and the size of the nuclei in a coupling pattern. The method was validated by the experimental data reported in the literature. The method may provide an effective way to estimate the size of nuclei on fractal surfaces, through which a number of promising applications in relative fields can be envisioned. Ó 2017 Elsevier B.V. All rights reserved.
1. Introduction One of the most important open questions in crystallization is the controls of nucleus size and structure by substrate geometry configurations with the auxiliary manipulations of nucleating rates [1–7]. While there have increasing experimental/computational methods for investigating homogeneous and heterogeneous nucleation, there are almost no equivalent developments for assessing the size of the nuclei in contact with complex surfaces structured, for example, in the pattern of self-similarity (fractal structure). Determining the size of the nuclei on complex surfaces is always a big challenge in aspects of biological, material and chemical engineering. In the last decades, there have been developed many methods that can directly measure the size of nuclei. Gasser et al. [8] reported a real–space imaging technique by laser scanning confocal microscopy for identifying and observing both the nucleation and growth of crystalline regions directly. Yau and Vekilov [9] used direct atomic force microscopy to determine the raftlike shape of apoferritin crystals (quasi-planar nuclei) and estimate their size. Andersen et al. [10] applied in-situ powder X-ray diffraction with Rietveld refinement and whole powder pattern modeling to measure the size of maghemite nanocrystals. Nanev et al. [11] used a microscopy-based method to separate the nucleation and growth stages of the crystallization process for insulin crystals and assess their size. Those successfully used techniques, however,
E-mail address: [email protected] http://dx.doi.org/10.1016/j.jcrysgro.2017.05.037 0022-0248/Ó 2017 Elsevier B.V. All rights reserved.
meet limitations by complex experimental constraints. For instance, when crystallization takes place in the pores of porous solids, the microscopy-based technique would fail to detect the nuclei. So, more frequently, the size of nuclei remains estimated approximately from (classic and advanced) thermodynamic analysis [12,13], the inverse analysis of experimental data that were not designed to measure the size [14], molecular simulations [7,15] and the combination of experiments and simulations [16]. Although most of the published methods for nuclei size determination might assess the size of nuclei successfully, there remains no perfect approach covering all the phase transition mechanisms and affecting factors, especially when nucleation occurs on fractal surfaces. Therefore, finding a simple way to determine the size of the nuclei heterogeneously nucleating on fractal surfaces is the main task of this study. A fractal surface is structured in the pattern of self-similarity that repeats at every scale theoretically [17] or in limited scales practically [18–20]. Previous study by Stolyarova et al. [21] indicated that the number of nucleation seeds on a fragment with fractal structures significantly exceeds that on the one with flat surfaces, which can be reasonably due to a sufficient local molecular concentration for nucleation generated by the fractal structure. Furthermore, the nucleation–promoting effect may hold for any substrate with fractal structures. The authors further presented an explicit model to link nucleation density to fractal parameters. Although the model in Ref. [21] contained a nucleus-volumerelated parameter, it remains requiring a clear and explicit model to elucidate the way in which the nucleation density is governed
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Q. Zeng / Journal of Crystal Growth 475 (2017) 49–54
by the size of nuclei and the fractal parameters of surfaces. This, in turn, may help to establish a method to estimate the nuclei size once the nucleation density and fractal parameters are determined. In what follows, we first presented a section to establish the relations between nucleation density and the size of the crystals that nucleate on fractal surfaces heterogeneously. We showed scale-depended regimes for calculating the contact areas between the nuclei and substrates. Those generated equations can be used to estimate the size of nuclei once nucleation density, fractal dimension and the up and low bounds of fractal regions were known. The simple method was tested by the data reported in the literature. The results may provide an effective way to estimate the size of nuclei on fractal surfaces. 2. Estimating method A heterogeneously nucleating process is often very complex especially when nuclei form on a rough surface with, for instance, fractal structures. By tracking a number of representative nuclei with definable ‘‘macro” properties from the parent phase (e.g., solutions, gases and melts), this process can be identified qualitatively and/or characterized quantitatively. However, the roughness of substrate may make it difficult to determine the contact area – one of the most important parameters for evaluating the nucleation density and the size of nuclei. Several assumptions were made in this study to obtain a simple and tough estimate. Firstly, a general assumption of spherical-cap shaped nuclei was adopted regardless of the geometry and surface properties of the contact substrate. This provides a practical way to calculate the contact areas between the spherical-cap shaped nuclei and the substrate with complex structure, and to link the size of the contact areas to the real size of the nuclei. Secondly, a Wenzel type contact between the nuclei and rough substrate was retained. This says that the areas of the contact surfaces are completely covered by the nuclei without empty voids involved. And this regime has been reported to account for the boiling and wetting behaviors on fractal hierarchical surfaces [22,23]. The debates of the relevant issues (e.g., the available range and physical limitation of the Wenzel
model) have been discussed in depth in the literature [24,25]. Thirdly, it retained the constraints employed extensively in the literature (e.g., [26–28]), that is, the application of macroscopic thermodynamic properties to very small nuclei (groups of molecules) is acceptable. This allows to eliminate the complexity raised by the changes of scale-related properties. This scheme has been used to expose the effect of size and curvature on the characteristics of heterogeneous nucleation on rough surfaces [26], although the deviations of the thermodynamic properties by scaling effect may be relatively large because the thermodynamic inequilibrium (e.g., supercooling or supersaturation) for nucleation is generally innegligible. Under the premises mentioned above, now we focused on the regimes by which a nucleus is in contact with a substrate. Let R stand for the radius of the spherical-caped nucleus, d the size of the contact area, h the contact angle between the nucleus (phase 2) agglomerated from the parent phase (phase 1) and the substrate (phase 3), Lx the scale for observing, and Lu and Li the up and low bounds of the fractal region respectively; see Fig. 1. For a smooth surface, the contact area of the embryo on the surface, a perfect circle, depends only on the diameter, d, without the scaling effect; see Fig. 1(a). The area, Se , can be expressed as,
Se1 ¼
p 4
d ¼ pR2 cos2 h; 2
D¼2
ð1Þ
The perfect contact between the spherical-cap shaped nucleus and substrate allows to link the area to the size of the nucleus with the help of the contact angle as shown in Eq. (1). For a fractal surface, the contact area, remaining a circle roughly, depends on its size, the fractal dimension, and the up and low bounds of fractal region. Three different regimes of the contact area between the nucleus and fractal substrate can be identified as illustrated in Fig. 1(b)–(d). If a large embryo forms on the fractal surface with a small up bound of fractal region, d > Lu , the contact area of the single nucleus over the fractal substrate, Se , is larger than that over the smooth surface by a factor ðLu =LÞðD2Þ , with L the minimum scale of measurement.
