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Mass Transfer Limitations at Crystallizing Interfaces in an Atomic Force Microscopy Fluid Cell: A Finite Element Analysis David Gasperino, Andrew Yeckel, Brian K. Olmsted, Michael D. Ward, and Jeffrey J. Derby* Department of Chemical Engineering and Materials Science, UniVersity of Minnesota, Minneapolis, Minnesota 55455-0132 ReceiVed March 3, 2006. In Final Form: May 12, 2006 Although atomic force microscopy (AFM) has emerged as the preeminent experimental tool for real-time in situ measurements of crystal growth processes in solution, relatively little is known about the mass transfer limitations that may impact these measurements. We present a continuum analysis of flow and mass transfer in an atomic force microscope fluid cell during crystal growth, using data acquired from calcium oxalate monohydrate (COM) crystal growth measurements as a comparison. Steady-state flows and solute concentration fields are computed using a three-dimensional, finite element method implemented on a parallel supercomputer. Steady-state flow results are compared with flow visualization experiments to validate the model. Computations of the flow field demonstrate how nonlinear momentum transport alters the spatial structure of the flow with increasing flow volume, altering mass transport conditions near the AFM cantilever and tip. The simulations demonstrate that the combination of solute depletion from crystal growth and mass transfer resistance lowers the solute concentration in the region between the tip and the crystal compared with the solute concentration at the inlet of the AFM cell. For example, using experimentally measured growth rates for COM, the solute concentration in this region is 3.1% lower than the inlet value because the solute consumed by crystal growth beneath the AFM tip cannot be replenished fully due to mass transport limitations. The simulations also reveal that increasing the flow rate through the cell does not affect this difference significantly because of the inherent shielding by the AFM tip in proximity with the crystal surface. Models such as the one presented here, used in conjunction with AFM measurements, promise more precise interpretations of measurement data.
1. Introduction Crystal growth from liquid solutions is employed for the production of a variety of materials, from bulk crystallization of organic molecules1,2 to the growth of large, single crystals of inorganic, nonlinear optical materials.3,4 Despite the long history of liquid-phase crystal growth, however, much remains poorly understood, especially the fundamental factors influencing growth rate, shape, size, purity, and perfection. Toward this goal, atomic force microscopy (AFM) has become a preeminent tool for realtime in situ visualization of the microscopic events associated with crystal growth of various organic and inorganic crystals,5-18 * Corresponding author. Fax: +1-612-626-7246. E-mail:
[email protected]. (1) Davey, R.; Garside, J. From Molecules to Crystallizers: An Introduction to Crystallization; Oxford University Press: Oxford, U.K., 2000. (2) Myerson, A., Ed.; The Handbook of Industrial Crystallization, 2nd ed.; Butterworth-Heinemann: Stoneham, MA, 2001. (3) Elwell, D.; Scheel, H. J. Crystal Growth from High-Temperature Solutions; Academic Press: London, U.K., 1975. (4) DeYoreo, J. J.; Burnham, A.; Whitman, P. Int. Mater. ReV. 2002, 13, 233. (5) Jung, T.; Sheng, X.; Choi, C. K.; Kim, W.-S.; Wesson, J. A.; Ward, M. D. Langmuir 2004, 20, 8587. (6) Hillner, P. E.; Manne, S.; Gratz, A. J.; Hansma, P. K. Ultramicroscopy 1992, 2, 1387-1393. (7) DeYoreo, J. J.; Land, T.; Dair, B. Phys. ReV. Lett. 1994, 73, 838. (8) DeYoreo, J J.; Rek, Z.; Zaitseva, N.; Woods, B. J. Cryst. Growth 1996, 166, 291. (9) Land, T.; DeYoreo, J. J.; Lee, J. Surf. Sci. 1997, 136-155. (10) Palmore, G.; Luo, T.; McBride, M.; Voong, N.; DeYoreo, J. J. Trans. ACA 1998, 33, 45. (11) Yau, S.; Petsev, D.; Thomas, B.; Vekilov, P. J. Mol. Biol. 2000, 303, 667-678. (12) A. Hillier, P. C.; Ward, M. J. Am. Chem. Soc. 1994, 116, 944. (13) Hillier, A.; Ward, M. Science 1994, 268, 1261. (14) Last, J. A.; Hillier, A. C.; Hooks, D. E.; Maxson, J. B.; Ward, M. D. Chem. Mater. 1998, 10, 422. (15) Yip, C.; Ward, M. Biophys. J. 1996, 71, 1071. (16) Yip, C. M.; Brader, M. L.; DeFelippis, M. R.; Ward, M. D. Biophys. J. 1998, 75, 1172. (17) Ward, M. D. Chem. ReV. 2001, 101, 1697. (18) Guo, S.; Ward, M.; Wesson, J. Langmuir 2002, 18, 4284.
