Communication pubs.acs.org/crystal
Running away from Thermodynamics To Promote or Inhibit Crystal Growth Cecília Ferreira,† Fernando A. Rocha,† Ana M. Damas,‡,§ and Pedro M. Martins*,†,‡,§ LEPAE, Departamento de Engenharia Química, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal S Supporting Information *
ABSTRACT: Impurities/additives may be either detrimental or beneficial to many different crystal growth applications. Determined to a great extent by thermodynamics, their effects are hardly avoided once supersaturation, temperature, pH, and impurity content are established. In this work we introduce the rate of supersaturation variation Rσ as a new variable that can dramatically influence crystal growth relatively to steadystate conditions. We show that the crystal growth of a model protein can be accelerated, retarded, or even suppressed by altering Rσ. Our results provide insight into the mechanism by which fast supersaturation variation prevents the adsorption equilibrium from being restored. When impurity adsorption onto kink sites gets delayed, crystal growth is enhanced and a “purifying” effect takes place. If, instead, impurity desorption from kink sites gets delayed, then a “poisoning” effect takes place. The same rationale is used to elucidate fundamental challenges that inspired this work. Included in this list are the nonlinear acceleration kinetics of growth layers and the growth rate hysteresis. While attenuating impurity incorporation, the purifying effect is expected to be important for the production of high quality lattices during single crystal growth. On the other hand, the poisoning effect opens new possibilities for crystal growth inhibition during pathological mineralization.
U
given by
nwanted effects of impurities in solutions are a critical aspect during the production of crystalline materials in pharmaceutics and industry,1−3 as well as on the production of high quality crystals for X-ray structure determination4,5 and for optic and electronic applications.6−9 On the other hand, physiological mineralization is known to be greatly affected by the presence of soluble additives.10−14 The supersaturation level σ, i.e., the thermodynamic driving force for crystallization, corresponds to the difference in chemical potential between the fluid and the crystal.10 We have found that the rate of supersaturation variation Rσ is an unexplored variable that can be used to dramatically promote or inhibit crystal growth of organic and inorganic materials. Crystal growth curves, representing the influence of the supersaturation level on the crystal growth rate G, are generally used to gauge the activity of a given impurity/additive. A variety of unexpected growth curve shapes have been interpreted since the presentation of the competitive adsorption model (CAM) in 2006.15 This is the case for type-2 growth curves, which do not show a critical supersaturation limit below which crystal growth ceases,15 and the inverted U-shaped curves resulting from the unsteady-state adsorption of impurities at the crystal surface.16,17 The CAM admits that impurities undergo two consecutive adsorption steps, first at the crystal surface and then at the active sites for growth (or kinks). The fraction of active sites occupied by impurities (θl) is related to the surface coverage (θS) by a proportionality constant (β = θl/θS) that is comparable to the “impurity effectiveness factor” predicted by pinning mechanisms.18−20 The growth rate lowering effect is © 2011 American Chemical Society
G = 1 − βθS G0
(1)
where G and G0 are the crystal growth rates in the presence and absence of impurities, respectively. Fundamental challenges that inspired this work, such as nonlinear acceleration kinetics of growth layers21−23 and crystal growth hysteresis,24,25 remain only partially understood. While trying to solve them, we went further into the fundamental meaning of the proportionality constant β. By definition, the value of θl increases with time as the number of kinks occupied with impurities (ni) increases, and/or the total number of kinks (nT) decreases:
dθl d(ni /nT ) d ln nT 1 d ni = = − θl dt dt nT dt dt
(2)
The definition of β can be used to rewrite this equation as
d ln θS d ln nT dβ 1 d ni +β = −β dt dt θSnT dt dt (3) The variation of ni results from the imbalance between instantaneous rates of adsorption (kinetic constant ki) and Received: October 11, 2011 Revised: November 16, 2011 Published: November 21, 2011 40
dx.doi.org/10.1021/cg201354s | Cryst. Growth Des. 2012, 12, 40−43
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Communication
desorption (kinetic constant k−i):
d ni = kinT θS − k−ini (4) dt A first definition of β follows from this equation assuming the steady-state condition:
βe =
θl θS
= e
ki k −i
(5)
where the subscript e stands for equilibrium condition. Combining the previous two equations leads to the following:
1 d ni = k−i(βe − β) θSnT dt
(6)