Fig. 1. Schematic illustration of heterogeneous nucleation on different surfaces: (a) nuclei are in contact with smooth surface and there is no scaling effect on the contacted area; (b) nuclei are in contact with fractal surface with the size of the contacted area significantly larger than the up bound of the fractal region, d > Lu ; (c) nuclei are in contact with fractal surface with the size of the contacted area significantly smaller than the low bound of the fractal region, d < Ll ; and (d) nuclei are in contact with fractal surface with the size of the contacted area between the low and up bounds of the fractal region, Ll < d < Lu . In this illustration, phase 1 represents the parent phase (liquid or gas), phase 2 denotes the nuclei or crystals and phase 3 is the substrate with smooth or rough surfaces.
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Q. Zeng / Journal of Crystal Growth 475 (2017) 49–54
D2
Se2 ¼
pd2 Lu 4
L
;
d > Lu > L P Ll
ð2Þ
Generally, L P Ll . If L < Ll ; Ll replaces L in Eq. (2). By contrast, if a crystal that heterogeneously nucleates on the fractal surface as displayed in Fig. 1(c), is even smaller than the low bound of the fractal region, d < Ll , the contact area, Se , is surprisingly independent of the fractal parameters, but identical to the expression of Eq. (1); Se3 ¼ Se1 ; d < Ll . A much complexer contact between the nucleus and substrate emerges when the size of contact area matches the fractal region, Ll < d < Lu . As demonstrated in Fig. 1(d), this regime shows the scaling features characterized by the fractal parameters, which gives,
D2
Se4 ¼
pd2 d
Ll
4
;
Ll < d < Lu
ð3Þ
Eqs. (1)–(3) present the contact areas of the embryos heterogeneously nucleating on smooth and fractal surfaces in different regimes. We then studied an overall observing area, A. For the cases mentioned above, the total area with the characteristic length of Lx can be given by,
A1 ¼ L2x ; D ¼ 2 D2 Lu ; A2 ¼ L2x L D2 Lu ; A3 ¼ L2x Ll D2 Lx ; ¼ L2x Ll D2 Lu ; A4 ¼ L2x Ll D2 Lx ; ¼ L2x Ll
ð4aÞ Lx > d > Lu > L P Ll
ð4bÞ
Lx P Lu > Ll > d
ð4cÞ
Lu > Lx > Ll > d
ð4dÞ
Lx P Lu > d > Ll
ð4eÞ
Lu > Lx > d > Ll
ð4fÞ
Note that the scaling competitions of Lx and Lu lead to two different expressions for the total observing areas in the last two contact regimes. When the observing scale is larger than and/or equal to the up bound of the fractal region, Lx P Lu , the factor of the contact area on a fractal surface to that on a smooth surface depends on all the fractal parameters; see Eqs. (4c) and (4e). Otherwise, Eqs. (4d) and (4f) hold. We here define a parameter n to represent the available number of sites where the implantation of nuclei may take place. It thus has,
n¼
A Se
ð5Þ
Note that not every site of a surface can be occupied by nucleation embryos, so it would be reasonable to assume that the effective sites for nucleation are proportional to the total number of the sites on the surface. One therefore obtains,
ne ¼ Kn ¼
KA Se
ð6Þ
where ne is the effective nucleation sites of the surface with the area of A; and K denotes the effective ratio with K 1. Due to the fact that the observing scales are different for different experiments, it is preferred to define a specific effective nucleation density, N, as follows,
N¼
ne L2x
¼
KA Se L2x
ð7Þ
Combining Eqs. (1)–(7), on can directly obtain the relationships between the specific nucleation density and the size of nuclei in different contact regimes,
4K
logðNi ji¼1;2 Þ ¼ 2 logðdÞ þ log ; Lx > d > Lu > L P Ll p Lu Lu d 4K logðN3 Þ ¼ D log 2 log þ log ; Lx P Lu > Ll > d Ll Ll p Lx Lx d 4K ¼ D log 2 log þ log ; Lu > Lx > Ll > d Ll Ll p Lu 4K logðN4 Þ ¼ D log ; Lx P Lu > d > Ll 2 log ðLu Þ þ log d p Lx 4K ¼ D log ; Lu > Lx > d > Ll 2 log ðLx Þ þ log d p
ð8aÞ ð8bÞ ð8cÞ ð8dÞ ð8eÞ
Eq. (8) can be abstracted in a simple form,
log N ¼ mðD 2Þ þ n
ð9Þ
where n ¼ 2 logðdÞ þ logð4K=pÞ for all the cases and m depends on the regimes of the contact between the nuclei and substrates: m ¼ ambient values for D ¼ 2 or d > Lu ; m ¼ log Lj jj¼u;x =Ll for d < Ll ; m ¼ log Lj jj¼u;x =d for Ll < d < Lu . Obviously, the nucleation density is associated with the size of nuclei, and the relationships are depending on the ways in which the nuclei and surfaces are contacted. Those can be summarized as follows: For non fractal surfaces or nuclei in large size over the up bound of fractal region, d > Lu , the number of crystals is independent of the fractal dimensions of the surfaces, but only charged by the contact areas between the crystals and substrates and the effective ratios for nucleation. The size of crystals can not be estimated from the plots of log N against D via Eq. (9). For very tiny nuclei under the low bound of fractal region, Ll > d, the nucleation density is associated with the fractal dimensions, the size of the contact areas between the crystals and substrate, and the low and up bounds (or the maximum observing scales Lx ) of the fractal region. The size of nuclei is involved in the Y-intercept of the plot of log N against D via Eq. (9). However, the effective nucleation ratio K remains unknown. For scale-matched nuclei and fractal surfaces, Lu > d > Ll , the size of nuclei can be directly obtained by analyzing the plotting slope of log N against D via Eq. (9). It has,
