Modeling flows in protein crystal growth

Transcription

1 Modeling flows in protein crystal growth D. N. Riahi Department of Theoretical and Applied Mechanics, 216 Talbot Laboratory, 104 S. Wright St., University of Illinois, Urbana, Illinois 61801, USA Telephone Number: Fax Number: ABSTRACT This is a review article on modeling convective flows due to either buoyancy force or surface tension gradient force during protein crystallization that have been studied in the past. The modeling and computational studies of such flows have provided useful results about the effects of the undesirable convection, which need to be minimized in order to produce protein crystal with higher quality and better order in the structure. Ramachandran et al. [1] developed analytical and numerical models to describe the flows and transport associated with the protein crystal growth. Lee and Chernov [2] carried out analytical and numerical studies of convective and diffusional mass transport to an isolated protein crystal growing from solution, with slow linear interface kinetics. Very recent modeling and computational studies by Bhattacharjee and Riahi [3, 4] of compositional convection during protein crystallization and under the external constraint of rotation indicated beneficial effects of rotation in reducing the effect of convection under certain range of the parameters values. The results by these authors also provided conditions under which convective flow transport during the protein crystallization can approach the diffusion limited transport, which is desirable for the production of higher quality protein crystals. KEYWORDS: convective flow, protein crystallization, convection, convection during crystallization, protein crystal growth INTRODUCTION It is known that space environment under microgravity condition has the advantage over the earth environment for the production of higher quality crystals since at least the component of the undesirable convective flow due to the buoyancy force is negligible under the microgravity condition of the space environment. A number of experiments involving protein crystal growth that have been carried out in space under microgravity condition, have been able to produce certain types of crystals that were relatively larger and better ordered as compared to those grown on earth under normal (earth) gravity (1g 0 level) condition [5-8]. The main motivation for growing protein crystals in space has been to achieve diffusion controlled growth condition, which was thought to produce higher quality crystal. However, there have also been a number of flight experiments that produced crystals with no noticeable improvement in the quality 1

2 and internal structure as compared to those grown under the normal gravity on earth. One main problem due to the space environment under the microgravity condition has been the dominance of the effect of surface tension gradient force, which is undesirable and can adversely affect the crystal-melt interface. Although this problem is difficult to be examined quantitatively at present due to the lack of sufficient information about the surface tension and its precise variation with respect to the solute concentration, qualitative information about ways that Marangoni effect can be reduced noticeably, is of value. The recent investigation in [4] determined such information for the case where the protein crystal grows in a microgravity environment and is under a rotational constraint. These authors detected some interesting results about the beneficial effects of the external constraint of rotation. In particular, they found certain values of the rotation rate under which the effect of the surface tension gradient force on the convective flow was reduced noticeably. In the engineering applications, a number of studies already documented the beneficial effects of rotation, under some conditions, on the growth of crystals other than those due to proteins. Lie et al. [9-10] studied centrifugal pumping with applications to semiconductor crystal growth and silicon crystal growth in float zone processes. Riahi [11-13] investigated asymptotically nonlinear compositional convection during alloy solidification in a high gravity environment, where the solidification system was subjected to an inclined rotational constraint, and determined range of values for the centrifugal and Coriolis parameters under which buoyancy driven convection may be as weak as possible. The problems of convective flows, subjected to the buoyancy or surface tension gradient force, are of considerable interest in geophysics and engineering [14]. A number of nonlinear studies have been carried out in the past [15-24]. The only work that takes into account the effects Coriolis force in a Benard-Marangoni type convection in a rotating fluid is apparently that of Riahi [25] who studied theoretically the weakly nonlinear case near the onset of motion and found, in particular, some stabilizing effect of the Coriolis force on the flow pattern. In regard to the effects of the surface tension gradient force on flows during nonprotein crystal growth, some results have been determined for flow in a circular cylindrical float zone [26] and for flow during a realistic float zone crystal growth [27]. For flow during protein crystal growth, very few investigations have been done so far to study the Marangoni effect [28]. In addition to the recent work [4], Resenberger [29] have made some qualitative points about the Marangoni effects on flow during protein crystal growth. Monti and Savino [30-31] investigated numerically some modeling aspects of fluid mechanics of vapour diffusion systems and studied qualitatively the Marangoni effects that could exist in drops and around growing crystals. Bhattacharjee and Riahi [4] investigated computationally and for the first time the effect of rotation on surface tension driven flow during protein crystallization under different microgravity environments. Their computational modeling and procedure was similar to that used by Ramachandran [1] for buoyancy driven convection during protein 2

