J. Phys. Chem. B 2000, 104, 1539-1545
1539
Atomic Force Microscopy under Defined Hydrodynamic Conditions: Three-Dimensional Flow Calculations Applied to the Dissolution of Salicylic Acid Shelley J. Wilkins, Marco F. Sua´ rez, Qi Hong, Barry A. Coles, and Richard G. Compton* Physical and Theoretical Chemistry Laboratory, Oxford UniVersity, South Parks Road, Oxford OX1 3QZ, United Kingdom
George E. Tranter and David Firmin Physical Sciences, GlaxoWellcome Medicines Research Centre, Gunnels Wood Road, SteVenage, Hertfordshire SG1 2NY, United Kingdom ReceiVed: August 12, 1999; In Final Form: October 27, 1999
The kinetics of the dissolution of salicylic acid in aqueous solutions is studied using in situ atomic force microscopy, using a novel liquid flow cell for which the full three-dimensional flow pattern is known. This allows the interpretation of dissolution rate in terms of an interfacial reaction mechanism, with excellent agreement between the theoretically predicted and the experimental results. The use of a three-dimensional simulation to obtain the flow velocities enables accurate prediction over a much wider range of flow rates than is possible using a simpler two-dimensional model for the flow pattern. The dissolution of the (110) face of salicylic acid in the presence of water and aqueous solutions containing sodium chloride has been studied as a function of flow rate and is found to be consistent with a model combining a constant rate of dissolution with some redeposition having a first-order dependence on the surface concentration [SA]0, with the flux J ) kf - kb[SA]0. The parameters for kf are found to be 2.04 × 10-8, 1.65 × 10-8, and 8.85 × 10-9 mol cm-2 s-1 for dissolution in water and 0.1 M and 1 M sodium chloride, respectively, at a cell temperature of 21 °C.
Introduction
Theory
The novel design of a flow cell which allows AFM to be carried out under defined flow conditions has been described in recent publications.1,2 This cell is based on the Topometrix3 liquid immersion cell, with the addition of an inlet tube shaped to allow a jet of fluid to be applied directly to the sample surface, flowing along the front of the cantilever support chip and over the cantilever. The rate of interfacial reaction is measured directly by averaging the absolute height of the imaged surface as indicated by the z-piezo voltage and by following the changes of this height obtained from successive scans. This cell established that AFM could be used to make quantitative determinations of heterogeneous reaction rates, discriminate between proposed interfacial reaction mechanisms and provide information about binding sites,4 analyze surface dissolution and passivation,5 and identify the mechanisms of dissolution.6 The flow pattern in this cell is complex, and in the initial work, this was solved using an approximate two-dimensional simulation with a finite-element fluid dynamics program. Threedimensional simulations of the flow patterns have now been completed, using the FIDAP 7.62 program7 on the Columbus Superscalar service at the Rutherford Appleton Laboratory. We have applied the velocity data from these simulations to investigate the dissolution processes of the (110) face of salicylic acid exposed to aqueous solutions of flows ranging from 2.5 to 17.4 µL s-1 corresponding to centerline jet velocities of 2.114.8 cm s-1. The use of the 3D model is shown to permit a precise description of the interfacial rate law.
Flow Simulation. FIDAP7 is a fluid dynamics program which uses finite-element methods to solve flow problems; these may be steady-state or transient and may include the effects of temperature and chemical species. The flow region is divided into elements, connected together at nodes, and appropriate boundary conditions may be specified for those nodes which are on an external surface. The equations of conservation of momentum, mass, and energy are solved for each element, and the values for velocities, pressure, temperature, and concentrations at each node are obtained. The volume simulated is shown in outline in Figure 1, with dimensions 0.414 cm × 0.09 cm × 0.1188 cm, and contained 80 794 nodes. The sloping section at the left represents the inside volume of the jet tube, which had a square cross section; the bottom of the volume represents the sample surface, and the AFM scanning cantilevers project into the volume from the back. Figure 1 shows an example of the particle path trajectories, starting from a vertical set of coordinates inside the jet tube, as the solution flows down the tube and then spreads out into the cell, for a fairly low flow rate. At higher flow velocities the jet of solution flows more closely along the sample surface with a smaller degree of spreading. For the simulation of chemical processes, it is not necessary to consider velocities remote from the sample, so data files have been prepared containing the x, y, and z components of velocity at 35 769 points in the smaller volume which starts from the mouth of the jet and encloses the scanning cantilevers (Figure 2), for 14 flow rates encompassing the practicable range of the cell. It should be noted that the crystal surface
* To whom all correspondence should be addressed. E-mail:
[email protected].
