DOI: 10.1021/cg101412p
Observation of the Kinetic Roughening of L-Arginine Trifluoroacetate (LATF) Crystals
2011, Vol. 11 791–795
Xiaojing Liu,*,† Peiji Wang,† Dong Xu,‡ and Xianqi Wei† †
School of Science, University of Jinan, Jinan 250022, Shandong Province, People’s Republic of China, and ‡State Key Laboratory of Crystal Materials, Institute of Crystal Materials, Shandong University, Jinan 250100, People’s Republic of China Received October 21, 2010; Revised Manuscript Received December 6, 2010
ABSTRACT: The transition from slow to fast growth kinetics, caused by kinetic roughening, which depends on the solution supersaturation levels, has been investigated on the {101} facets of L-arginine trifluoroacetate (LATF) crystals at 296.15 K. Two different growth mechanisms (i.e., two-dimensional (2D) nucleation and continuous growth) are assumed to describe this growth transition phenomenon accurately. Experimental results indicate that the growth mode transition occurs at a critical supersaturation of σc = 1.10, and then the effective step energy barrier can be determined to be γ = 3.35 10-14 erg/molecule. The analysis of the experimental results for σc and γ provide the free-energy barrier at maximum (ΔG* = 7.84 10-14 erg/molecule) and the number of macromolecules in the critical nucleus (N* = 1.92). The energy barrier for the continuous addition of the growth units is examined to be Ec = 8.04 10-14 erg/molecule.
1. Introduction Recently, much progress has been made in the development of nonlinear optical (NLO) organic materials for second-harmonic generation (SHG). The discovery of the outstanding NLO properties of L-arginine phosphate monohydrate (LAP) has played an important role in working out the conception of semiorganic crystals.1-3 L-Arginine trifluoroacetate (LATF) is such a promising semiorganic NLO material, which possesses excellent optical, thermal, and mechanical properties; in particular, it has high optical nonlinearity and large optical damage threshold.4-6 LATF belongs to the monoclinic system with space group P21, and the lattice parameters are a = 10.581 A˚, b = 5.7100 A˚, c = 10.861 A˚, and β = 106.81°. Previously, under varied growth conditions, we have performed experimental measurements concerning the surface morphologies and growth mechanisms that are believed to contain invaluable information.7-9 Recently, two main growth mechanisms categories are generally considered in our theoretical and experimental work for understanding the growth kinetics of LATF crystals. The first one is the slow growth of flat, stepped crystal faces. At an atomic scale, such a surface is smooth and has a positive-step free energy, and a nucleation barrier exists for the generation of a new growth layers. The second category is the fast growth of “rough” crystal surfaces with zero-step free energy. Here, the growth rate is governed by a Wilson-Frenkel-type law.10 The atomically rough surfaces are characteristic for surfaces that are highly misoriented from a flat face, for crystal surfaces with bond energies weak compared with kT (thermal roughening) and for high supersaturations (kinetic roughening). The nature of the thermal roughening (or roughening transition), which is related to a change in the growth mechanism and morphology, is well-established, both theoretically11-13 and experimentally.14-16 Generally speaking, the roughening transition that occurs on a crystal surface at the roughening temperature (TR) plays a key role in the growth of crystals.16-18 To our knowledge, to identify the character of the roughening *Corresponding author. E-mail address:
[email protected]. r 2011 American Chemical Society
transition, the critical behavior of the edge free energy of a step (γ) must be studied, because it not only reflects the physical state of the surface but also controls the growth mechanism. When T g TR, the crystal surface is rough on an atomic scale and the edges between different faces of a crystal become rounded off.18 The computer simulations show γ = 0 and the barriers for twodimensional (2D) nucleation disappear, which implies that the growth rate depends linearly on the supersaturation (σ). Now the crystal is growing continuously. When T < TR and σ < σc (where σc is the critical supersaturation), the crystal is bounded by low-index areas, which are almost smooth on an atomic scale. The standard growth of crystals takes place