3 crystal growth and in the absence of rotation. Due to the complexity of the their problem, which was compounded by the presence of the externally imposed inclined rotation, the authors restricted their study to the simplest case of a planar model for the protein system where the governing system of equations was satisfied and the effects of centrifugal force, Coriolis force and the surface tension gradient force were all taken into account. This has been the first study of the surface tension driven convection during protein crystal growth and under the inclined rotational effect. In the present review article, one of the main sections is on the problems and the results presented in [3, 4], and the other main section is about the case treated in [2]. FLOW SYSTEM The work in [3, 4] considered a small cylindrical crystal growing in a cylindrical chamber whose axis coincides with that of the crystal. The authors employed the modeling data similar to those used in [1] for the protein crystal, the chamber and the fluid within the chamber boundary and the crystal. Hence, diameter size of the crystal was chosen to be 0.1 cm, length of the height L of the crystal was taken as 0.1 cm, and the crystal was suspended with its center 0.25 cm from the bottom of the chamber. The values of the kinematic viscosity ν, solute diffusivity D and density ratio ρ/ρ (ρ - ρ i )/ρ, were taken [1] respectively as 0.01 cm 2 /s, cm 2 /s and Here ρ i is the density at the crystal-melt interface, which was assumed to be a constant, and ρ is the reference density of the fluid away from the crystal interface. Following [1], the moving boundary of the crystal was ignored since the crystal growth velocity was sufficiently slow. No-slip conditions for flow velocity were used on the crystal surface and on the container s solid boundary in contact with the fluid, while impermeable and free surface conditions were assumed at the top surface of the fluid, which was not in contact with the container s boundary. Additional boundary conditions for the solute concentration C were that C=C i on the crystal interface and C=C at the container s boundary in contact with the fluid, where C was assumed to be a constant. As explained in [1], the dependence of C i on the growth system has been generally unknown, for most of the protein growth systems, and, thus, C i was also taken as a constant. Adopting Boussinesq approximation [32], the density difference ρ was then directly proportional to the difference in the solute concentration. The governing equations for momentum, continuity and solute concentration were considered in a rotating frame embedded in the crystal, and the crystal growth system was assumed to be rotating at some constant angular velocity Ω about the rotation axis, which considered to be either anti-parallel or parallel with respect to the normal gravity vector, and the axis of the protein crystal made an angle γ with respect to the rotation axis. For 0 γ<90, the rotation vector Ω was assumed to be anti-parallel to the normal gravity vector, while Ω was assumed to be parallel to the normal gravity vector for 90 γ<180. The coordinate system, which was assumed embedded in the crystal growth system, was, thus, rotating about the rotation axis. The momentum equation, thus, contains terms due to inertial, pressure gradient, viscous, buoyancy, centrifugal and Coriolis forces. In the momentum equation g =ng 0 is acceleration due to gravity, where 3

4 g 0 is the normal (earth) acceleration due to gravity (981.0 cm/s 2 ) and n denotes the gravity level used in the present study. The governing equations and the boundary conditions were non-dimensionalized using L, ν/l, L 2 /ν, C and ρ ν 2 /L 2 as scales for length, velocity, time, solute concentration and pressure, respectively, where C C C i. The governing system then contains the non-dimensional parameters S, T, G and M. Here S=ν/D is the Schmidt number, which had the value of in [3, 4] study, T=4Ω 2 L 4 /ν 2 is the Taylor number, G =g( ρ/ρ )L 3 /ν 2 is the Grashof number, which had the form G r =(01 s 2 /cm)g in [3, 4], the ratio ρ/ρ had the value of 0.01 [3, 4], and M= C L(dξ/dC)/(Dµ) is the Marangoni number, where ξ is the surface tension and µ is the dynamic viscosity. The numerical code used in [3, 4] was a finite-difference Navier-Stokes code for non-staggered Cartesian grids. After doing the required sensitivity tests, a grid of was used throughout the simulations, with the grid lines clustered towards the walls. The code used a finite-difference scheme in two-dimensional Cartesian coordinates on nonstaggered grid. The convective terms were discretized using positive coefficient skew upwinding and for diffusive terms central differencing was used. The code was benchmarked with pure natural convection case and with the simulations in [1] and was found to give good agreement. The case of g= g 0 was used as the baseline case for the simulations, and each case was run for several different angles of inclination γ. The parameters of interest were the maximum V max of the magnitude of velocity vector u in the protein growth system, which was also referred to as global maximum velocity, the Sherwood number S h, and the local velocity V loc, which was defined as the maximum value of the magnitude of velocity vector u in a domain of two non-dimensional units away from the crystal surface. A brief discussion of the results obtained in [4] is as follows. First, the results are discussed for the cases of g= g 0 and γ=0.0 with the free surface level very close to the upper face of the crystal, where the non-dimensional perpendicular distance d between the free surface and the upper face of the crystal was taken the value of and either M was zero or the effects of the surface tension driving force on the crystal growth was significant for a given non-zero M. For M=γ=0, It was found that for very small values of the rotation rate, there was a slight decrease in the velocity scales, which was already very small (~ cm/s) after which the velocity increased sharply with the rotation rate. Thus it was concluded that moderate and strong rotation increased the convection at zero Marangoni number. The calculation results for the Sherwood number indicated that S h remained nearly a constant for the studied range of T. For M= and γ=0.0, it was found that the local velocity scales were significantly lower than the maximum velocity scales over most of the domain for T that had been investigated. In this case, though the local velocity scale increased almost linearly with T, the maximum velocity scale decreased for T in the range 0 T 50.0, after which it increased and for T >80.0 its value became higher than that of the local velocity scale. The calculation 4