10.1021/jp992858k CCC: $19.00 © 2000 American Chemical Society Published on Web 01/28/2000
1540 J. Phys. Chem. B, Vol. 104, No. 7, 2000
Wilkins et al. BIFD calculations. The derivatives in eq 1 are approximated to
∂[C] [C]j,k - [C]j,k-1 ) ∂x ∆x
(2)
∂[C] [C]j+1,k - [C]j-1,k ) ∂y 2∆y
(3)
∂2C [C]j+1,k - 2[C]j,k + [C]j-1,k ) ∂y2 (∆y)2
(4)
Combining eqs 1-4, we obtain Figure 1. Schematic view of the simulated portion of the AFM cell, showing the jet tube, at the left, and particle path trajectories as the solution spreads out into the cell. The sample surface is at the bottom.
should not normally extend back into the jet itself due to the high velocities and gradients present. The surface should be inert in this position by coating part of the crystal or embedding the crystal in an inert matrix. These data files are now available and may be freely downloaded8 from the World Wide Web. Simulations of chemical processes in the cell were carried out by finite difference methods in two dimensions and required a much finer mesh than is normally used for flow simulation. To assist with this, two Fortran-77 programs were prepared. The first of these finds the solution flow path which passes over the scanning tip position, and it generates a two-dimensional grid file of velocities in a surface which follows the flow path. The second program performs a bicubic spline interpolation from this two-dimensional grid to provide the velocity components in a user-specified grid which may contain any number of points. Further details are available in a paper9 published on the World Wide Web, which may be downloaded along with the programs and the data files. Backward-Implicit Finite Difference Simulation. The mass transport from the crystal surface to the bulk solution is described by the following time-dependent convective diffusion equation at steady state:
∂[SA] ∂[SA] ∂[SA] ∂2[SA] - Vx(x,y) ) DSA - Vy(x,y) )0 2 ∂t ∂x ∂y ∂y (1) where x is the direction of the flow over the crystal surface, y is the direction perpendicular to the surface, DSA is the diffusion coefficient of salicylic acid, and Vx and Vy represent the solution velocities in the x and y directions, respectively, at the point (x,y). We assumed that salicylic acid is in equilibrium with the salicylate ion and H+. Experimental data were compared with different theoretical models using rate constants as adjustable parameters. To apply the backward implicit finite difference (BIFD) method, the xy plane is covered with a two-dimensional finite difference grid. Increments in the x direction are ∆x and in the y direction are ∆y.
yj ) j∆y xk ) k∆x
j ) 0,1, . . . J where ∆y ) 2h/J
k ) 0,1, . . . K where ∆x ) (xe - xs)/K
where xs and xe are the start and finish coordinates of the BI grid and 2h is the maximum distance from the surface for the
D
Cj+1,k - 2Cj,k + 2Cj-1,k (∆y)
2
Cj,k - Cj,k-1 - Vx ∆x Cj,k - Cj-1,k ) 0 (5) Vy ∆y
The equation can be rewritten as
λVxCj,k-1 + (λy + λVy)Cj-1,k - (2λy + λVx + λVy)Cj,k + λyCj+1,k ) 0 (6) where
λy )
Vx Vy D , λVx ) , and λvy ) 2 ∆x ∆y (∆y)
If a ) -(λy + λVy)/λVx, b ) (2λy + λVx + λVy)/λVx, and c ) -λy/λVx, it follows that
Cj,k ) aCj-1,k+1 + bCj,k+1 + cCj+1,k+1 Cj,k-1 ) aCj-1,k + bCj,k + cCj+1,k
(7)
Equation 7 is the final general equation for BIFD simulation. The BIFD method only involves vector calculations. The vectors describe concentrations in the y direction for different values of x. The calculation proceeds downstream, each vector enabling the calculation of the next, starting from the vector defining the boundary conditions specified upstream of the reacting crystal of the concentration at the mouth of the inlet jet. The boundary conditions used were as follows:
D
(∂C∂y )
J,k
) 0, CNJ-1,k ) CNJ,k
Thus,
CNJ-1,k-1 ) aCNJ-2,k + bCNJ-1,k + cCNJ,k CNJ-1,k-1 ) aCNJ-2,k + (b + c)CNJ-1,k At the solid salicylic acid surface, assuming that salicylic acid dissolves from the solid at a constant rate and partly precipitates on the surface,
-D
[∂C∂y ]
j)0
) kf - kbC0,k
C1,k - C0,k ) C0,k )
kb∆y ∆y k + C D f D 0,k
DC1,k + kf∆y D + kb∆y
where kf and kb are rate constants for dissolution and precipita-
Dissolution of Salicylic Acid
J. Phys. Chem. B, Vol. 104, No. 7, 2000 1541
Figure 2. The solution volume included in the velocity data files.