including imaging of terraces, ledges, and kinks and the measurement of kink and step velocities. Such measurements permit estimates of the thermodynamic properties of crystal surfaces and the kinetics of growth.19,20 Despite the direct visualization of crystal growth enabled by in situ AFM, there are currently no independent means of determining the flow conditions present during AFM measurements of solution crystal growth. There exists a need to improve the understanding of limitations associated with mass transport of solute molecules to the crystal surface. Such limitations have been identified clearly in previous AFM studies of crystallizing systems. For example, Jung et al.5 used an AFM fluid cell to probe crystallization of calcium oxalate monohydrate (COM) and the role of macromolecule additives in COM growth inhibition. COM is a major component of kidney stones, and much work has been performed to understand its growth18,21-24 and growth inhibition.5,25,26,27 Jung et al.5 found that reliable measurement of step velocities required that flow through the cell be high enough to avoid mass transfer limitations on growth. This system, shown along with a scanning electron microscope (SEM) image the COM (010) face in Figure 1, will be analyzed here. (19) Malkin, A. J.; Kuznetsov, Y. G.; Lucas, R. W.; McPherson, A. J. Struct. Biol. 1999, 127, 35. (20) Gliko, O.; Reviakine, I.; Vekilov, P. G. Phys. ReV. Lett. 2003, 90, 225503. (21) Gvozdev, N.; Petrova, E.; Chernevich, T.; Shustin, O.; Rashkovich, L. J. Cryst. Growth 2004, 261, 539-548. (22) Deganello, S.; Piro, O. Neues Jahrb Mineral Monatsh 1981, 2, 81. (23) Nancollas, G.; Gardner, G. Journal of Cryst. Growth 1974, 21, 267-276. (24) Zauner, R.; Jones, A. G. Chem. Eng. Sci. 2000, 55, 4219-4232. (25) Antinozzi, P.; Brown, C.; Punch, D. J. Cryst. Growth 1992, 125, 215222. (26) Shirane, Y.; Kurokawa, Y.; Miyashita, S.; Komatsu, H.; Kagawa, S. Urol. Res. 1999, 27, 426-431. (27) Ward, M. D.; Sheng, X.; Jung, T.; Wesson, J. A. Polym. Mater. Sci. Eng. 2004, 90, 273.
10.1021/la060592k CCC: $33.50 © 2006 American Chemical Society Published on Web 06/22/2006
Mass Transfer Limitations at Crystallizing Interfaces
Figure 1. (a) Photograph of a Digital Instruments AFM fluid cell. (b) SEM image of a COM crystal viewed nearly normal to the (010) face (from ref 5). (c) Computational domain of the AFM fluid cell. During actual experiments the components illustrated in panels a and c are flipped upside down with respect to these images, with the gasket forming a seal over a glass slide on which the crystals grow and subsequently are scanned.
Computational models can complement experimental observations by providing a detailed picture of flows and mass transfer in the AFM system, particularly near the measurement tip. Coles et al.28,29,30 described a two-dimensional simulation based on a customized AFM cell configuration that aided the interpretation of kinetic factors inferred from the AFM measurements of reacting surfaces. Wilkins et al.31 conducted three-dimensional simulations for a simplified geometry representing a portion of the Coles AFM cell, enabling accurate predictions over a much wider range of flow rates. These reports suggest that a more detailed understanding of flow and mass transfer would be beneficial for unambiguous interpretation of real-time in situ AFM measurements of crystal growth. We describe here finite element analyses of the continuum transport processes (fluid dynamics and mass transfer) that occur during experimental measurements of crystal growth performed in an AFM fluid cell. In conjunction with AFM experiments, these analyses provide a quantitative model that can lead to more precise interpretations of measurement data and self-consistent evaluation of fundamental properties, such as kinetic rate constants.
2. Model Formulation The overall goal of this work is the development of a modeling capability that allows simulation of fluid flow and mass transfer (28) Coles, B.; Compton, R.; Booth, J.; Hong, Q.; Sanders, G. Chem. Commun. 1997, 619-620. (29) Hong, Q.; Suarez, M.; Coles, B.; Compton, R. J. Phys. Chem. B 1997, 101, 5557-5564. (30) Coles, B.; Compton, R.; Suarez, M.; Booth, J.; Hong, Q.; Sanders, G. Langmuir 1998, 14, 218-225. (31) Wilkins, S.; Suarez, M.; Hong, Q.; Coles, B.; Compton, R. J. Phys. Chem. B 2000, 104, 1539-1545.