5 results for the Sherwood number at these values of M and g, indicated that, as T increased from zero value, S h decreased first and then increased monotonically, a minimum was reached at about T=20.0. The maximum velocity scale was also found to reach a minimum value at this value of T. As was noted by the authors, the values of the velocity scales were about times more than the corresponding velocity scales values for the M=0.0 case, and the rotation rate required to bring a notable effect corresponded to values of T about 10 times larger than the T values found in the M=0.0 case. The M = case was considered next. The results for the local and maximum velocity scales indicated that the values of these velocities were equal for zero rotation rate case, but with increasing T, the local velocity scale decreased while the maximum velocity scale increased, reached a peak at T about 25.0 and then began to decrease. At T about 75.0 the minimum for both the maximum and local velocity scales were found, the minimum being almost 50% of the value at zero rotation. Further increase of T was found to increase the velocity scales only marginally. The velocity vectors determined at their maximum at the boundary, and these vectors were found to shorten in length with increasing T. Also the pattern of convection was determined. For small rotation rates, the convection was limited mainly to the left and right walls of the crystal, in a counter clockwise direction on the left wall. With increasing T, this convection was pushed upwards and new convection is found to develop in the lower boundary of the crystal this one in a clockwise sense. The Sherwood number as found to decrease by 50%, to a minimum at T corresponding to minimum velocity scales. The beneficial aspects of rotation were found best for this case. The Sherwood number variation was also determined. There were steep gradients in the concentration for the low rotation case, but with increasing rotation, these contours of constant concentration were found to spread out yielding lower S h. When the contours were perfectly spread out as in the case of pure diffusion, the Sherwood number was minimized. Rotation was found to drive the contours towards that ideal condition. For the case of M= , the values of the velocity scales were about 1.5 cm/s or less. The maximum and local velocity scales decreased monotonically for increasing T. For the highest T simulations, the maximum velocity scale was found to equal the local velocity scale of 0.35, thus a decrease in the maximum velocity scale of about 4.2 times was found. It was thus postulated that the beneficial effects of decreasing Marangoni convection was better for higher values of M. The results for S h in this case and for cases of M= and indicated that S h increased with M. Next, the cases of g=0.0001g 0 and γ=0.0 with the free surface level again very close to the upper face of the crystal (d=0.074) were calculated. The computation in [4] was carried to determine maximum and local velocity scales, Sherwood number, velocity contours, etc, for different M and T. For M= , the maximum velocity scale was found to be larger than the local velocity scale for zero rotation. As T increased beyond its zero value, both velocity scales decreased and reached a minimum value at about T=100.0 beyond which the velocity scales increased strongly with the rotation rate. 5

6 Larger values of g were found to require larger values of the rotation rate in order for the velocity scales to reach their minimum values. This result is consistent with the physical aspect of the destabilizing effects of the buoyancy force. The effects of the location of the free surface relative to the location of the crystal were studied for cases where the free surface level was somewhat away by the distance d=1.486 parallel to the x-axis from the upper crystal face. It was found that as d increased from the previously chosen value 0.074, then flow adjacent to the crystal face stabilized as expected physically. Some typical results about the velocity vectors were obtained for g= g 0, γ=0.0, M = , d=1.486 and for several different values of T. For small rotation, convection was found to be limited to the region between the free surface and the crystal face parallel to the free surface and forming two circulations with opposite directions. With increasing T, these flow circulations extended over the whole flow domain. The authors also generated data for the velocity scales and the Sherwood number in this case. The calculated data for the Sherwood number in this case indicated qualitatively the same behavior as in the d=0 case, except that the minimum value of S h corresponded at a lower value of T if d was larger. The results for the velocity scales indicated that the value of the local velocity scale decreased with increasing d, while the value of the maximum velocity scale increased with d for small rotation and slightly decreased with increasing d for large rotation. In regard to the effect of the angle of inclination γ, it was found that its effect on the solute flux, local velocity scale and convection was generally not significant. However, its effect on the maximum velocity was found to be noticeable. For example, for g = g 0, M= and T=120.0, it was found that the maximum velocity scale remained nearly a constant about for γ less than about 30.0 degrees, while it increased with γ for 30.0 <γ 90.0 reaching a maximum value of about and then decreased with further increase in γ for 90.0 <γ For <γ 180.0, V max again remained nearly a constant value of about THE CASE WITH INTERFACE KINETICS The work in [2] considered buoyancy driven convection during protein crystallization and in the absence of both rotation and surface tension gradient effects. The authors considered a cylindrical protein crystal sitting at the bottom of a cylindrical chamber filled with its solution containing an impurity. Initially at time t=0, the uniform density of the solution was ρ s, and the uniform concentrations of the crystallizing protein and impurity were, respectively, ρ 1 0 and ρ 2 0. For t>0, the concentrations depended, in general, on time and space variables, and were designated by ρ 1 and ρ 2. The mass diffusivities for protein and impurity were designated by D 1 and D 2, respectively. It was assumed that there was slight contamination such that ρ 2 0 <<ρ 1 0. Crystal-solution equilibrium was reached when ρ 1 =ρ 1 e. Inside the crystal structure, the density for the protein was ρ 1 c and for impurity was ρ 2 c. Next, the boundary condition for ρ 1 at a crystal surface was related to the growth rate of that surface. The growth of the crystal took place both in tangential and normal 6