tion, respectively.
C1,k-1 ) aC0,k + bC1,k + cC2,k )
[
]
akf∆y aD + + b C1,k + cC2,k D + kb∆y D + kb∆y
( ) ( )( )
These simultaneous equations may be expressed as a (J 1) × (J - 1) matrix equation:
{d} ) [T]{u}
b(1) c(1) a2 b2 · · · · ) · · a(j)
d1 d2 · · · · · · dj
dNJ-1
c2 · · · · · · b(j) · · ·
c(j) · · a·
(NJ-1)
where
d1 ) C1,k-1 )
[
u1,k u2,k · · · · · · uj,k
b(NJ-1) uNJ-1,k
]
akf∆y aD + + b C1,k + cC2,k D + kb∆y D + kb∆y
dj ) Cj,k-1 ) aCj-1,k + bCj,k + cCj+1,k dNJ-1 ) CNJ-1,k-1 ) aCNJ-2,k + (b + c)CNJ-1,k The tridiagonal form of the matrix enables the Thomas algorithm10 to be used. The boundary condition Cj,0 ) Cinlet supplies the vector {d}0 from which {u}1 is calculated. In the absence of homogeneous chemical complications, {d}k ) {u}k, so {u}k is calculated from {d}k-1 and so on until {u}K is obtained. Thus, all the values Cj,k (j ) 1, 2, . . . J - 1, k ) 1, 2, . . . K) are evaluated. Materials and Methods A Topometrix TMX 2010 Discoverer atomic force microscope operating in contact mode with SFM probes type 152000 and a 75 µm scanner type 5590-00 was used to obtain the images. The novel flow cell described earlier1,2 was used for in situ AFM imaging. The rates of dissolution of the surfaces of salicylic acid were measured using the following procedure. A
Figure 3. Salicylic acid crystal habit after growth from a saturated solution of salicylic acid in ethanol.
single crystal of salicylic acid was fixed in the flow cell exposing the (110) face for imaging. Solution was flowed through the liquid cell using various flow rates ranging from 0.0025 to 0.0174 cm3 s-1. A series of images of the dissolving surfaces scanning an area of 20 µm × 20 µm was then recorded continuously at ca. 60 s intervals. In addition to conventional topographical images, plots of the absolute z-piezo voltage were recorded. The change of the mean piezo voltage calculated from each scan and therefore the change of the mean surface level can be derived using the calibration factor 0.333 V µm-1. Note that, although the thickness of the crystal is changing, the piezo support moves to maintain the top surface of the crystal at a constant height since the cantilever deflection is being maintained constant by the AFM feedback. The flow geometry and velocities therefore do not change as dissolution proceeds. The flow system was gravity fed with solution flowing from a reservoir to the flow cell then to an outlet whose height was adjusted relative to the reservoir. Measurements were taken at a temperature of 21 °C. Salicylic acid crystals were grown from a saturated solution in ethanol by gradually decreasing the temperature from 30 to 25 °C over 200 h, yielding large numbers of good quality crystals averaging 3 mm × 4 mm × 30 mm in size (Figure 3). Salicylic acid crystals were indexed and the crystal faces assigned using standard methods with a ENRAF NONIUS FR 590 diffractometer using Mo KR radiation. The total solubilities of salicylic acid in water and solutions containing 0.1 M and 1 M NaCl, respectively, were determined by UV spectrophotometric analysis. An excess of salicylic acid
1542 J. Phys. Chem. B, Vol. 104, No. 7, 2000
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Figure 4. In situ images of the (110) surface of salicylic acid during the dissolution process in 0.1 M NaCl flowing at a rate of 2.5 µL s-1. Sequence after (a) 0, (b) 118, and (c) 236 s.