via a spiral growth mechanism or a 2D nucleation mechanism (such as “birth and spread”), because of the 2D nucleation barrier (γ > 0), and a nonlinear relationship between the growth rate and the supersaturation will be found. However, when the crystal face is growing at T < TR, but at a supersaturation σ g σc, despite the nonzero-step edge free energy, it becomes microscopically rough again and the growth rate becomes linear in supersaturation again. This transition phenomenon from a slow, layer-by-layer growth regime to a fast, continuous growth regime is known as “kinetic roughening”.19-21 In literature, kinetic roughening often has been observed for the case of crystallization from the vapor phase, melt, and solution.22-24 In this paper, data are provided to lend support to the hypothesis that LATF crystal growth kinetics exhibit the kinetic roughening transition. The experimentally observed changes in crystal growth rates for varying supersaturation can be interpreted in terms of the well-established theoretical expressions, and thus the numerical data on parameters such as edge free energy, critical nucleus size, and the effective step energy barrier for rough growth can be obtained. 2. Experimental Materials and Theory 2.1. Preparation of LATF Crystal. In accordance with the previous experimentally determined solubility curve,9 LATF was synthesized by dissolving equimolar quantities of L-arginine (BR grade) and trifluoroacetate acid (AR grade) in deionized water. The synthesized salt is Published on Web 01/27/2011
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further purified by repeated recrystallization and used for the growth of LATF crystals. The solution was filtered through a series of 0.45-μm membrane filters and injected into the crystallizing vessel. Crystal samples investigated here were obtained using a slow-cooling method and the supersaturation is defined as σ = ln(c/ceq), ranging between 1.45 and 0.5 at 23 °C, where c and ceq refer to the actual and equilibrium concentrations, respectively. 2.2. Face Growth Rate Measurements and Theory. The investigations of face growth rate were performed using a programmable computer-controlled microscopy system with a constant temperature bath, and a thermocouple that had been placed into the bath was used to read the temperature of the solution. Accordingly, observations of the changes in linear dimensions of the appointed facets of crystals versus time can be recorded. The slope of the linear dimension versus time determines the growth rate of crystals. The effect of kinetic transition is the same as thermal roughening. However, the latter is a real transition and has a firm thermodynamic basis. In contrast, the former is not an actual phase transition, but rather a kinetic phenomenon; it lacks a precise definition. To mark the onset of kinetic roughening, various criteria have been put forward to identify it: (i) in terms of the driving force and step edge free-energy data, the two classical 2D nucleation criteria (N1 and N2) state that a surface will grow kinetically roughened either when the Gibbs free energy of the critical 2D nucleus (ΔG*) is less than kT or the critical nucleus radius r* is smaller than half a growth unit;16,25 (ii) the two experimental criteria (E1 and E2) indicate that kinetic roughening coincides with either the observed transition from flat to rounded crystal facets or the growth curve consists of a linear part at σ g σc,16,26 (the evaluation of this criteria requires geometrical and growth rate data); and (iii) the two statistical mechanical criteria (S1 and S2) focus either on the interface width or on the surface roughness (detailed atomistic pictures of the surface are required under this criteria).27 As mentioned previously, in the layer-by-layer mode of growth, the crystal surface advances via the generation of steps and the subsequent incorporation of growth units at the step edges or kinks. As is well-known, the step generation occurs as a result of the formation of new layers, either by spiral dislocation or 2D nucleation.28 For the latter case, the 2D nucleation free-energy barrier ΔG on any smooth surface is defined in terms of the step free energy and the driving force; such an expression is given as ΔGðrÞ ¼ Arγ - Br2 Δμ
ð1Þ