7 directions to each surface of the crystal. It is known that the tangential growth is much faster than the normal growth, and, so, each surface was considered as advancing uniformly in the normal direction like a plane at a rate V, which in general depended on time t. Applying the mass conservation for the protein, the normal diffusion flux should be proportional to V, which was assumed to be proportional to the super-saturation ρ 1 - ρ 1 e. Hence, D 1 ρ 1 / n=ρ 1 c V=β(ρ 1 -ρ 1 e ), where β is the kinetic coefficient characterizing the rate at which the molecules are incorporated by the growing face, and n is normal distance pointing towards the solution from the crystal surface. Since the growing crystal face was assumed to be flat, V did not depend on the position along the face. The authors in [2] assumed β to be a constant, and the advancement of the crystal growing surfaces was not taken into account in their modeling even though V had a finite value. Using the mass conservation for the impurity at the growing interface, it was found that D 2 ρ 2 / n=(k-1)ρ 2 V, where k=ρ 2 c /ρ 2, in which ρ 2 was evaluated at the interface, was the segregation coefficient, which was assumed to be constant. The work in [2] then assumed stable flow conditions, which, in particular, excluded striations in the protein crystal. The authors considered axisymmetric flow and used cylindrical coordinates for governing equations for the vorticity, continuity for the protein and continuity for the impurity. The boundary conditions at the growing crystal surfaces were those two described above for ρ 1 and ρ 2 plus assumed zero velocity vector u=0 at the crystal surfaces. The boundary conditions at the chamber walls were u= ρ 1 / n = ρ 2 / n= 0, and initially it was assumed that u=0 and ρ 1 =ρ 2 =1 in the solution. The authors developed a generic scaling model analytically and solved the flow system numerically using a finite difference scheme. They determined the contribution of the convection and diffusion to the mass transport and also studied the effects of various factors on the growth kinetics and protein crystal purity. Their results indicated, in particular, that the impurity distribution within the protein crystal was not homogenous due to convective flow effect. The authors determined the level of residual gravity, below which there is diffusion-controlled transport for crystals sitting at a chamber wall. This could be useful for proper design of space experiments, provided Marangoni convection effects also accounted for, which was not considered in [2]. REMARKS It appears from the work in [3, 4] and the subsequent results about the effects of the inclined rotation on the buoyancy or surface tension driven flow (Marangoni flow) during protein crystallization that such flow can be weakened noticeably for the moderate rates of rotation, while such flow can be enhanced for sufficiently large rotation rates. Furthermore, we conclude that stronger Marangoni flow can be weakend by applying rotation rate of larger magnitude and the effect of the angle of inclination γ on the Marangoni flow is rather weak in such flow system. Unfortunately no experimental results are available at present for a rotating Marangoni flow during crystal growth in either micogravity or normal gravity environment in order to be able to make at least 7

8 some comparison between the results of the work in [3, 4] and those due to the experimental observation. However, the numerical results due to [3, 4] can stimulate experimental studies for further understanding the effects of rotation on such flows, and, in particular, on the Marangoni flow, which significantly dominate over buoyancy driven flows in a microgravity environment. The results on the effects of rotation on the convective flow during protein crystal growth [3, 4] indicated that the centrifugal force appears to dominate over the Coriolis force for T a >1.0, while the Coriolis force dominates over the centrifugal force for 0.0<T a <1.0. Since it was found that Marangoni flow is weakest for a T a value, which is generally larger than 1.0, we can conclude that in the regime for higher quality production of the protein crystals, centrifugal force dominates over the Coriolis force. In regard to the insignificant effect of the angle of inclination γ on the Sherwood number and the convective flow in the vicinity of the crystal face detected in the microgravity system [4], it should be noted that it may be related to the fact that here the surface tension gradient force dominates over the gravity force and the angle of inclination γ is the angle between the crystal axis and the rotation axis, which is either parallel or anti-parallel to the weak gravity force, so that both the magnitude and direction of the weak gravity force are insignificant in the present system. It should be noted that, as partly explained earlier in the introduction section, the protein crystal growth system was studied in [4] for a two-dimensional planar model and in microgravity environments only. It is quite possible that similar system could be studied experimentally in future first under normal gravity condition where both buoyancy and Marangoni forces are significant and the system actually is in threedimension. Hence, it would be quite appropriate and interesting to investigate such a three-dimensional extension of the work in [4] under both microgravity and normal gravity conditions in order to determine the results, which can be compared to possible future experimental data as well as to detect possible beneficial roles that rotation can play to weaken either or both of the buoyancy and surface tension gradient forces. As can be concluded from the work in [2] and [4] reviewed in this article that a useful extension of either of these two studies can be an investigation of the problem of the type treated in [4] but with the consideration of surface kinetic effects and the corresponding boundary conditions for the density of the crystallizing protein and the density of the impurity at the protein crystal surfaces and at the chamber walls. Finally, it should be concluded that convection control during protein crystal growth is possible by applying a rotation vector of appropriate magnitude and direction on the crystal growth system. In addition, as the results in [13] for the alloy crystal suggest, application of both a magnetic field and a rotation vector on the protein crystallization system may well lead to a more effective way to control the undesirable convection and thereby producing higher quality protein crystal. It is then recommended future studies of the flow during protein crystallization and in the presence of both 8