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during AFM measurements of crystal growth in solution. To evaluate the simulations under relevant conditions, the model is applied specifically to the growth of COM crystals, for which the rates of crystal growth on specific crystal faces have been measured using AFM.5 Specifically, we formulate a finite element model to describe incompressible flow and solute transport occurring during measurements in a Digital Instruments (DI) AFM fluid cell (part #150-000-002). The mathematical domain for the model is defined by the geometry of the AFM cell, wafer, cantilever, and tip (Figure 1), using specifications from DI and scanning electron microscopy (SEM). The SEM measurements of the cantilever and tip were accurate to 0.1 µm (Supporting Information, Figure S1). AFM experiments typically are performed with the cantilever situated above the crystal being scanned, with the AFM tip pointing downward, and the inlet and outlet ports on the top of the cell. However, the computational and experimental flow visualizations here employ an inverted view (i.e., the cell is upside-down) to better visualize the flows and mass transfer throughout the AFM cell. 2.1. Governing Equations. To construct a mathematical model for this system, we assume that (a) the fluid flowing through the cell is incompressible, which is a reasonable approximation for liquids, (b) the system is isothermal (at 25 °C), so that buoyant flow effects can be ignored, (c) the transport properties are independent of temperature and composition, (d) the scan rate is slow and its amplitude small compared to the fluid flow through the system and the dimensions of the system, allowing stage movement effects to be ignored, and (e) the mass transfer can be described by the convection and diffusion of a dilute single species (the solute) dissolved in water (the solvent). Admittedly, mass transfer within the actual COM solutions is more complicated as the solute comprises both associated and disassociated calcium and oxalate ions that interact with water molecules. The contributions of these species to mass transport and their role in crystal growth is not understood sufficiently for rigorous treatment in the model, and their detailed representation would involve extraneous model complications. We also invoke a quasi-steady-state assumption for flow and mass transfer. The characteristic time for convective transport is L/U ∼ O(0.1 s), where L is the thickness of the fluid cell and U is the centerline velocity at the fluid cell inlet. Similarly, the time scale for diffusive transport near the crystal is Lc2/D ∼ O(1 s), where Lc is the length of a growing crystal and D is the diffusion coefficient for the solute. Whereas the characteristic time for COM growth is on the order of 1 h,5 the time scale for continuum transport is virtually instantaneous compared with that for crystal growth, justifying the use of the steady-state form of the governing equations. The flow of liquid through the AFM cell is governed by momentum conservation and continuity, as described by the steady-state, Navier-Stokes equations for an incompressible fluid. Choosing the AFM fluid cell as a stationary frame of reference, we write the following equations in dimensionless vector form as
Re(v‚∇v) ) ∇‚(∇v + ∇vT) - ∇p
(1)
∇‚v ) 0
(2)
where the dimensionless quantity Re is the Reynolds number, defined as Re ≡ FUL/µ, where U and L are defined as above, F and µ are the density and viscosity, respectively, “‚” represents the dot product, ∇ denotes the gradient operator, and the superscript T indicates a transpose operation. The dimensionless velocity, v, is a vector field consisting of three components, one
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in each coordinate direction, defined as Vi ≡ V˜ i/U, where the subscript i denotes a coordinate direction and the tilde denotes a dimensional quantity. The dimensionless variable p denotes the dynamic pressure (in which the hydrostatic pressure field has been lumped), nondimensionalized by p ≡ p˜ /(µU/L). The nondimensional Reynolds number, which characterizes the ratio of inertial forces to viscous forces, is illustrative of the physical nature of the flow. Flows with small Reynolds numbers are dominated by viscous effects and are laminar. As the Reynolds number grows larger, nonlinear inertial effects become important, making the flow more complicated. Turbulent flows are typically exhibited at very high values of Re. Appropriate boundary conditions are specified for the NavierStokes equations along all surfaces of the computational domain. A parabolic flow is specified at the cross-section of the tube leading to the fluid cell chamber, consistent with a well-developed, laminar flow through the inlet. At the cross section of the fluid cell outlet, an outflow condition is applied
n‚∇v ) 0
(3)
which represents a well-developed flow downstream of the computational domain.32 No-slip boundary conditions
n‚v ) 0
(4)
are set at all other boundaries of the fluid cell model domain. Conservation of the solute species is governed by the steadystate convective-diffusion equation, written in dimensionless form as
Pe(v‚∇σ) ) ∇2σ
(5)
where Pe is the dimensionless Peclet number, which represents the ratio of mass transfer via convection to that by diffusion. The Peclet number is defined as Pe ≡ UL/D, v represents the dimensionless velocity vector, and σ is a dimensionless concentration, defined as the relative supersaturation
σ)
C - Ceq Ceq
(6)
Here C is equal to the concentration of the solute and Ceq is the equilibrium value of the solute concentration in a saturated solution. Strictly speaking, the driving force for crystallization of a solute from a solution is the difference in the chemical potential of solute in the supersaturated solution to that of the solute in the saturated solution (which is in equilibrium with the solid crystal). This thermodynamic driving force, termed the supersaturation and denoted here as σT, is defined as σT ) ∆µ/kT ) ln(a/aeq), where ∆µ is the difference in the chemical potential described above, k is Boltzmann’s constant, T is the absolute temperature, and a and aeq are the activities of the solute in the supersaturated and saturated solutions, respectively.33 For the case of near-unity activity coefficients, σT ≈ ln(C/Ceq), and using eq 6 and a Taylor series expansion, we obtain
1 1 σT ≈ ln(σ + 1) ≈ σ - σ2 + σ3 - ... 2 3
(7)
For a relative supersaturation σ much less than one, eq 7 reduces to σT ≈ σ and the thermodynamic supersaturation is nearly equal to the relative supersaturation. Note that the effect of temperature (32) de Santos, J. Ph.D. Thesis, University of Minnesota: Minneapolis, MN, 1991. (33) Mohana, R.; Myerson, A. Chem. Eng. Sci. 2002, 57, 4277-4285.