d ¼ Lj jj¼u;x 10m
ð10Þ
Eq. (10) provides an alternatively simple way to estimate the average size of nuclei. 3. Application and discussion Based on several assumptions of heterogeneous nucleation on complex surfaces that are featured by fractal structure, simple equations were developed to estimate the size of the nuclei (or micro/macro crystals) in contact with the fractal surfaces. Although extensive tests on the fractal properties of various artificial and natural materials (e.g., silica porous materials, cementbased porous materials, rocks, catalysts and membranes [18,19,21,29,30]), very few reports (both experimentally and theoretically) on heterogeneous nucleation on the complex surfaces with different fractal dimensions have been documented. Here we adopted the experimental data reported by Mu et al. [31] to demonstrate the relationships between the nucleation density and fractal dimension, and to estimate the size of nuclei from the established fractal model. The authors of Ref. [31] prepared six
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Q. Zeng / Journal of Crystal Growth 475 (2017) 49–54
groups of magnesium surfaces with different fractal dimensions measured by a differential box-counting method, and measured the number of nucleation sites of water drops under three different supercooling extents (i.e., DT ¼ 1:8 C, 2.3 °C and 4.6 °C) condensed on the magnesium surfaces by an image analysis method; see Ref. [31] for the experimental details and results. Fig. 2 shows the measured data of nucleation density (points) versus fractal dimension. Obviously, it shows linear plots of log N against ðD 2Þ for all the samples under different supercooling degrees, suggesting that increasing the fractal dimensions can heavily promote the nucleation density observed. This also confirms the nucleation facilitation by rough surfaces reported in the literature [1,4,6,32–34]. Generally, this observation is probably owning to the following factors: (a) the promotion of nucleation possibility by increasing the available surface areas [21,33] and (b) the energy barrier reduction by changing the configurations of the contact between the nuclei and substrate when nucleation takes place on the fractal surfaces [27,28]. For the surfaces with a certain fractal dimension, the higher is the supercooling extent, the higher is the nucleation density observed, which is reasonably due to the enhanced probability of nucleation under a severer supercooling [7,35]. A fitting procedure was conducted to all the datasets through Eq. (9), so lines were obtained and shown in Fig. 2. Note that for the dataset of DT ¼ 1:8 °C, one data (in dashed circle) that deviates from the linear plot of logðNÞ against ðD 2Þ was abandoned during fitting. The good agreements between the lines and the experimental data in turn valid the applicability of Eq. (9) proposed in this study. From Fig. 2, the slopes can be read directly, so it is possible to determine the size of the nuclei in contact with fractal surfaces through Eq. (10). A key parameter for determining the size of nuclei can be the up bound of fractal region, Lu , or the observing scale, Lx . Ref. [31] did not present the values of Lu of the magnesium surfaces, but provided the minimum analyzed scale (around 1 lm). As the analyzed areas of the magnesium surfaces were always fractal, Lx < Lu , so Eq. (8e) maybe the correct equation for representing the relationship between the nucleation site density and fractal dimension. Fig. 3 compares the average size of nuclei estimated by Eq. (10) and measured by the image analysis reported in [31]. From Fig. 3, it shows that the measured sizes of nuclei are comparable with the ones by the estimation at the supercoolings of 1.4 °C and 3.6 °C, but the deviation between the two methods somehow becomes significant at the supercooling of 2.3 °C. As observed in Fig. 3, increasing the supercooling extent tends to decrease the size
Fig. 2. Effect of fractal dimension on nucleation sites: Raising the nucleation density with the fractal dimension increasing. Points: experimental data by Mu et al. [31], and lines: fitting curves by Eq. (9). We excluded one data (in dashed circle) in the dataset of DT ¼ 1:8 C that deviates from the linear plot of logðNÞ against ðD 2Þ.