9 rotation and magnetic field in order to detect any possible relevant parameter regimes where the convective flow can be weakened as much as possible. REFERENCES [1] Ramachandran, N., Baugher, C. R. and Naumann, R. J Modeling flows and transport in protein crystal growth. Microgravity Science and Technology 8, [2] Lee, C. P. and Chernov, A. A Solutal convection around growing protein crystals and diffusional purification in space. J. Crystal Growth 240, [3] Bhattacharjee, P. and Riahi, D. N Computational studies of the effect of rotation on convection during protein crystallization. Submitted for publication. [4] Bhattacharjee, P. and Riahi, D. N Effect of rotation on surface tension driven flow during protein crystallization. Microgravity Science and Technology 14, [5] Delucas, L. J., Smith, C. D., Smith, W., Vijay-Kumar, S., Senadhi, S. E., Ealick, S. E., Carter, D. C., Snyder, R. S., Weber, P. C., Raymond Salemme, F., OhAlendorf, D. H., Einspahr, H. M., Clancy, L. L., Navia, M. A., McKeever, B. M., Nagabhushan, T. L., Nelson,G., McPherson, A., Kozelak, S., Taylor,G., Stammers, D., Powel, K., Darby, G., Bugg, C. E Protein Crystal Growth Results for Shuttle Flights. Journal of Crystal Growth, 110, [6] Day, J., McPherson, A Macromolecular Crystal Growth Experiments on International Microgravity Laboratory. Protein Science 1, [7] Delucas, L. J., Long, M. M., Moore, K. M., Rosenblum, W. M., Bray, T. L., Smith, C., Carson, M., Narayana, S. V. L., Harrington, M. D., Clark Jr., A, D., Nanni, R. G., Ding, J., Jacobo-Molina A., Kamer, G., Hughes, H. S., Arnold, E., Einspahr, H. M., Clancy, L. L., Rao, G. S. J., Cook, P. F., Harris, B. J., Munson, S. H., Finzel, B. C., McPherson, A., Weber, P. C., Lewandowski, F. A., Nagabhushan, T. L., Trotta, P. P., Reichert, P., Navia, M. A., Wilson, K. P., Thomson, J. A., Richards, R. N., Bowersox, K. D., Meade, C. J., Baker, E. S., Bishop, S. P., Dumbar, B. J., Trinh, E., Prahl, J., Sacco Jr., A., Bugg, C. E Recent Results and New Hardware Developments for Protein Crystal Growth in Microgravity. Journal of Crystal Growth 135, [8] Long, M. M., Delucas, L. J., Smith, C., Carson, M., Moore, K., Harrington, M. D., Pillion, D. J., Bishop, S. P., Rosenblum, W. M., Naumann, R. J., Chait, A., Prahl, J., Bugg, C. E Protein Crystal Growth in Microgravity-Temperature induced Large Scale Crystallization of Insulin. Microgravity Science and Technology, 7, [9] Lie, K. H., Walker, J. S., Riahi, D. N Centrifugal pumping in a liquid cylinder in an axial magnetic field with application to semi-conductor crystal growth. Proceedings of the First Congress in Fluid Dynamics 3,

10 [10] Lie, K. H., Walker, J. S., Riahi, D. N Centrifugal pumping in the float-zone growth of silicon crystal with an axial magnetic field. Journal of Physico-Chemical Hydrodynamics 10, [11] Riahi, D. N Effects of centrifugal and Coriolis forces on chimney convection during alloy solidification. Journal of Crystal Growth 179, [12] Riahi, D. N Effect of rotation on a non-axiymmetric chimney convection during alloy solidification. Journal of Crystal Growth 204, [13] Riahi, D. N Effects of centrifugal and Coriolis forces on a hydromagnetic chimney convection in a mushy layer. Journal of Crystal Growth 226, [14] Davis, S. H Thermocapillary Instabilities. Annual Review of Fluid Mechanics 19, [15] Scanlon, J., Segel, L Finite Amplitude Cellular Convection Induced by Surface Tension. Journal of Fluid Mechanics 30, [16] Lebon, G., Cloot, A Buoyancy and Surface Tension Driven Instabilities in Presence of Negative Rayleigh and Marangoni Numbers. Acta Mechanica 43, [17] Lebon, G., Cloot, A A Nonlinear Stability Analysis of the Benard-Marangoni Problem. Journal of Fluid Mechanics 145, [18] Riahi, D. N Thermocapillary Instability of an Infinite Prandtl Number Fluid with Negligible Gravitational Effects. Acta Mechanica 64, [19] Riahi, D. N Nonlinear Benard-Marangoni Convection. Journal of the Physical Society of Japan 56, [20] Riahi, D. N Thermocapillary Instability of a Low Prandtl Number Fluid with Negligible Gravitational Effects. Acta Mechanica 71, [21] Riahi, D. N Hexagon Pattern Convection for Benard-Marangoni Problem. International Journal of Engineering Sciences 27, [22] Ramachandran, N Thermal Buoyancy and Marangoni Convection in a Two Fluid Layered System. Journal of Thermophysics and Heat Transfer 7, [23] Kliakhandler, I. L., Nepomnyashchy, A. A., Simanovski, I. B., Zaks, M. A Nonlinear Marangoni waves in Multilayer System. Physical Review E 58, [24] Bragard, J., Velarde, M. G Benard-Marangoni Convection: Planforms and Related Theoretical Predictions. Journal of Fluid Mechanics 368,