on relative supersaturation is accounted for in the value of Ceq employed in eq 6. The model presented here assumes that the temperature does not change during the conduct of the AFM experiments, so that Ceq is constant. Returning to the transport equation for solute in the system, eq 5, suitable boundary conditions must be supplied. A constant value of the relative supersaturation, σ0, is set at the inlet tube cross section. Along all other surfaces, the condition
n‚∇σ ) 0
(8)
is applied, representing a no-flux condition for the solute. The same mathematical condition, applied at the flow outlet, describes a well-developed concentration field exiting the computational domain. The growing crystals are defined as two-dimensional lozenges on the surface opposing the AFM tip. In an experiment in the AFM cell, the stochastic nature of heterogeneous nucleation leads to some variation in position and size of the growing crystals. To faithfully depict the effects of these growing crystals, we pattern our model-crystal surface patches with a uniform, but representative, size and spacing across regions of the plate where they are observed to grow during the experiments. The dimensionless flux boundary condition applied to these surface patches is
n·∇σ )
Fcβσ FDC/eq
(1 - C/eq(σ + 1))
(9)
where the left-hand side represents the flux of the crystallizing solute to the surface of the crystal, with n as a unit vector defined normal to the crystal surface, β as the normal growth kinetic coefficient, and C/eq as a mass fraction of the solute.34,35 2.2. Implementation. The governing equations and associated boundary conditions are solved approximately using the finite element method.36 The Galerkin least squares method is applied to the Navier-Stokes equations,37 and the streamline-upwind Petrov-Galerkin method is applied to the solute conservation equation.38 Four-noded, tetrahedral elements with linear basis functions are used to represent all field variables. The resulting nonlinear set of algebraic equations is solved iteratively using Newton’s method. The algorithm is implemented in a parallel code that employs MPI communications constructs.39,40 At each Newton step, the linear system of equations is solved using the SPOOLES direct solver package accessed through the PETSc scientific computing library.41 Newton iterations were deemed convergent when the L2 norm of the residuals vector was reduced below 10-8. Typically, four Newton iterations were required for solution convergence. (34) Vartak, B.; Kwon, Y.; Yeckel, A.; Derby, J. J. Cryst. Growth 1999, 210, 704-718. (35) Vartak, B. Three-Dimensional modeling of solution crystal growth Via the finite element method, Thesis, University of Minnesota: Minneapolis, MN, 2001. (36) Gresho, P.; Sani, R. Incompressible Flow and the Finite Element Method; John Wiley and Sons: West Sussex, U.K., 1998. (37) Hughes, T. J. R.; Franca, L. P.; Hulbert, G. M. Comput. Methods Appl. Mech. Eng. 1989, 73, 173-189. (38) Hughes, T. J. R.; Brooks, A. Comput. Methods Appl. Mech. Eng. 1982, 32, 199-259. (39) Salinger, A. G.; Xiao, Q.; Zhou, Y.; Derby, J. J. Comput. Methods Appl. Mech. Eng. 1994, 119, 139-156. (40) Zhou, H.; Derby, J. Parallel Implementation of Finite Element Method with MPI: Application to Three-Dimensional Free Surface Stokes Flow. In; 2000 International Conference on Parallel and Distributed Processing Techniques and Applications: Las Vegas, NV, 2000. (41) Balay, S.; Eijkhout, V.; Gropp, W. D.; McInnes, L. C.; Smith, B. F. Efficient Management of Parallelism in Object Oriented Numerical Software Libraries; Birkhauser Press: Boston, 1997.
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Figure 2. (a) View of the inside of the AFM fluid cell; only surface mesh elements are shown. (b) View of the large cantilever mesh. (c) Crystal boundary mesh, used for mass transfer simulations.