Fig. 3. Comparative plot of the nucleus size from the estimation proposed in this study and experimental measurement in Ref. [31]. For better illustration, the full Y error bar of the dataset of DT ¼ 1:8 C by the estimation in this study was not shown in the figure, as the maximum Y error values (63 nm) are significant larger than the average ones (20 nm).
of the nuclei that were required for condensing on the fractal surfaces in steady state. This is reasonable, because, according to classic nucleation theory, the critical nucleus size is inversely proportional to the third power of the free energy differences (directly associated with temperature) between the crystalline cluster and the parent liquid phase [7]. A study by Govindarajan and Lindow [35] reported the logarithmic increases of the size of ice nuclei with decreasing the supercooling temperature from 12 °C to 2 °C for bacterial catalysts, which is much severer than the results shown in Fig. 3. It is noteworthy that the present model for estimating the size of the nuclei on fractal surface is sensitive to the relationships between the nucleation site density and fractal dimension that are both determined experimentally, because d is proportional to the mth power of 10. So the very minor deviations of m displayed in Fig. 2 can lead to the significant errors of the average size of nuclei. For example, it shows the maximum errors of 63 nm for the dataset of DT ¼ 1:8 °C presented in Fig. 3. This would also yield significant discrepancy between the estimated size of nuclei and the measured one (e.g., the data of DT ¼ 2:3 °C displayed in Fig. 3). Here we emphasized that the size estimated in the present study and determined in Ref. [31] measures the contact interface between nuclei and fractal surface, but not the ‘‘real” size of the nuclei; see Fig. 1 for much clearer demonstration. So to measure the ‘‘real” size of nuclei, one needs to know the contact angle between the nuclei (or condensed droplets) and the substrate and the patterns in which the fractal structure impacts the contact angle. A very recent classic-nucleation-based theoretical study by the same corresponding author revealed that the deviations of the size of the nuclei between fractal and smooth surfaces are significant for the hydrophilic nucleations (acute contact angles), but negligible (<5%) for the hydrophobic ones (obtuse contact angles) [27]. A literature survey by Liu et al. [36] indicated that the apparent contact angles between the condensed droplets and substrates are general obtuse, so the influence of fractal dimension on the size of nuclei may be negligible. Therefore, the size of the contract interface estimated by Eq. (10) truly inflects the real size of nuclei via the simple relation, R ¼ d=ð2 cos hÞ; see Fig. 1(a). There are several limits of the present estimating method. Firstly, the fractal parameters, i.e., fractal dimensions and the up/low bounds of fractal regions, should be determined accurately. It is not easy to achieve this aim for complex porous materials, for example, cement-based porous materials show two or more fractal
Q. Zeng / Journal of Crystal Growth 475 (2017) 49–54
regions, and the up/low bounds of the fractal regions vary upon the material structures [18,19,37]. Secondly, it is also difficult to obtain the number of the nuclei in contact with fractal surfaces. In Ref. [31], the number of nucleation sites was measured by the imaging analysis, of which the shortages can not be avoid in this analysis either. Last but not least, the method proposed in the present study is limited to where the classic surface physic laws (e.g., the Young and Wenzel equations) will approximately apply. The cases of typical heterogeneous nucleation on hydrophobic surfaces, where the Cassie-Baxter-type contact interpretation should be applied, are beyond the estimating method developed. This may be also an important factor for the deviations between the estimated size of nuclei and the measured one. 4. Conclusion In this study, we aimed to establish a simple method to estimate the size of the nuclei in contact with fractal surfaces, and to present a validation of the method by the published data. The study also helps to understand how the fractal nature of substrates influences the physical interactions between crystalline clusters (or droplets) and surfaces. A simple model was developed to calculate the nucleation site density on fractal surfaces. The nucleation site density is associated with the size of nuclei in different regimes due to the different scales of the nuclei and fractal surfaces. Three regimes were found to link the nucleation site density to the size of nuclei, fractal dimensions and/or the bounds of fractal regions: When d > Lu (large nucleus over the up bound of fractal region) and D ¼ 2, the nucleation site density is independent of fractal dimension, but only proportional to the contact area between the nuclei and substrates. When Li > d (tinny nucleus under the low bound of fractal region), the nucleation site density is governed by all the scale-associated parameters: fractal dimension, the size of contact area, and the low and up bounds of fractal region (or the observing scale Lx ). The Y-intercept of the plot of log N against D 2 contains the size parameter. When Lu > d > Li (nucleus in fractal region), the nucleation site density is determined by all the scale-associated parameters as well. The slope of the plot of log N against D 2 contains the size parameter. The developed model for estimating the size of nuclei was validated by analyzing the experimental data reported in the literature and comparing the results between the two methods. The good agreements between the results by the two methods indicate that the proposed method is useful for assessing the size of the embryos that heterogeneously nucleate on fractal surfaces. Further tests on the robustness of the method and rigorous investigations on determining the nucleation site density and size accurately are required in the future. Acknowledgement The research is supported by the National Natural Science Foundation of China (No. 51408536). The authors also appreciate the efforts of the anonymous reviewers to improve the quality of this study. References [1] J.L. Holbrough, J.M. Campbell, F.C. Meldrum, H.K. Christenson, Topographical control of crystal nucleation, Cryst. Growth Des. 12 (2012) 750–755. [2] A.J. Page, R.P. Sear, Crystallization controlled by the geometry of a surface, J. Am. Chem. Soc. 131 (2009) 17550–17551.