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13 List of Recent TAM Reports No. Authors Title Date 959 Kuznetsov, I. R., and D. S. Stewart 960 Zhang, S., K. J. Hsia, and A. J. Pearlstein 961 Sharp, K. V., R. J. Adrian, J. G. Santiago, and J. I. Molho Modeling the thermal expansion boundary layer during the combustion of energetic materials Combustion and Flame, in press (2001) Potential flow model of cavitation-induced interfacial fracture in a confined ductile layer Journal of the Mechanics and Physics of Solids, 50, (2002) Liquid flows in microchannels Chapter 6 of CRC Handbook of MEMS (M. Gad-el-Hak, ed.) (2001) 962 Harris, J. G. Rayleigh wave propagation in curved waveguides Wave Motion 36, (2002) 963 Dong, F., A. T. Hsui, A stability analysis and some numerical computations for thermal and D. N. Riahi convection with a variable buoyancy factor Journal of Theoretical and Applied Mechanics 2, (2002) 964 Phillips, W. R. C. Langmuir circulations beneath growing or decaying surface waves Journal of Fluid Mechanics (submitted) 965 Bdzil, J. B., D. S. Stewart, and T. L. Jackson 966 Bagchi, P., and S. Balachandar Program burn algorithms based on detonation shock dynamics Journal of Computational Physics (submitted) Linearly varying ambient flow past a sphere at finite Reynolds number: Part 2 Equation of motion Journal of Fluid Mechanics 481, (2003) (with change in title) The evolution equation for a disclination in a nematic fluid Proceedings of the Royal Society A 458, 1 20 (2002) 967 Cermelli, P., and E. Fried 968 Riahi, D. N. Effects of rotation on convection in a porous layer during alloy solidification Chapter 12 in Transport Phenomena in Porous Media (D. B. Ingham and I. Pop, eds.), (2002) 969 Damljanovic, V., and R. L. Weaver Elastic waves in cylindrical waveguides of arbitrary cross section Journal of Sound and Vibration (submitted) 970 Gioia, G., and Two-phase densification of cohesive granular aggregates Physical A. M. Cuitiño Review Letters 88, (2002) (in extended form and with added co-authors S. Zheng and T. Uribe) 971 Subramanian, S. J., and Calculation of a constitutive potential for isostatic powder P. Sofronis compaction International Journal of Mechanical Sciences (submitted) 972 Sofronis, P., and Atomistic scale experimental observations and micromechanical/ I. M. Robertson continuum models for the effect of hydrogen on the mechanical behavior of metals Philosophical Magazine (submitted) 973 Pushkin, D. O., and Self-similarity theory of stationary coagulation Physics of Fluids 14, H. Aref (2002) 974 Lian, L., and Stress effects in ferroelectric thin films Journal of the Mechanics and N. R. Sottos Physics of Solids (submitted) 975 Fried, E., and Prediction of disclinations in nematic elastomers Proceedings of the R. E. Todres National Academy of Sciences 98, (2001) 976 Fried, E., and Striping of nematic elastomers International Journal of Solids and V. A. Korchagin Structures 39, (2002) 977 Riahi, D. N. On nonlinear convection in mushy layers: Part I. Oscillatory modes of convection Journal of Fluid Mechanics 467, (2002) 978 Sofronis, P., I. M. Robertson, Y. Liang, D. F. Teter, and N. Aravas 979 Fried, E., M. E. Gurtin, and K. Hutter Recent advances in the study of hydrogen embrittlement at the University of Illinois Invited paper, Hydrogen Corrosion Deformation Interactions (Sept , 2001, Jackson Lake Lodge, Wyo.) A void-based description of compaction and segregation in flowing granular materials Continuum Mechanics and Thermodynamics, in press (2003) Oct Nov Nov Jan Jan Jan Jan Feb Apr Apr May 2001 May 2001 June 2001 June 2001 July 2001 Aug Aug Aug Sept Sept Sept. 2001