The meshes employed here are depicted in Figure 2. The finer mesh was employed for the mass transfer computations and consists of 250 517 tetrahedral elements with a total of 285 145 degrees of freedom. This mesh was carefully constructed and tested for accuracy, especially in regions near the cantilever (Figure 2b) and along the top surface of each crystal (Figure 2c). The two-dimensional crystals in the simulation cover 5% of the area of the top surface of the cell and are constructed with an arbitrary morphology, here depicted as a six-sided lozenge. This morphology is not intended to represent any particular real crystal habit or orientation. For example, it should not be confused with the similar shape of the (100) face of COM that is sometimes observed.42 It also should not be construed as a faithful portrayal of the actual COM (010) face, which has a distinctly different morphology, albeit with six sides. Rather it is meant simply to provide a representative surface upon which to apply solute flux conditions describing growth. Solutions of this model required approximately 5.4 min of CPU time per processor per Newton iteration on the IBM SP at the University of Minnesota Supercomputing Institute.
3. Model Verification and Validation For any computational model, strict verification and validation exercises are important to build confidence in its predictions. Verification addresses the question of whether our computer code is accurately solving the mathematical equations posed by the model. Validation addresses the question of whether our mathematical model is faithfully representing the behavior of the system of interest. This is typically accomplished via the comparison of model predictions to experimental measurements. An accurate model also depends on reliable values of the relevant physical properties and system parameters used in the governing equations. These values for COM in an aqueous solution at 25 °C are provided in Table 1. The macroscopic kinetic coefficient for growth, β, was estimated from the data obtained during our previous measurements of COM crystal growth.5 Step speed, height, and lattice spacings for the (010) COM face were used to compute the normal growth rate of the crystal, Vf, which is of the same order of magnitude as the normal growth rate found experimentally by others.21 This value was then used to compute an average flux (42) Qiu, S.; Wierzbicki, A.; Orme, C.; Cody, A.; Hoyer, J.; Nancollas, G.; Zepeda, S.; Yoreo, J. J. Proc. Natl. Acad. Sci. U.S.A. 2004, 101, 1811-1815.
Table 1. Physical Properties and System Parameters Used to Model the AFM Fluid Cell System symbol
value
units
description
C/eq
1.4 × 10-4
M (mol/L)
D
8.78 × 10-6
cm2/s
L Lc
0.06367 0.0032
cm cm
U
2.96
cm/s
β
1.55 × 10-8
cm/s
µ F Fc σ0
1.0 × 10-2 0.997 2.2 0.42857
g/cm‚s g/cm3 g/cm3
concentration of COM in a saturated aqueous solution diffusion coefficient for solute, estimated using the Nernst equation for dilute electrolyte solutions fluid cell thickness assumed length of the (010) face of the COM crystal inlet centerline velocity; varied in computations; base case value used in experiment corresponds to Re ) 18.8 kinetic coefficient; see text for evaluation viscosity of solution density of solution density of crystal inlet supersaturation used in ref 5
of solute to the crystal via a mass balance.43,35 Accordingly, the boundary condition described by eq 9 was replaced with a uniform mass flux condition, and a complete finite element computation of system flow and mass transfer was performed for the experimental flow rate. From this simulation, the average supersaturation level along the top face of the crystal being scanned, σavg, was computed, and the kinetic coefficient was determined as
β ≡ σavg/Vf
(10)
This value of β, reported in Table 1, was employed in all subsequent mass transfer computations. 3.1. Verification. Our finite element code has been verified for a number of known solutions to the Navier-Stokes equations.39 We also have performed careful mesh refinement studies to assess and control numerical error for the simulations, which is important for the application of the finite element method to systems with complex geometries and high Reynolds or Peclet numbers. (43) Westphal, G. H.; Rosenberger, F. J. Cryst. Growth 1978, 43, 687-693.
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Figure 3. Snapshots of pathline trajectories in the DI AFM fluid cell during flow visualization experiments (on the left) and computational simulations (on the right) through the global domain (a and b) and near the leading edge of the wafer (c and d). The flowrate was 9 × 10-3 mL/s. Flow is traveling from bottom to top in all images.
A series of computations were performed on successively finer meshes using the properties listed in Table 1. A preliminary study in which only the flow equations were solved established that a mesh of 176,407 elements produced flow solutions of sufficient accuracy. Accurate solution of the associated mass transfer characteristics, however, proved to be more challenging because the Peclet number for mass transfer is quite large, Pe ∼ O(104). Global mass conservation was assessed by numerical integration of the solute flux over the inlet, outlet and all crystal boundaries. A series of six meshes was assessed, and the final mesh of 250,517 elements and 285,145 degrees of freedom (Figure 2) resulted in a global mass flux error of less than 0.2%. This was deemed sufficiently accurate and was used for all computations involving mass transport. 3.2. Validation. To validate our model, we conducted flow visualization experiments in the AFM cell used to construct the computational domain and employed in the previous experimental work.5 Flow visualization has been used successfully for over a century to characterize flow patterns in a wide variety of systems.44,45 Here, we use a particle-path tracing technique46 that relies on video microscopy of light reflected from the surfaces of rheoscopic tracer particles. To minimize discrepancy between experiment and model, care was taken to avoid structural irregularities on the upstream side of the scanning wafer that sometimes appeared when it was sheared from its mother wafer (see the Supporting Information, Figure S1). The experimental flow visualizations were performed in the DI AFM fluid cell, mounted in a custom polystyrene fixture affixed with a glass window located at the imaging substrate. Frames were acquired with an Olympus 57H10 Stereo microscope equipped with an Insight QE model #4.2 digital camera and (44) Merzkirch, W. Flow Visualization; Academic Press: Orlando, FL, 1987. (45) Yang, W. Handbook of Flow Visualization; Taylor and Francis: New York, 2001. (46) Dyke, M. V. An Album of Fluid Motion; The Parabolic Press: Stanford, CA, 1982.