The constants A and B are geometric factors (A = 2B = 2π for a circular nucleus and A = 2B = 8 for a square nucleus). The size of the critical 2D nucleus is obtained by solving ∂ΔG(r)/∂r = 0 for r. This gives the critical radius to be γ ð2Þ r ¼ Δμ and the 2D nucleation activation barrier at maximum to be ! 2 A γ ΔG ¼ 2 Δμ
ð3Þ
Many attempts have been made to derive an analytical expression for the steady-state growth rates in the case of a 2D nucleation mechanism:29-33 !1=3 πIvstep 2 R2D ¼ d ð4Þ 3 Here, d is the layer thickness and vstep is the step velocity, which is dependent on the kinetic constant (β), according to30 Δμ -1 ð5Þ vstep ¼ β exp kT I is the frequency of stable nucleus formation per unit area, which is the product of the equilibrium surface concentration of critical nuclei (c(r*)), the Zeldovich factor (Z), and the frequency of addition (v) of one extra growth unit to make a critical nucleus stable: I ¼ vZcðrÞ
ð6Þ
For a circular critical nucleus with perimeter 2πr* and direct addition of the growth units from the mother phase (i.e., no surface diffusion), Δμ v ¼ 2πrDceq exp ð7Þ kT cðrÞ ¼ ceq exp
Z ¼
Δμ ΔG exp kT kT
Δμ Δμ 1=2 2πγ kT
ð8Þ
ð9Þ
where D is the diffusion coefficient. Combining the expressions given in eqs 2-9 gives the growth rates generated by 2D nucleation and advancement of circular steps (A = 2π in eq 3)33-35 " #2=3 ! Δμ 2Δμ Δμ 1=6 - πγ2 -1 R2D ¼ K exp exp exp kT 3kT kT 3ΔμkT ð10Þ Here, the constant K accounts for the frequency at which a macromolecule attempts to overcome a barrier for step generation and advancement; it is related not only to surface diffusive processes but also to the solution transport processes.33 For solution growth, the dimensionless driving force (Δμ/kT) for crystallization in eq 10 is related to c and ceq, via Δμ c ¼ σ ¼ ln ð11Þ kT ceq Note that eq 10 fails to predict the measured surface kinetics when the solution supersaturation or the solute concentration exceeds σc or cR. Under such conditions, the continuous mechanism growth rate (Rc) is described by the following expression:33 NA Da2 Ec ð12Þ Rc ¼ ðc c ðTÞÞ exp R kT 1:6Mw j - 1=3 where a2 is the area of the lattice site, Mw the molecular weight, j the volume fraction, cR(T) the roughening concentration, and Ec the energy barrier for the continuous addition. 2.3. Atomic Force Microscopy (AFM) Investigations. To preserve the nascent surface structure, the samples that were grown into acceptable sizes were drawn from solution by passage slowly through a layer of an immiscible nonsolvent (n-hexane) and then dried with paper tissue immediately. The {101} facet of the crystals then was mounted parallel to the sample holder and imaged. All atomic force microscopy (AFM) images were obtained in contact mode by employing a Nanoscope IIIa MultiMode AFM instrument from Digital Instruments (DI) Incorporation, with DI square pyramidal Si3N4 tips on V-shaped cantilevers 100 μm in length, with spring constants of 0.06 or 0.12 N/m. To minimize surface damage due to the AFM scanning, when a crystal face was chosen, the area was zoomed in on immediately to obtain images before the area had been rastered too many times.
3. Results and Discussion 3.1. Growth Rate Measurements. Figure 1 depicts a typical {101} facet growth rate versus supersaturation, which can be divided into three distinctly different growth areas. In area I, the growth rate equals zero, probably because of the effect of impurities that were being distributed on the surface.36 The distinct transition from slow to fast kinetics (area II to III) implies that two different growth mechanisms exist in our entire measurements. Repetitions of this growth rate investigations confirm that this transition occurs at σc > Δμ/kT = 1.10. To our knowledge, area II can be seen as an intermediate growth region, because, for σ < σc, the layer-by-layer growth mechanism is present, whereas, for σ > σc (area III), the continuous growth mechanism is dominant.
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3.2. Nonlinear Growth Kinetics below the Transitioning Supersaturation. The growth mechanism of LATF at low to moderate supersaturation (area II in Figure 1) was verified by AFM. Figure 2 shows that the dominating layer generation mechanism is 2D nucleation and growth. Figures 2a-c shows some 2D nuclei randomly distributed at the step edges on the larger terraces. The heights of the nuclei are in the range of 0.60-1.05 nm,
Figure 1. Growth rate of the {101} facet of LATF, as a function of supersaturation at t = 23 °C.