14 List of Recent TAM Reports (cont d) No. Authors Title Date 980 Adrian, R. J., S. Balachandar, and Z.-C. Liu Spanwise growth of vortex structure in wall turbulence Korean Society of Mechanical Engineers International Journal 15, (2001) 981 Adrian, R. J. Information and the study of turbulence and complex flow Japanese Society of Mechanical Engineers Journal B, in press (2002) 982 Adrian, R. J., and Observation of vortex packets in direct numerical simulation of Z.-C. Liu fully turbulent channel flow Journal of Visualization, in press (2002) 983 Fried, E., and Disclinated states in nematic elastomers Journal of the Mechanics R. E. Todres and Physics of Solids 50, (2002) 984 Stewart, D. S. Towards the miniaturization of explosive technology Proceedings of the 23rd International Conference on Shock Waves (2001) 985 Kasimov, A. R., and Spinning instability of gaseous detonations Journal of Fluid Stewart, D. S. Mechanics (submitted) 986 Brown, E. N., Fracture testing of a self-healing polymer composite Experimental N. R. Sottos, and Mechanics (submitted) S. R. White 987 Phillips, W. R. C. Langmuir circulations Surface Waves (J. C. R. Hunt and S. Sajjadi, eds.), in press (2002) 988 Gioia, G., and Scaling and similarity in rough channel flows Physical Review F. A. Bombardelli Letters 88, (2002) 989 Riahi, D. N. On stationary and oscillatory modes of flow instabilities in a rotating porous layer during alloy solidification Journal of Porous Media 6, 1 11 (2003) 990 Okhuysen, B. S., and Effect of Coriolis force on instabilities of liquid and mushy regions D. N. Riahi during alloy solidification Physics of Fluids (submitted) 991 Christensen, K. T., and R. J. Adrian Measurement of instantaneous Eulerian acceleration fields by particle-image accelerometry: Method and accuracy Experimental Fluids (submitted) 992 Liu, M., and K. J. Hsia Interfacial cracks between piezoelectric and elastic materials under in-plane electric loading Journal of the Mechanics and Physics of Solids 51, (2003) 993 Panat, R. P., S. Zhang, Bond coat surface rumpling in thermal barrier coatings Acta and K. J. Hsia Materialia 51, (2003) 994 Aref, H. A transformation of the point vortex equations Physics of Fluids 14, (2002) 995 Saif, M. T. A, S. Zhang, A. Haque, and K. J. Hsia 996 Fried, E., and M. E. Gurtin Effect of native Al 2 O 3 on the elastic response of nanoscale aluminum films Acta Materialia 50, (2002) A nonequilibrium theory of epitaxial growth that accounts for surface stress and surface diffusion Journal of the Mechanics and Physics of Solids 51, (2003) 997 Aref, H. The development of chaotic advection Physics of Fluids 14, (2002); see also Virtual Journal of Nanoscale Science and Technology, 11 March Christensen, K. T., and The velocity and acceleration signatures of small-scale vortices in R. J. Adrian turbulent channel flow Journal of Turbulence, in press (2002) 999 Riahi, D. N. Flow instabilities in a horizontal dendrite layer rotating about an inclined axis Journal of Porous Media, in press (2003) 1000 Kessler, M. R., and S. R. White Cure kinetics of ring-opening metathesis polymerization of dicyclopentadiene Journal of Polymer Science A 40, (2002) Point defects in nematic gels: The case for hedgehogs Proceedings of the National Academy of Sciences (submitted) 1001 Dolbow, J. E., E. Fried, and A. Q. Shen 1002 Riahi, D. N. Nonlinear steady convection in rotating mushy layers Journal of Fluid Mechanics 485, (2003) Sept Oct Oct Oct Oct Oct Nov Nov Nov Nov Dec Dec Dec Jan Jan Jan Jan Jan Jan Feb Feb Feb Mar. 2002

15 List of Recent TAM Reports (cont d) No. Authors Title Date 1003 Carlson, D. E., E. Fried, The totality of soft-states in a neo-classical nematic elastomer Mar and S. Sellers Journal of Elasticity 69, (2003) with revised title 1004 Fried, E., and R. E. Todres Normal-stress differences and the detection of disclinations in nematic elastomers Journal of Polymer Science B: Polymer Physics 40, (2002) June Fried, E., and B. C. Roy Gravity-induced segregation of cohesionless granular mixtures July 2002 Lecture Notes in Mechanics, in press (2002) 1006 Tomkins, C. D., and Spanwise structure and scale growth in turbulent boundary Aug R. J. Adrian layers Journal of Fluid Mechanics (submitted) 1007 Riahi, D. N. On nonlinear convection in mushy layers: Part 2. Mixed oscillatory and stationary modes of convection Journal of Fluid Mechanics (submitted) Sept Aref, H., P. K. Newton, Vortex crystals Advances in Applied Mathematics 39, in press (2002) Oct M. A. Stremler, T. Tokieda, and D. L. Vainchtein 1009 Bagchi, P., and Effect of turbulence on the drag and lift of a particle Physics of Oct S. Balachandar Fluids, in press (2003) 1010 Zhang, S., R. Panat, and K. J. Hsia Influence of surface morphology on the adhesive strength of aluminum/epoxy interfaces Journal of Adhesion Science and Technology 17, (2003) Oct Carlson, D. E., E. Fried, On internal constraints in continuum mechanics Journal of Oct and D. A. Tortorelli Elasticity 70, (2003) 1012 Boyland, P. L., M. A. Stremler, and H. Aref 1013 Bhattacharjee, P., and D. N. Riahi 1014 Brown, E. N., M. R. Kessler, N. R. Sottos, and S. R. White 1015 Brown, E. N., S. R. White, and N. R. Sottos 1016 Kuznetsov, I. R., and D. S. Stewart 1017 Dolbow, J., E. Fried, and H. Ji Topological fluid mechanics of point vortex motions Physica D 175, (2002) Computational studies of the effect of rotation on convection during protein crystallization Journal of Crystal Growth (submitted) In situ poly(urea-formaldehyde) microencapsulation of dicyclopentadiene Journal of Microencapsulation (submitted) Microcapsule induced toughening in a self-healing polymer composite Journal of Materials Science (submitted) Burning rate of energetic materials with thermal expansion Combustion and Flame (submitted) Chemically induced swelling of hydrogels Journal of the Mechanics and Physics of Solids, in press (2003) 1018 Costello, G. A. Mechanics of wire rope Mordica Lecture, Interwire 2003, Wire Association International, Atlanta, Georgia, May 12, Wang, J., N. R. Sottos, Thin film adhesion measurement by laser induced stress waves and R. L. Weaver Journal of the Mechanics and Physics of Solids (submitted) 1020 Bhattacharjee, P., and Effect of rotation on surface tension driven flow during protein D. N. Riahi crystallization Microgravity Science and Technology 14, (2003) 1021 Fried, E. The configurational and standard force balances are not always statements of a single law Proceedings of the Royal Society (submitted) 1022 Panat, R. P., and K. J. Hsia 1023 Fried, E., and M. E. Gurtin 1024 Dong, F., D. N. Riahi, and A. T. Hsui Experimental investigation of the bond coat rumpling instability under isothermal and cyclic thermal histories in thermal barrier systems Proceedings of the Royal Society of London A, in press (2003) A unified treatment of evolving interfaces accounting for small deformations and atomic transport: grain-boundaries, phase transitions, epitaxy Advances in Applied Mechanics, in press (2003) On similarity waves in compacting media Horizons in Physics, in press (2003) Oct Feb Feb Feb Mar Mar Mar Apr Apr Apr May 2003 May 2003 May 2003