Figure 4. Global pathline portraits of the computed flow in the AFM fluid cell, with flow traversing from bottom to top in each image. Pathlines are colored according to velocity (cm/s) for Reynolds number equal to (a) 18.8 and (b) 200.
Insight Spot software. The concentration of the tracer particles (AQ-1000 concentrate by Kalliroscope Corporation) was varied from 1% to 5%. The AQ-1000 concentrate is a dilute suspension of highly anisotropic polymeric flakes that align along streamlines,
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Figure 5. (a and d) Velocity magnitude from the computational simulations are projected on a clip plane of the global domain of the fluid cell that intersects the larger of the two AFM cantilevers. (b and e) Pathline portraits of flow past the sample-scanning side of the large AFM cantilever. (c and f) Pathline portraits of flow past the scanning tip attached to the large cantilever. Pathlines for the bottom two visualizations are colored by the same velocity scale. Image groups are for Reynolds number equal to (a,b,c) 18.8 and (d,e,f) 200.
reflecting light according to their orientation. The liquid cell was configured with both the long (200 µm) and short (120 µm), wide-legged (36 µm width) V-shaped Si3N4 cantilevers positioned as close to the viewing window as possible without incurring contact. Flow was produced by either gravity feed or a Cole Parmer MasterFlex C/L peristaltic pump. The camera shutter speed and the location of bifurcate light sources were adjusted to optimize imaging of real-time flow trajectories. Figure 3 shows pathline portraits through the experimental (left) and computational (right) DI AFM fluid cell. The flow rate is 9 × 10-3 mL/s, corresponding to a Reynolds number of Re ) 14.35 (and therefore laminar flow, not turbulent), with flow traversing the figure from bottom to top. The streaks trace the pathlines caused by light reflected from the rheoscopic tracer particles. Figure 3a,b depicts pathlines for flow at the global level of the system. These flows enter the cell through the hole at the bottom of the figure and exit through the hole at the top. The pathlines around the lip of both holes have similar orientation. Following the trajectories from the entrance to the cell, pathlines trace the same arc around the entire front edge of the wafer in
both the computational and experimental images, showing very good agreement. Figure 3c,d displays local flows around the front edge of the wafer near the cantilevers, where imaging occurs and resolution of the flow lines is especially important. The trajectories of the pathlines entering the bottom right of each image are similar, as are the pathline profiles around the cantilevers. This agreement between experiment and model at both the global level and near the AFM cantilevers validate the model.
4. Results and Discussion 4.1. Fluid Flow. Our previous measurement of COM crystallization rates5 was performed using a flow rate corresponding to a Reynolds number of Re ) 18.8. Figure 4 depicts the pathline portraits of steady flows inside the AFM fluid cell under this condition and for a flow rate more than 1 order of magnitude greater with Re ) 200. The value of the flow speed is indicated along each pathline by color. Both visualizations are constructed with pathlines having common origins, entering through the inlet at the bottom and exiting the outlet at the top.
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Figure 6. Visualizations of the computed supersaturation, σ, in the fluid cell for Reynolds number equal to (a) 0.02, (b) 0.2, (c) 18.8, and (d) 100, looking into the cell (in the negative z direction). The top surface has been made transparent, and the two-dimensional crystal surfaces are surrounded by yellow and green depletion fields. The cell inlet is located in the bottom right of each image, and pathlines are shown in white. The white arrow points to the large cantilever.