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Journal of Crystal Growth journal homepage: www.elsevier.com/locate/crys
A simple method for estimating the size of nuclei on fractal surfaces Qiang Zeng Institute of Advanced Engineering Structures and Materials, College of Civil Engineering and Architecture, Yuhangtang Rd 866, Zhejiang University, 310058 Hangzhou, PR China
a r t i c l e
i n f o
Article history: Received 2 November 2016 Received in revised form 23 May 2017 Accepted 31 May 2017 Available online 3 June 2017 Communicated by T.F. Kuech Keywords: Nucleation Size Fractal Scale
a b s t r a c t Determining the size of nuclei on complex surfaces remains a big challenge in aspects of biological, material and chemical engineering. Here the author reported a simple method to estimate the size of the nuclei in contact with complex (fractal) surfaces. The established approach was based on the assumptions of contact area proportionality for determining nucleation density and the scaling congruence between nuclei and surfaces for identifying contact regimes. It showed three different regimes governing the equations for estimating the nucleation site density. Nuclei in the size large enough could eliminate the effect of fractal structure. Nuclei in the size small enough could lead to the independence of nucleation site density on fractal parameters. Only when nuclei match the fractal scales, the nucleation site density is associated with the fractal parameters and the size of the nuclei in a coupling pattern. The method was validated by the experimental data reported in the literature. The method may provide an effective way to estimate the size of nuclei on fractal surfaces, through which a number of promising applications in relative fields can be envisioned. Ó 2017 Elsevier B.V. All rights reserved.
1. Introduction One of the most important open questions in crystallization is the controls of nucleus size and structure by substrate geometry configurations with the auxiliary manipulations of nucleating rates [1–7]. While there have increasing experimental/computational methods for investigating homogeneous and heterogeneous nucleation, there are almost no equivalent developments for assessing the size of the nuclei in contact with complex surfaces structured, for example, in the pattern of self-similarity (fractal structure). Determining the size of the nuclei on complex surfaces is always a big challenge in aspects of biological, material and chemical engineering. In the last decades, there have been developed many methods that can directly measure the size of nuclei. Gasser et al. [8] reported a real–space imaging technique by laser scanning confocal microscopy for identifying and observing both the nucleation and growth of crystalline regions directly. Yau and Vekilov [9] used direct atomic force microscopy to determine the raftlike shape of apoferritin crystals (quasi-planar nuclei) and estimate their size. Andersen et al. [10] applied in-situ powder X-ray diffraction with Rietveld refinement and whole powder pattern modeling to measure the size of maghemite nanocrystals. Nanev et al. [11] used a microscopy-based method to separate the nucleation and growth stages of the crystallization process for insulin crystals and assess their size. Those successfully used techniques, however,
E-mail address: [email protected] http://dx.doi.org/10.1016/j.jcrysgro.2017.05.037 0022-0248/Ó 2017 Elsevier B.V. All rights reserved.
meet limitations by complex experimental constraints. For instance, when crystallization takes place in the pores of porous solids, the microscopy-based technique would fail to detect the nuclei. So, more frequently, the size of nuclei remains estimated approximately from (classic and advanced) thermodynamic analysis [12,13], the inverse analysis of experimental data that were not designed to measure the size [14], molecular simulations [7,15] and the combination of experiments and simulations [16]. Although most of the published methods for nuclei size determination might assess the size of nuclei successfully, there remains no perfect approach covering all the phase transition mechanisms and affecting factors, especially when nucleation occurs on fractal surfaces. Therefore, finding a simple way to determine the size of the nuclei heterogeneously nucleating on fractal surfaces is the main task of this study. A fractal surface is structured in the pattern of self-similarity that repeats at every scale theoretically [17] or in limited scales practically [18–20]. Previous study by Stolyarova et al. [21] indicated that the number of nucleation seeds on a fragment with fractal structures significantly exceeds that on the one with flat surfaces, which can be reasonably due to a sufficient local molecular concentration for nucleation generated by the fractal structure. Furthermore, the nucleation–promoting effect may hold for any substrate with fractal structures. The authors further presented an explicit model to link nucleation density to fractal parameters. Although the model in Ref. [21] contained a nucleus-volumerelated parameter, it remains requiring a clear and explicit model to elucidate the way in which the nucleation density is governed
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by the size of nuclei and the fractal parameters of surfaces. This, in turn, may help to establish a method to estimate the nuclei size once the nucleation density and fractal parameters are determined. In what follows, we first presented a section to establish the relations between nucleation density and the size of the crystals that nucleate on fractal surfaces heterogeneously. We showed scale-depended regimes for calculating the contact areas between the nuclei and substrates. Those generated equations can be used to estimate the size of nuclei once nucleation density, fractal dimension and the up and low bounds of fractal regions were known. The simple method was tested by the data reported in the literature. The results may provide an effective way to estimate the size of nuclei on fractal surfaces. 