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corresponding to the distance between the {101} facets (0.64 nm). The appearance of 2D nuclei provides step sources for further growth, and the step terraces can also become the nucleation sites. The nucleation of 2D islands can be influenced by the step profile; on the other hand, the presence of large 2D islands can destabilize the morphology of terraces and steps. As shown in Figure 2d, terraces cannot be resolved, because the surface is mainly covered by irregularly shaped 2D islands, some of which exhibit other structures nucleated on their surface. After substitution of eq 11 into eq 10, the following general expression can be found: R2D πγ2 1 ¼ ln K ln ð13Þ 3ðkTÞ2 σ ðexp σ - 1Þ2=3 expð2σ=3Þσ1=6 The plot of ln[R2D(exp σ - 1)-2/3 exp(-2σ/3)σ-1/6] against σ-1 (Figure 3) shows linear character in the range of 0.91 e σ-1 e 2.00 (0.50 e σ e 1.10), where the kinetic data were best described by the birth and spread model of the 2D nucleation (goodness of fit, R2 = 0.9892). From the slope of Figure 3, we can obtain the effective step energy barrier γ = 3.35 10-14 erg/molecule. Note that eq 13 fails to predict face kinetics beyond σ = 1.10, where the significant deviations between measured and theoretically predicted growth rate data values exist. Using the experimental estimated value of γ and σc, the freeenergy barrier at maximum and the number of macromolecules in the critical nucleus at the transition supersaturation can be determined to be given as ΔG* = πγ2/(kTσc) ≈ 7.84 10-14 erg/molecule and N* = πγ2/(kTσc)2 ≈ 1.92, respectively. This agrees well with the assumption that the kinetic roughening is
Figure 2. Two-dimensional (2D) nucleation and growth mechanism at low supersaturation, imaged using atomic force microscopy (AFM). Scanning areas are 8 μm 8 μm, 9 μm 9 μm, 3 μm 3 μm, and 3 μm 3 μm for images (a)-(d), respectively.
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Figure 3. Dependence of ln[R2D(exp σ - 1)-2/3 exp(-2σ/3)σ-1/6] as a function of low supersaturation levels, which characterizes crystal growth via a 2D nucleation mechanism.
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Figure 5. Roughening of the LATF crystal surface.
the magnitudes of Ec to be 8.04 10-14 erg/molecule (goodness of fit R2 = 0.9873). This linear relationship is a strong indication that kinetic roughening is a likely explanation for the observed surface kinetics. In addition, as theoretically predicted at high supersaturations,16,26 experimental investigations of rough and rounded crystal facets (Figure 5) are also believed to corroborate the kinetic roughening hypothesis. 4. Conclusions
Figure 4. Dependence of growth rate on the reduced concentration (c-cR) at high supersaturation; the straight line indicates a theoretical fit based on eq 12.
characterized by a critical nucleus with the size of a few molecules or smaller. In addition, as mentioned previously, it is assumed that the magnitude of ΔG* is on the order of kT for the system that exhibits kinetic roughening. If indeed the assumption was valid (i.e., eq 3 is rearranged into ΔG* = πγ2/(kTσc)=kT), we can obtain γ=2.41 10-14 erg/molecule, which is smaller than the estimation value (3.35 10-14 erg/ molecule), using eq 13. The probable reason for the overestimation of γ may due to the inherent assumption of a monomer addition to a step edge or kink used in the derivation of eq 13. 3.3. Linear Growth Kinetics beyond the Transitioning Supersaturation. In the above analyses, a causal relationship between the critical transitioning supersaturation and the effective step energy barrier was realized. Of significant concern is then an independent estimation of the dependence of not only the critical roughening concentration (cR), but also the energy barrier associated with continuous growth (Ec). The straight line in Figure 4 represents the fit to data obtained at high supersaturation levels (σ > 1.10), using the parametric relation for eq 12, enabling the determination of
In this paper, the growth rate versus supersaturation curve, which exhibits a transition from slow to fast growth kinetics, has been presented and discussed. It is assumed that this experimental phenomenon coincides with the kinetic roughening hypothesis. The experimental results indicate that growth was regulated by the layer-by-layer model in which growth occurs by two-dimensional (2D) nucleation in the supersaturation range of 0.50 e σ e 1.10, while the growth mechanism changes to continuous growth mode at a supersaturation of σc ≈ 1.10 at 23 °C. Morphological rounding of the crystal facets, which is considered to be direct support for the kinetic roughening hypothesis, was also observed. From the analysis of the 2D nucleation model of layer growth, the effective step energy barrier can be determined to be γ = 3.35 10-14 erg/molecule, which corresponds to 0.82 KT. In addition, on the basis of the linear kinetics at σ > σc, the energy barrier of Ec =8.04 10-14 erg/molecule associated with continuous growth is also obtained, which is comparable to the free-energy barrier at maximum, ΔG*=7.84 10-14 erg/molecule. Acknowledgment. This work was supported by the Doctoral Foundation of University of Jinan (Nos. XBS0920, XBS0833), the National Natural Science Foundation of China (Nos. 60608010, 50872067), and Provincial Natural Science Foundation of Shandong (Y2008A21).
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