Several notable characteristics of the global flow profiles are evident. The flow at Re ) 18.8 is much more viscous in nature than the flow at Re ) 200, with pathlines that are uniformly spaced and aligned smoothly with the system geometry. The faster flows at Re ) 200 are affected more strongly by inertia. The pathlines do not spread evenly across the cell; instead, there are areas of “dead space” that are passed by the surrounding flow. The system geometry also has longer-range and nonlinear effects on the flow. For example, upon encountering the cantilever wafer after entering the cell, the pathlines for the Re ) 200 flow rebound away from the wafer (see the area near the lower-left corner of the wafer). This occurs because of the inertia of the faster flow, in contrast to the smoother changes of direction exhibited by the slower flow. The net result is that the flow changes in speed and orientation with respect to the AFM cantilever and tip with changes in overall flow rate. Flows near the wafer, cantilever, and tip are visualized in Figure 5 under the same conditions as those shown in Figure 4. The flow field is well resolved even at the much smaller scale associated with the AFM tip, shown in Figure 5c,f. Figure 5a,d illustrates the speed of the flow through a clip plane that cuts through the wafer and the center of the large cantilever for Re ) 18.8 and 200, respectively. The flow speed dramatically decreases in the gap between the wafer and the top plate, especially near the cantilever. Also, the faster portion of the flow within the outer gap (in red, to the left of the wafer) moves away from the wafer with increasing Reynolds number. As depicted in Figure 5b,e, for Re ) 18.8, the flow approaches the tip of the cantilever from the base of the cantilever toward the tip (from lower right to upper left in the figure), whereas for Re ) 200, the flow is nearly parallel with the base as it approaches the tip of the cantilever. These changes in the flow are also evident in Figure 5c,f, where pathlines arch differently around the AFM tip. 4.2. Solute Transfer. The impact of flow rate and crystal growth on the relative supersaturation field is assessed via coupled computations of flow and mass transfer for different rates of liquid flow through the system, as expressed by the dimensionless
Reynolds number, Re. Visualizations in Figure 6 are produced from the vantage point looking down into the AFM fluid cell, with the exterior walls of the cell having been made semitransparent, allowing one to see the inlet, wafer and cantilevers through the top surface of the cell. The surfaces inside the fluid cell, including the top surface of the fluid cell and crystals lying on this surface, are color coded according to the value of σ. In addition, a clip plane parallel to the top surface is created, lying just underneath this surface. Pathlines are calculated on this clip plane, with the length of each pathline indicating the magnitude of the velocity at that point scaled relative to a common factor. In this view, the inlet appears in the bottom part of the image and the flow sweeps upward and across the cell. Depictions in Figure 6 show the concentration and velocity profiles in the computational domain for several flow rates ranging from Re ) 0.02 to 100. Immediately evident are solute-depleted regions that surround each crystal. These regions appear as green and yellow spots centered on each crystal, indicating lower values of σ. The depleted solute concentration in the vicinity of the crystal reflects the inability of diffusive and convective mass transport to replenish the solute consumed by crystal growth. As the flow rate increases, however, convection is better able to replenish the solute at the active crystal growth surfaces, to the extent that the depletion regions become less prominent, as depicted in Figure 6d (Re ) 100). Solute depletion due to crystal growth that is not compensated fully by mass transport also is evident in the region between the wafer and the top surface of the AFM cell, especially at the slow flows corresponding to Re ) 0.02 and 0.2 (Figure 6). Flow of liquid is diminished between the wafer and the top surface, especially as the gap narrows toward the cantilever mounting edge (Figure 5a,d). Notable in all cases shown in Figure 6 are prominent depletion regions around the crystals mounted above the AFM cantilevers (these crystals are located at the end of the white arrows in each figure). These features signal that the reduced convective mass transport cannot replenish the solute at a rate that compensates for the rate of solute consumed by crystal growth
Mass Transfer Limitations at Crystallizing Interfaces
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Figure 7. Visualizations of the supersaturation predicted by the computational model, σ, along the plane containing the growing crystals, below the large cantilever, for Reynolds number equal to (a) 0.02, (b) 0.2, (c) 18.8, and (d) 100. The crystal (outlined in black) is located directly under the large cantilever (semi-transparent). Pathlines (colored white) lying in a plane between the cantilever and the crystal surface are included. Flow is from right to left in all images.