2. Estimating method A heterogeneously nucleating process is often very complex especially when nuclei form on a rough surface with, for instance, fractal structures. By tracking a number of representative nuclei with definable ‘‘macro” properties from the parent phase (e.g., solutions, gases and melts), this process can be identified qualitatively and/or characterized quantitatively. However, the roughness of substrate may make it difficult to determine the contact area – one of the most important parameters for evaluating the nucleation density and the size of nuclei. Several assumptions were made in this study to obtain a simple and tough estimate. Firstly, a general assumption of spherical-cap shaped nuclei was adopted regardless of the geometry and surface properties of the contact substrate. This provides a practical way to calculate the contact areas between the spherical-cap shaped nuclei and the substrate with complex structure, and to link the size of the contact areas to the real size of the nuclei. Secondly, a Wenzel type contact between the nuclei and rough substrate was retained. This says that the areas of the contact surfaces are completely covered by the nuclei without empty voids involved. And this regime has been reported to account for the boiling and wetting behaviors on fractal hierarchical surfaces [22,23]. The debates of the relevant issues (e.g., the available range and physical limitation of the Wenzel
model) have been discussed in depth in the literature [24,25]. Thirdly, it retained the constraints employed extensively in the literature (e.g., [26–28]), that is, the application of macroscopic thermodynamic properties to very small nuclei (groups of molecules) is acceptable. This allows to eliminate the complexity raised by the changes of scale-related properties. This scheme has been used to expose the effect of size and curvature on the characteristics of heterogeneous nucleation on rough surfaces [26], although the deviations of the thermodynamic properties by scaling effect may be relatively large because the thermodynamic inequilibrium (e.g., supercooling or supersaturation) for nucleation is generally innegligible. Under the premises mentioned above, now we focused on the regimes by which a nucleus is in contact with a substrate. Let R stand for the radius of the spherical-caped nucleus, d the size of the contact area, h the contact angle between the nucleus (phase 2) agglomerated from the parent phase (phase 1) and the substrate (phase 3), Lx the scale for observing, and Lu and Li the up and low bounds of the fractal region respectively; see Fig. 1. For a smooth surface, the contact area of the embryo on the surface, a perfect circle, depends only on the diameter, d, without the scaling effect; see Fig. 1(a). The area, Se , can be expressed as,
Se1 ¼
p 4
d ¼ pR2 cos2 h; 2
D¼2
ð1Þ
The perfect contact between the spherical-cap shaped nucleus and substrate allows to link the area to the size of the nucleus with the help of the contact angle as shown in Eq. (1). For a fractal surface, the contact area, remaining a circle roughly, depends on its size, the fractal dimension, and the up and low bounds of fractal region. Three different regimes of the contact area between the nucleus and fractal substrate can be identified as illustrated in Fig. 1(b)–(d). If a large embryo forms on the fractal surface with a small up bound of fractal region, d > Lu , the contact area of the single nucleus over the fractal substrate, Se , is larger than that over the smooth surface by a factor ðLu =LÞðD2Þ , with L the minimum scale of measurement.
Fig. 1. Schematic illustration of heterogeneous nucleation on different surfaces: (a) nuclei are in contact with smooth surface and there is no scaling effect on the contacted area; (b) nuclei are in contact with fractal surface with the size of the contacted area significantly larger than the up bound of the fractal region, d > Lu ; (c) nuclei are in contact with fractal surface with the size of the contacted area significantly smaller than the low bound of the fractal region, d < Ll ; and (d) nuclei are in contact with fractal surface with the size of the contacted area between the low and up bounds of the fractal region, Ll < d < Lu . In this illustration, phase 1 represents the parent phase (liquid or gas), phase 2 denotes the nuclei or crystals and phase 3 is the substrate with smooth or rough surfaces.
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D2
Se2 ¼
pd2 Lu 4
L
;
d > Lu > L P Ll
ð2Þ
Generally, L P Ll . If L < Ll ; Ll replaces L in Eq. (2). By contrast, if a crystal that heterogeneously nucleates on the fractal surface as displayed in Fig. 1(c), is even smaller than the low bound of the fractal region, d < Ll , the contact area, Se , is surprisingly independent of the fractal parameters, but identical to the expression of Eq. (1); Se3 ¼ Se1 ; d < Ll . A much complexer contact between the nucleus and substrate emerges when the size of contact area matches the fractal region, Ll < d < Lu . As demonstrated in Fig. 1(d), this regime shows the scaling features characterized by the fractal parameters, which gives,
D2
Se4 ¼
pd2 d
Ll
4
;
Ll < d < Lu
ð3Þ
Eqs. (1)–(3) present the contact areas of the embryos heterogeneously nucleating on smooth and fractal surfaces in different regimes. We then studied an overall observing area, A. For the cases mentioned above, the total area with the characteristic length of Lx can be given by,
A1 ¼ L2x ; D ¼ 2 D2 Lu ; A2 ¼ L2x L D2 Lu ; A3 ¼ L2x Ll D2 Lx ; ¼ L2x Ll D2 Lu ; A4 ¼ L2x Ll D2 Lx ; ¼ L2x Ll
ð4aÞ Lx > d > Lu > L P Ll
ð4bÞ
Lx P Lu > Ll > d
ð4cÞ
Lu > Lx > Ll > d
ð4dÞ
Lx P Lu > d > Ll
ð4eÞ
Lu > Lx > d > Ll
ð4fÞ
Note that the scaling competitions of Lx and Lu lead to two different expressions for the total observing areas in the last two contact regimes. When the observing scale is larger than and/or equal to the up bound of the fractal region, Lx P Lu , the factor of the contact area on a fractal surface to that on a smooth surface depends on all the fractal parameters; see Eqs. (4c) and (4e). Otherwise, Eqs. (4d) and (4f) hold. We here define a parameter n to represent the available number of sites where the implantation of nuclei may take place. It thus has,
n¼
A Se
ð5Þ
Note that not every site of a surface can be occupied by nucleation embryos, so it would be reasonable to assume that the effective sites for nucleation are proportional to the total number of the sites on the surface. One therefore obtains,
ne ¼ Kn ¼
KA Se
ð6Þ
where ne is the effective nucleation sites of the surface with the area of A; and K denotes the effective ratio with K 1. Due to the fact that the observing scales are different for different experiments, it is preferred to define a specific effective nucleation density, N, as follows,