Figure 5. In situ images of the (110) surface of salicylic acid during the dissolution process in 0.1 M NaCl flowing at a rate of 12.5 µL s-1. Sequence after (a) 0, (b) 59, and (c) 118 s.
was added to 10 mL flasks of each of the solutions. The flasks were agitated for 24 h before removing a small aliquot from each flask and centrifuging to separate the excess salicylic acid from solution. The solutions were diluted 100 fold with phosphate buffer of pH 7.429, and the solubilities were calculated from the UV absorption spectra. The solubilities were found to be 14.9 mM in water, 13.88 mM in 0.1 M NaCl, and 10.79 mM in 1 M NaCl. The diffusion coefficient for salicylic acid was calculated to be DSA ) 7.1 × 10-6 cm2 s-1 using the Wilke-Chang equation,11 and the dissociation constant of salicylic acid is 1 × 10-3 M at 25 °C. The solubility values used for the mathematical simulation were the concentration of undissociated salicylic acid in a saturated solution, calculated from the known solubility and the dissociation constant. Solutions were made up using triply deionized water of resistivity > 107 Ω cm (from an Elgastat system, High Wycombe, UK). The chemicals used were salicylic acid 99+% and sodium chloride 99.9% (BDH Ltd., Poole, Dorset). The buffer solution for solubility measurements contained 0.0303 M Na2HPO4 and 0.0068 M KH2PO4 (BDH Ltd., Poole, Dorset). Results and Discussion Crystals of salicylic acid were imaged under a range of flow conditions using the modified AFM liquid cell. AFM dissolution experiments of the (110) plane were conducted using flow rates of 2.5, 7.5, 12.5, and 17.4 µL s-1 for dissolution in pure water and solutions of 0.1 M and 1 M NaCl. The rate of dissolution was determined for the different flow rates by monitoring the change in the mean z-piezo voltage from successive scans at 59 s intervals. During the course of each experiment some erosion of the crystal occurred in the jet region. To prevent interference of this erosion in the flow dynamics, the crystal was shifted in the z direction between each sequence of images. The (110) face was found to be flat with minor defects typically less than 150 nm in height before exposure to the aqueous solutions. Naturally grown faces were used; it was not found possible to prepare faces by cleavage due to the fragile nature of the crystals. Exposure to the three solutions at slow flow rates was found to induce the formation of small pyramidal growths on the surfaces, as demonstrated in Figure 4. Analysis
of the z-piezo voltage plots showed the crystal height steadily decreasing due to dissolution, with the small growths increasing in height before falling again as dissolution proceeded. At slow flow rates, these growths are observed where the salicylic acid has dissolved in the region close to the jet then precipitated on the surface downstream. The orientation and shape of these growths are not clear from these images since the growths may be sharper than the pyramidal AFM probes. However, the growths typically form in areas of defect on the surface such as steps and pits. At higher flow rates, the small growths were absent and the sequences of images clearly showed the formation of steps and terraces on the crystal surface during dissolution, as shown in Figure 5. The steps and pits grow in two directions, the most rapid dissolution occurring in the (001) direction with steps also dissolving perpendicular to the (001) direction but at a slower rate. The height, size, and velocity of retreat varied considerably between experiments; however, the overall dissolution of the crystal was found to vary linearly with time. An example of this is shown in Figures 6 and 7. In Figure 6, the major step height in the first image is approximately 3200 nm; the step height of the major step in Figure 7 is approximately 800 nm. The larger step can be seen to retreat more slowly than the smaller step. However, the overall flux remains linear since the amount of salicylic acid dissolved from the surface is similar in each case. Experiments with the two NaCl solutions showed a suppression in the rate of dissolution compared with the dissolution in water. The dissolution rates for the salicylic acid (110) plane are shown as a function of the solution flow rate together with the theoretical curves in Figure 8. The dependence of the flux from the flow rate is a function of the heterogeneous rate constants kf and kb. The relationship between solubility and the rate constants is shown in eq 8.