beneath the AFM cantilever and tip, where images are collected. This effect is more pronounced in this region than in other parts of the system. The behavior of the depletion regions below the tip and cantilever is illustrated further in Figure 7, which presents the supersaturation field on the plane containing the crystal surface, viewed through a semi-transparent cantilever. The pathlines in this figure (colored white) lie in the same clip plane as those in Figure 6, with flow in the cell moving right to left. The depletion region beneath the cantilever is apparent for all flow rates (Re ) 0.2-100), extending over nearly the entire crystal surface. As mentioned above, this effect is a consequence of mass transport limitations associated with shielding by the cantilever and tip, which reduces the rate of replenishment of solute that has been consumed by crystal growth. At slow flow rates, such as Re ) 0.02 in Figure 7a, the depletion layer around the crystal surface is almost symmetrical. As the flow rate increases, however, the depletion layer slightly shifts along the crystal surface in the direction of flow, resulting in a more solute-rich region near the upstream side of the crystal, as illustrated in Figure 7b-d. The effect of the competition between solute consumption by crystal growth and solute replenishment by convective mass transport and to a lesser extent, diffusive transport, is summarized in Figure 8. The inlet solute concentration, expressed in terms of the relative supersaturation of the feed solution, is independent of Re, as expected. Above an actively growing crystal surface located midway between the AFM cantilever and the cell outlet, that is, unobstructed by the cantilever, σ is reduced by approximately 3.2% at low Re values because the solute cannot be replenished at a rate that compensates for solute consumption due to crystal growth. However, as the flow rate is increased, σ above the unobstructed crystal surface increases, achieving a value that is within 1.3% of the inlet concentration at Re ) 100. In contrast, the crystal surface obstructed by the AFM tip exhibits a lower and nearly unchanging value of σ for the entire range of Re examined here, reflecting the mass transport shielding introduced by the wafer/cantilever/tip assembly. This shielding
Figure 8. Comparison of effect of Re on the relative supersaturation at selected points in the system: Dashed line without symbols indicates the inlet; circles indicate the center of an actively growing crystal surface unobstructed by the wafer/cantilever/tip assembly, and squares show values above an actively growing crystal surface in proximity with the AFM cantilever and tip. These computations are based on previously measured growth kinetics for COM. The unobstructed crystal surface is located midway between the cantilever and cell outlet. The reduced relative supersaturation above the unobstructed crystal and the obstructed crystal in the proximity of the AFM cantilever reflects the inability of mass transport to compensate fully for the consumption of the solute by crystal growth. This effect is more severe beneath the cantilever due to the shielding effect of fluid flows, which is evident in the simulations.
prevents any significant increases in convective mass transport to the crystal which normally would arise from larger volumetric flows. With little help from increased convection to replenish the consumption of solute by crystal growth, the level of supersaturation near the crystal remains relatively unaffected by increasing Re. At the Re ) 18.8 value used previously for AFM
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measurements of COM growth, the shielding reduces σ above the growing crystal surface by 3.1% compared with σ0 at the inlet. Although this is not a dramatic difference, it is reasonable to suggest that this modest decrease should be taken into account when using AFM measurements to derive quantities that depend on local supersaturation, such as kinetic factors. The essentially unchanging value of σ in the region between the crystal surface and the cantilever from Re ) 0.02-100 indicates that this effect cannot be removed substantially by simply increasing the flow rate.
5. Conclusions We have described herein a steady-state, three-dimensional finite element model for the detailed analysis of fluid flow and mass transfer in an AFM fluid cell. These simulations reveal significant interactions between the inherent nonlinearity of fluid flow and the complicated system geometry, resulting in changing conditions with different volumetric flows through the system. The DI cell configuration that serves as the computational domain in our simulation creates spatial heterogeneities in the transport of solute molecules to crystals growing in the cell. Overall, our simulations reveal that the spatial structure of the flow field in the system changes significantly with the volumetric flow rate and that the effectiveness of increasing the flow for reducing mass transfer limitations varies for different regions of the flow cell. Under all operating conditions much of the flow is directed away from measurement regions. The flow tends to channel more strongly through open parts of the system as the Reynolds number increases, and the wafer/cantilever/tip assembly acts as a shroud that shields nearby crystal surfaces from faster flows. Consequently, solute molecules consumed by crystal growth cannot be replenished completely by convective mass transport, thus lowering the relative supersaturation above the imaging region by several percent.
Gasperino et al.
The simulations indicate that mass transfer limitations generally will exist in the narrow gap between the AFM measurement apparatus (wafer and cantilever) and the surface to be scanned, and it is unlikely that flow alone will ever be sufficient to eliminate local mass transfer limitations, at least under practical volumetric flow rates. Although the reduction in the relative supersaturation for the system studied here (COM) is not overly large, the mass transfer limitations may prove more severe for crystalline materials that exhibit faster growth rates (such as inorganic crystals) or crystals composed of larger molecules that diffuse more slowly through the liquid phase (such as proteins). This argues for an approach that combines the modeling of flow and mass transfer in conjunction with AFM measurements to achieve self-consistent evaluation of fundamental properties such as kinetic rate constants. We also note that at higher flow rates inertial effects change the direction of the flow near the AFM cantilever, which could alter the orientation and magnitude of the flow-induced shear stress on the cantilever. We are currently examining the importance of these effects during imaging. Acknowledgment. This work was supported in part by the National Science Foundation, under Grants CTS-0121467 and CTS-0323696, the MRSEC Program under Award Number DMR0212302, and the University of Minnesota Supercomputing Institute. The authors thank Xiaoxia Sheng for significant input to this work. Supporting Information Available: Computer-aided design (CAD) depiction of the DI AFM fluid cell with component descriptions and scanning electron microscope (SEM) images of AFM wafer and tip with comparisons to the matching components in the computational model. This material is available free of charge via the Internet at http://pubs.acs.org. LA060592K