N¼
ne L2x
¼
KA Se L2x
ð7Þ
Combining Eqs. (1)–(7), on can directly obtain the relationships between the specific nucleation density and the size of nuclei in different contact regimes,
4K
logðNi ji¼1;2 Þ ¼ 2 logðdÞ þ log ; Lx > d > Lu > L P Ll p Lu Lu d 4K logðN3 Þ ¼ D log 2 log þ log ; Lx P Lu > Ll > d Ll Ll p Lx Lx d 4K ¼ D log 2 log þ log ; Lu > Lx > Ll > d Ll Ll p Lu 4K logðN4 Þ ¼ D log ; Lx P Lu > d > Ll 2 log ðLu Þ þ log d p Lx 4K ¼ D log ; Lu > Lx > d > Ll 2 log ðLx Þ þ log d p
ð8aÞ ð8bÞ ð8cÞ ð8dÞ ð8eÞ
Eq. (8) can be abstracted in a simple form,
log N ¼ mðD 2Þ þ n
ð9Þ
where n ¼ 2 logðdÞ þ logð4K=pÞ for all the cases and m depends on the regimes of the contact between the nuclei and substrates: m ¼ ambient values for D ¼ 2 or d > Lu ; m ¼ log Lj jj¼u;x =Ll for d < Ll ; m ¼ log Lj jj¼u;x =d for Ll < d < Lu . Obviously, the nucleation density is associated with the size of nuclei, and the relationships are depending on the ways in which the nuclei and surfaces are contacted. Those can be summarized as follows: For non fractal surfaces or nuclei in large size over the up bound of fractal region, d > Lu , the number of crystals is independent of the fractal dimensions of the surfaces, but only charged by the contact areas between the crystals and substrates and the effective ratios for nucleation. The size of crystals can not be estimated from the plots of log N against D via Eq. (9). For very tiny nuclei under the low bound of fractal region, Ll > d, the nucleation density is associated with the fractal dimensions, the size of the contact areas between the crystals and substrate, and the low and up bounds (or the maximum observing scales Lx ) of the fractal region. The size of nuclei is involved in the Y-intercept of the plot of log N against D via Eq. (9). However, the effective nucleation ratio K remains unknown. For scale-matched nuclei and fractal surfaces, Lu > d > Ll , the size of nuclei can be directly obtained by analyzing the plotting slope of log N against D via Eq. (9). It has,
d ¼ Lj jj¼u;x 10m
ð10Þ
Eq. (10) provides an alternatively simple way to estimate the average size of nuclei. 3. Application and discussion Based on several assumptions of heterogeneous nucleation on complex surfaces that are featured by fractal structure, simple equations were developed to estimate the size of the nuclei (or micro/macro crystals) in contact with the fractal surfaces. Although extensive tests on the fractal properties of various artificial and natural materials (e.g., silica porous materials, cementbased porous materials, rocks, catalysts and membranes [18,19,21,29,30]), very few reports (both experimentally and theoretically) on heterogeneous nucleation on the complex surfaces with different fractal dimensions have been documented. Here we adopted the experimental data reported by Mu et al. [31] to demonstrate the relationships between the nucleation density and fractal dimension, and to estimate the size of nuclei from the established fractal model. The authors of Ref. [31] prepared six
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groups of magnesium surfaces with different fractal dimensions measured by a differential box-counting method, and measured the number of nucleation sites of water drops under three different supercooling extents (i.e., DT ¼ 1:8 C, 2.3 °C and 4.6 °C) condensed on the magnesium surfaces by an image analysis method; see Ref. [31] for the experimental details and results. Fig. 2 shows the measured data of nucleation density (points) versus fractal dimension. Obviously, it shows linear plots of log N against ðD 2Þ for all the samples under different supercooling degrees, suggesting that increasing the fractal dimensions can heavily promote the nucleation density observed. This also confirms the nucleation facilitation by rough surfaces reported in the literature [1,4,6,32–34]. Generally, this observation is probably owning to the following factors: (a) the promotion of nucleation possibility by increasing the available surface areas [21,33] and (b) the energy barrier reduction by changing the configurations of the contact between the nuclei and substrate when nucleation takes place on the fractal surfaces [27,28]. For the surfaces with a certain fractal dimension, the higher is the supercooling extent, the higher is the nucleation density observed, which is reasonably due to the enhanced probability of nucleation under a severer supercooling [7,35]. A fitting procedure was conducted to all the datasets through Eq. (9), so lines were obtained and shown in Fig. 2. Note that for the dataset of DT ¼ 1:8 °C, one data (in dashed circle) that deviates from the linear plot of logðNÞ against ðD 2Þ was abandoned during fitting. The good agreements between the lines and the experimental data in turn valid the applicability of Eq. (9) proposed in this study. From Fig. 2, the slopes can be read directly, so it is possible to determine the size of the nuclei in contact with fractal surfaces through Eq. (10). A key parameter for determining the size of nuclei can be the up bound of fractal region, Lu , or the observing scale, Lx . Ref. [31] did not present the values of Lu of the magnesium surfaces, but provided the minimum analyzed scale (around 1 lm). As the analyzed areas of the magnesium surfaces were always fractal, Lx < Lu , so Eq. (8e) maybe the correct equation for representing the relationship between the nucleation site density and fractal dimension. Fig. 3 compares the average size of nuclei estimated by Eq. (10) and measured by the image analysis reported in [31]. From Fig. 3, it shows that the measured sizes of nuclei are comparable with the ones by the estimation at the supercoolings of 1.4 °C and 3.6 °C, but the deviation between the two methods somehow becomes significant at the supercooling of 2.3 °C. As observed in Fig. 3, increasing the supercooling extent tends to decrease the size
Fig. 2. Effect of fractal dimension on nucleation sites: Raising the nucleation density with the fractal dimension increasing. Points: experimental data by Mu et al. [31], and lines: fitting curves by Eq. (9). We excluded one data (in dashed circle) in the dataset of DT ¼ 1:8 C that deviates from the linear plot of logðNÞ against ðD 2Þ.