Now, if we admit that the σ is linearly increasing/decreasing with time at a constant rate of Rσ = dσ/dt, the combination of eqs 3 and 6 leads to
⎛ R σ dβ R ⎞ = βe − β⎜1 + σ ϕ⎟ k −i d σ k −i ⎠ ⎝
(7)
where ϕ is the result of the contrasting influence of σ on nT and on θS:
d ln θS d ln nT + (8) dσ dσ This parameter takes positive values when the positive influence of σ on the formation of new active sites is stronger than the negative influence on the surface adsorption of impurities. Equation 7 predicts that changing σ can have a “poisoning” effect, characterized by transient impurity effectiveness factors higher than βe, or a “purifying” effect characterized by β < βe. Sufficiently fast σ variation will prevent the adsorption equilibrium from being restored; if impurity desorption from kink sites is delayed (Rσϕ < 0 in eq 7), then a poisoning effect will take place (Figure 1A, middle cartoon sequence). If, instead, impurity adsorption onto kink sites is delayed (Rσϕ > 0), then a purifying effect will take place (Figure 1A, bottom cartoon sequence). While attenuating impurity incorporation, the purifying effect is expected to be of great use in the production of high quality lattices during single crystal growth.4−9 Moreover, crystal growth rates can be accelerated to the point of complete annihilation of the inhibiting effect of contaminants (Figure 1B, green solid line). This behavior, which we call the “type-3 growth curve”, has been observed in a variety of systems,23 including the recovery of surfaces from impurity poisoning,22 and physiological crystal formation.21 For high values of σ, the theoretical G-curves predicted by classical pinning mechanisms run approximately parallel to the G0-curve15 and, therefore, do not account for type-3 curves.24 Continuously increasing σ with time as a way to achieve the purifying effect requires that the value of ϕ is greater than 0; that is, that the rate of kink formation is a strong function of σ (see eq 8). In the mentioned literature examples, nT increases significantly with σ due to macrostep generation mechanisms.21−23 Our attempts to induce a similar effect during the crystallization of egg-white lysozyme (EWL) suggest that in this case ϕ < 0; that is, the nT increase with σ is smaller than the surface coverage θS decrease with increasing σ. This conclusion is drawn from the observations of the purifying effect ϕ=
Figure 1. Schematic illustration of poisoning and purifying effects provoked by increasing σ with time (Rσ > 0). (A) Sequence of cartoons representing the densities of free kink sites (green shadows) and adsorbed impurities (red circles) as σ increases. Red circles are predominantly located at the edges of the growing layers to symbolize highly active impurities. The top sequence represents the surface of a crystal growing at constant supersaturation levels σ1, σ2, and σ3. The middle sequence represents the poisoning effect resulting from delayed desorption of impurities from kink sites. The bottom sequence represents the purifying effect resulting from delayed adsorption of impurities onto newly formed kink sites. In both cases, a sufficiently fast σ increase does not allow adsorption equilibrium to be restored. (B) Schematic crystal growth curves in the absence (dashed line) and in the presence (solid lines) of impurities. Solid lines correspond to the three sequences of cartoons represented in part A.
when σ decreases with time, while increasing σ leads to crystal growth inhibition (Figure 2B and C). Lysozyme dimers and other protein aggregates are expected to be the dominant impurity during our crystal growth experiments.26−28 Advanced microscopic techniques have recently confirmed the growth suppressor effect of covalently bonded dimers during lysozyme crystallization.29,30 SDS-PAGE analysis of low protein concentration, low ionic-strength EWL solutions confirmed the presence of 0) the temperature of the EWL solutions. Long heating and cooling periods (i.e., low RΔT values) did not 41
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Figure 2. Crystal growth kinetics of the {101} faces of EWL during heating (red circles) and cooling periods (blue squares) at supercooling changing rates RΔT of (A) ∓2.00 °C × h−1 and (B, C) ∓ 2.67 °C × h−1. Top diagrams: normal growth length ΔL as a function of supercooling ΔT and time; arrows indicate the sequence of steps. Bottom diagrams: growth rate measurements at constant ΔT (purple triangles, partial results) fitted by a power-law equation (purple solid line; Ge-curve) for a 95% confidence interval (purple dashed lines); black lines represent G-curves determined by curve fitting of a power-law equation to the experimental ΔL variations (symbols are used only to identify each line)see the Supporting Information for details. (A) Slow heating/cooling rates did not provoke significant variations relatively to Ge. (B, C) Fast heating/cooling rates induced the purifying effect during heating periods and the poisoning effect during cooling periods.