[SA]sol )
kf kb
(8)
Therefore, when the solubility is known, the relationship between flux and flow rate is only dependent on one of the heterogeneous rate constants and can be determined by the best
Dissolution of Salicylic Acid
J. Phys. Chem. B, Vol. 104, No. 7, 2000 1543
Figure 6. In situ images of the (110) surface of salicylic acid during the dissolution process in 1 M NaCl flowing at a rate of 2.5 µL s-1. Sequence after (a) 0, (b)118, (c) 236, and (d) 354 s.
Figure 7. In situ images of the (110) surface of salicylic acid during the dissolution process in 1 M NaCl flowing at a rate of 2.5 µL s-1. Sequence after (a) 0, (b) 118, (c) 236, and (d) 354 s.
fit of the experimental curves by the model described. The parameters for kf were found to be 2.04 × 10-8, 1.65 × 10-8,
and 8.85 × 10-9 mol cm-2 s-1 at 21 °C for dissolution of salicylic acid in water, 0.1 M NaCl, and 1 M NaCl, respectively.
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Figure 8. Plot of dissolution rate vs flow rate for the dissolution of the (110) face of salicylic acid in water (9), 0.1 M NaCl ([), and 1 M NaCl (b) showing the theoretical curve using the 3D simulation compared to the experimental data.
Figure 9. Plot of dissolution rate vs flow rate for the dissolution of the (110) face of salicylic acid in water (9), 0.1 M NaCl ([), and 1 M NaCl (b) showing the theoretical curve using the 2D simulation compared to the experimental data.
Figure 9 shows, for comparison, the theoretical curves using a 2D simulation of the flow velocities. This gives less good agreement to the experimental data at low flow rates and hence demonstrates the importance of using a full 3D simulation for the flow. As the concentration of sodium chloride in the flow cell is increased, there is a reduction in the rate of dissolution. The 3D theoretical model is in good agreement with the experimental data. Conclusion
also provide details of mechanisms of dissolution. This is of considerable importance for example to pharmaceutical companies in understanding the fundamental factors governing the dissolution behavior of pharmaceutical drugs under different conditions. The merits of hydrodynamic AFM for quantitative studies of interfacial kinetics is evident; the only limitations of the method arise since, as the upstream zone of the crystal influences the kinetic behavior in the downstream imaged area at low flow rates, it is necessary to assume that the interfacial reaction mechanism remains unchanged over the full length of the sample.
It has been shown that an accurate calculation of the dissolution flux of salicylic acid in the flow cell under different flow conditions can be made using a 3D finite element fluid dynamics simulation of the complex flow field. The use of the novel flow cell reveals not only information about reaction rate constants of solid/liquid reactions but the topographic images
Acknowledgment. We thank GlaxoWellcome and BBSRC for support for S.J.W. via a CASE award, COLCIENCIAS for a scholarship for M.F.S., and EPSRC for the grant of supercomputer facilities (Grant GR/L34167).
Dissolution of Salicylic Acid References and Notes (1) Coles, B. A.; Compton, R. G.; Booth, J.; Hong, Q.; Sanders, G. H. W. Chem. Comm. 1997, 619. (2) Coles, B. A.; Compton, R. G.; Sua´rez, M.; Booth, J.; Hong, Q.; Sanders, G. H. W. Langmuir 1998, 14, 218. (3) Thermo Microscopes Ltd., Bicester, Oxon, UK (http://www.thermomicroscopes.com). (4) Hong, Q.; Sua´rez, M.; Coles, B. A.; Compton, R. G. J. Phys. Chem. B 1997, 101, 5557.
J. Phys. Chem. B, Vol. 104, No. 7, 2000 1545 (5) Booth, J.; Compton, R. G.; Atherton, J. H. J. Phys. Chem. B 1998, 102, 3980. (6) Sua´rez, M.; Compton, R. G. J. Phys. Chem. B 1998, 102, 7156. (7) Fluent Europe Ltd., Sheffield, UK (http://www.fluent.com). (8) ftp://joule.pcl.ox.ac.uk/pub/rgc/afmvmaps/. (9) Coles, B. A.; Compton, R. G. http://physchem.ox.ac.uk:8000/ research/afm/afmflow.shtml. (10) Lapidus, L.; Pinder, G. F. Numerical Solution of Partial Differential Equations in Science and Engineering; Wiley: New York, 1982. (11) Wilke, C. R.; Chang, P. AIChE J. 1955, 1, 264.