Fig. 3. Comparative plot of the nucleus size from the estimation proposed in this study and experimental measurement in Ref. [31]. For better illustration, the full Y error bar of the dataset of DT ¼ 1:8 C by the estimation in this study was not shown in the figure, as the maximum Y error values (63 nm) are significant larger than the average ones (20 nm).
of the nuclei that were required for condensing on the fractal surfaces in steady state. This is reasonable, because, according to classic nucleation theory, the critical nucleus size is inversely proportional to the third power of the free energy differences (directly associated with temperature) between the crystalline cluster and the parent liquid phase [7]. A study by Govindarajan and Lindow [35] reported the logarithmic increases of the size of ice nuclei with decreasing the supercooling temperature from 12 °C to 2 °C for bacterial catalysts, which is much severer than the results shown in Fig. 3. It is noteworthy that the present model for estimating the size of the nuclei on fractal surface is sensitive to the relationships between the nucleation site density and fractal dimension that are both determined experimentally, because d is proportional to the mth power of 10. So the very minor deviations of m displayed in Fig. 2 can lead to the significant errors of the average size of nuclei. For example, it shows the maximum errors of 63 nm for the dataset of DT ¼ 1:8 °C presented in Fig. 3. This would also yield significant discrepancy between the estimated size of nuclei and the measured one (e.g., the data of DT ¼ 2:3 °C displayed in Fig. 3). Here we emphasized that the size estimated in the present study and determined in Ref. [31] measures the contact interface between nuclei and fractal surface, but not the ‘‘real” size of the nuclei; see Fig. 1 for much clearer demonstration. So to measure the ‘‘real” size of nuclei, one needs to know the contact angle between the nuclei (or condensed droplets) and the substrate and the patterns in which the fractal structure impacts the contact angle. A very recent classic-nucleation-based theoretical study by the same corresponding author revealed that the deviations of the size of the nuclei between fractal and smooth surfaces are significant for the hydrophilic nucleations (acute contact angles), but negligible (<5%) for the hydrophobic ones (obtuse contact angles) [27]. A literature survey by Liu et al. [36] indicated that the apparent contact angles between the condensed droplets and substrates are general obtuse, so the influence of fractal dimension on the size of nuclei may be negligible. Therefore, the size of the contract interface estimated by Eq. (10) truly inflects the real size of nuclei via the simple relation, R ¼ d=ð2 cos hÞ; see Fig. 1(a). There are several limits of the present estimating method. Firstly, the fractal parameters, i.e., fractal dimensions and the up/low bounds of fractal regions, should be determined accurately. It is not easy to achieve this aim for complex porous materials, for example, cement-based porous materials show two or more fractal
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regions, and the up/low bounds of the fractal regions vary upon the material structures [18,19,37]. Secondly, it is also difficult to obtain the number of the nuclei in contact with fractal surfaces. In Ref. [31], the number of nucleation sites was measured by the imaging analysis, of which the shortages can not be avoid in this analysis either. Last but not least, the method proposed in the present study is limited to where the classic surface physic laws (e.g., the Young and Wenzel equations) will approximately apply. The cases of typical heterogeneous nucleation on hydrophobic surfaces, where the Cassie-Baxter-type contact interpretation should be applied, are beyond the estimating method developed. This may be also an important factor for the deviations between the estimated size of nuclei and the measured one. 4. Conclusion In this study, we aimed to establish a simple method to estimate the size of the nuclei in contact with fractal surfaces, and to present a validation of the method by the published data. The study also helps to understand how the fractal nature of substrates influences the physical interactions between crystalline clusters (or droplets) and surfaces. A simple model was developed to calculate the nucleation site density on fractal surfaces. The nucleation site density is associated with the size of nuclei in different regimes due to the different scales of the nuclei and fractal surfaces. Three regimes were found to link the nucleation site density to the size of nuclei, fractal dimensions and/or the bounds of fractal regions: When d > Lu (large nucleus over the up bound of fractal region) and D ¼ 2, the nucleation site density is independent of fractal dimension, but only proportional to the contact area between the nuclei and substrates. When Li > d (tinny nucleus under the low bound of fractal region), the nucleation site density is governed by all the scale-associated parameters: fractal dimension, the size of contact area, and the low and up bounds of fractal region (or the observing scale Lx ). The Y-intercept of the plot of log N against D 2 contains the size parameter. When Lu > d > Li (nucleus in fractal region), the nucleation site density is determined by all the scale-associated parameters as well. The slope of the plot of log N against D 2 contains the size parameter. The developed model for estimating the size of nuclei was validated by analyzing the experimental data reported in the literature and comparing the results between the two methods. The good agreements between the results by the two methods indicate that the proposed method is useful for assessing the size of the embryos that heterogeneously nucleate on fractal surfaces. Further tests on the robustness of the method and rigorous investigations on determining the nucleation site density and size accurately are required in the future. Acknowledgement The research is supported by the National Natural Science Foundation of China (No. 51408536). The authors also appreciate the efforts of the anonymous reviewers to improve the quality of this study. References [1] J.L. Holbrough, J.M. Campbell, F.C. Meldrum, H.K. Christenson, Topographical control of crystal nucleation, Cryst. Growth Des. 12 (2012) 750–755. [2] A.J. Page, R.P. Sear, Crystallization controlled by the geometry of a surface, J. Am. Chem. Soc. 131 (2009) 17550–17551.
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