16 List of Recent TAM Reports (cont d) No. Authors Title Date 1025 Liu, M., and K. J. Hsia Locking of electric field induced non-180 domain switching and phase transition in ferroelectric materials upon cyclic electric fatigue Applied Physics Letters, in press (2003) 1026 Liu, M., K. J. Hsia, and M. Sardela Jr. In situ X-ray diffraction study of electric field induced domain switching and phase transition in PZT-5H Journal of the American Ceramics Society (submitted) 1027 Riahi, D. N. On flow of binary alloys during crystal growth Recent Research Development in Crystal Growth, in press (2003) 1028 Riahi, D. N. On fluid dynamics during crystallization Recent Research Development in Fluid Dynamics, in press (2003) 1029 Fried, E., V. Korchagin, Biaxial disclinated states in nematic elastomers Journal of Chemical and R. E. Todres Physics 119, (2003) 1030 Sharp, K. V., and Transition from laminar to turbulent flow in liquid filled R. J. Adrian microtubes Physics of Fluids (submitted) 1031 Yoon, H. S., D. F. Hill, Reynolds number scaling of flow in a Rushton turbine stirred tank: S. Balachandar, Part I Mean flow, circular jet and tip vortex scaling Chemical R. J. Adrian, and Engineering Science (submitted) M. Y. Ha 1032 Raju, R., S. Balachandar, D. F. Hill, and R. J. Adrian 1033 Hill, K. M., G. Gioia, and V. V. Tota Reynolds number scaling of flow in a Rushton turbine stirred tank: Part II Eigen-decomposition of fluctuation Chemical Engineering Science (submitted) Structure and kinematics in dense free-surface granular flow Physical Review Letters, in press (2003) 1034 Fried, E., and S. Sellers Free-energy density functions for nematic elastomers Journal of the Mechanics and Physics of Solids, in press (2003) 1035 Kasimov, A. R., and On the dynamics of self-sustained one-dimensional detonations: D. S. Stewart A numerical study in the shock-attached frame Physics of Fluids (submitted) 1036 Fried, E., and B. C. Roy Disclinations in a homogeneously deformed nematic elastomer Nature Materials (submitted) 1037 Fried, E., and The unifying nature of the configurational force balance Mechanics M. E. Gurtin of Material Forces (P. Steinmann and G. A. Maugin, eds.), in press 1038 Panat, R., K. J. Hsia, and J. W. Oldham 1039 Cermelli, P., E. Fried, and M. E. Gurtin 1040 Yoo, S., and D. S. Stewart 1041 Dienberg, C. E., S. E. Ott-Monsivais, J. L. Ranchero, A. A. Rzeszutko, and C. L. Winter 1042 Kasimov, A. R., and D. S. Stewart 1043 Kasimov, A. R., and D. S. Stewart 1044 Panat, R., K. J. Hsia, and D. G. Cahill (2003) Rumpling instability in thermal barrier systems under isothermal conditions in vacuum Philosophical Magazine (submitted) Sharp-interface nematic isotropic phase transitions without flow Archive for Rational Mechanics and Analysis (submitted) A hybrid level-set method in two and three dimensions for modeling detonation and combustion problems in complex geometries Combustion Theory and Modeling (submitted) Proceedings of the Fifth Annual Research Conference in Mechanics (April 2003), TAM Department, UIUC (E. N. Brown, ed.) Asymptotic theory of ignition and failure of self-sustained detonations Journal of Fluid Mechanics (submitted) Theory of direct initiation of gaseous detonations and comparison with experiment Proceedings of the Combustion Institute (submitted) Evolution of surface waviness in thin films via volume and surface diffusion Journal of Applied Physics (submitted) 1045 Riahi, D. N. Steady and oscillatory flow in a mushy layer Current Topics in Crystal Growth Research (submitted) 1046 Riahi, D. N. Modeling flows in protein crystal growth Current Topics in Crystal Growth Research (submitted) May 2003 May 2003 May 2003 July 2003 July 2003 July 2003 Aug Aug Aug Sept Nov Nov Dec Dec Dec Feb Feb Feb Mar Mar Mar Mar. 2004