that the σ-changing steps are taken consecutively. In these cases, no growth rate hysteresis will take place. A number of practical and fundamental implications arise from the development of the CAM here proposed: (i) The supersaturation changing rate emerges as an important process variable that can be used to promote or inhibit crystal growth. (ii) According to the value of parameter ϕ, precise guidelines are given in order to induce the purifying or poisoning effects. On the basis of the dynamics of active sites formation and impurity adsorption, insight is provided into (iii) type-3 G-curves (in which growth rate values G reach those obtained for the pure case G0), (iv) nonlinear acceleration kinetics of crystal growth, and (v) growth rate hysteresis. All the conclusions were validated against crystal growth rate data taken from the literature and/or measured by us during the crystallization of EWL.
induce significant variations of G relatively to constant temperature growth rates Ge in the temperature range 18.0−22.0 °C (e.g., Figure 2A). Although distinct kinetic constants were obtained, all the G-curves determined at supercooling changing rates |RΔT| of 0.67, 1.14, and 1.33 °C h−1 were within the 95% confidence interval of the Ge-curve. The observed crystal-to-crystal variability of kinetic coefficients is likely to be explained by the constant crystal growth model of growth rate dispersion,31 since invariable G-curves were obtained during the heating and cooling periods of the same crystal. Growth rate dispersion is also the likely cause for the different magnitudes of growth promoting/inhibiting effects observed in Figure 2B and C. As predicted by eq 7, as |RΔT| increased, the transient values of β became farther from the steady-state value βe. This is illustrated by the occurrence of the purifying and poisoning effects at RΔT ≥ 2.00 °C h−1 (Figure 2B and C): during the heating period G was significantly higher than Ge, while during the cooling period crystals grew very slowly (Figure 2B) or even ceased to grow (Figure 2C). Some variability of results was identified at RΔT = 2.00 °C h−1, with growth promoting/ inhibiting effects being either present or absent depending on the growth rate history of each crystal (see Supporting Information). For this reason, all the crystal growth measurements were preceded by a preliminary 80-min stabilization period at 18 °C. Different crystal growth rates when ΔT (and σ) is increasing or decreasing is a phenomenon of hysteresis originally described by the group of Kubota.24,25 Identified limitations of the existing theories in explaining growth rate hysteresis24 can be overcome by the CAM. In fact, distinct transient values of β are expected from eq 7 for positive and negative values of Rσϕ. The differences tend to be more marked for |Rσϕ| values close to 1. Very high |Rσϕ| values are expected to originate equivalent growth promoting and inhibiting effects, provided
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ASSOCIATED CONTENT S Supporting Information * Further experimental details and results. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION Corresponding Author *ICBASInstituto de Ciências Biomédicas Abel Salazar Universidade do Porto, Largo Prof. Abel Salazar, 2, 4099-003 Porto, Portugal. E-Mail:
[email protected]. Telephone. (+351) 222062287. Fax: (+351) 222062232. Present Addresses † LEPAE, Departamento de Engenharia Química, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal. 42
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‡
(30) Yoshizaki, I.; Kadowaki, A.; Iimura, Y.; Igarashi, N.; Yoda, S.; Komatsu, H. J. Synchrotron Radiat. 2004, 11, 30. (31) Judge, R. A.; Forsythe, E. L.; Pusey, M. L. Cryst. Growth Des. 2010, 10, 3164.
ICBASInstituto de Ciências Biomédicas Abel Salazar, Universidade do Porto, Largo Prof. Abel Salazar, 2, 4099-003 Porto, Portugal. § IBMCInstituto de Biologia Molecular e Celular, Rua do Campo Alegre, 823, 4150-180 Porto, Portugal.
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ACKNOWLEDGMENTS We acknowledge the cooperation of F. Silva (UP3, IBMC) during the analytical analysis of protein solutions. C.F. gratefully acknowledges Grant No. SFRH/BD/74174/2010 from Fundaçaõ para a Ciência e a Tecnologia (FCT), Portugal. This work was supported by Project No. PTDC/BIA-PRO/ 101260/2008 from FCT, Portugal.
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