- Email: [email protected]
Magnetic suppression of convection in protein crystal growth processes
Journal of Crystal Growth 232 (2001) 132–137
Magnetic suppression of convection in protein crystal growth processes Jianwei Qia,b, Nobuko I. Wakayamaa,b,*, Mitsuo Atakab,c a
Institute for Advanced Industrial Science and Technology (AIST), 1-1 Higashi, Tsukuba, Ibaraki 305-8565, Japan b Core Research in Evolutional Science and Technology (CREST), JST, Japan c AIST, Kansai, 1-8-31 Midorigaoka, Ikeda, Osaka 563-8579, Japan
Abstract Magnetization force caused by a magnetic field gradient ðFm Þ is a body force and can cause buoyancy. We numerically simulate the natural convection arising from the depletion of protein concentration around a growing protein crystal when an upward magnetization force acts on the solution. The numerical predictions reveal that an upward magnetization force can damp convection. When a magnetic field gradient m20 HðdH=dzÞ ¼ 685 T2/m is applied, the maximum flow velocity is reduced by about 50% and the velocity in the vicinity of the crystal is reduced by about 24%. Due to the low electric conductivity of the solution, the contribution of the Lorentz force is negligible. When m20 HðdH=dzÞ ¼ 1370 T2 =m, the upward magnetization force (Fm rg) damps convection completely. Our study shows a new method of controlling convection in the process of protein crystal formation. # 2001 Elsevier Science B.V. All rights reserved. PACS: 44.25; 47.62; 81.10.Dn; 87.14.Ee Keywords: A1. Biocrystallization; A1. Convection; A1. Magnetic fields; A2. Growth from solutions; A2. Microgravity conditions; B1. Proteins
1. Introduction Producing protein crystals of adequate size and quality is often the ‘‘bottleneck’’ for three-dimensional X-ray structure analysis of protein molecules. Recent crystallization experiments conducted on board of a spaceshuttle have indicated that crystals grown in microgravity environment may be larger, display more uniform *Corresponding author. Institute for Advanced Industrial Science and Technology, 1-1 Higashi, Tsukuba, Ibaraki 3058565, Japan. Tel.: +81-298-61-4519; fax: +298-61-4487. E-mail address: [email protected] (N.I. Wakayama).
morphologies, and yield diffraction data of significantly higher resolutions than the best crystals of these grown on earth [1,2]. Since an obvious difference between the space- and earth-based experiments is the magnitude of gravity and buoyancy force, the role of solute convection around a growing crystal has been of great interest. In this paper, we will report a new method of damping natural convection around a growing protein crystal by applying a magnetic field. Generally, magnetization force acting on a unit volume of material is [3] 1 1 F m ¼ m0 wrH 2 ¼ m0 rwg rH 2 ; ð1Þ 2 2
0022-0248/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 1 ) 0 1 1 4 7 - 2
J. Qi et al. / Journal of Crystal Growth 232 (2001) 132–137
where m0 is the permeability of vacuum (4 107 Henry/m), w is the volume magnetic susceptibility, H is the magnitude of magnetic field strength (H), r is the density and wg is the mass magnetic susceptibility. Magnetization force along a vertical axis z is 1 q F m ¼ m0 rwg H 2 ez ; ð2Þ 2 qz where ez is the unit vector along z-axis. As shown in Eqs. (1) and (2), it is characteristic that magnetization force is a body force and is proportional to density (r) . Most materials such as water, protein crystals, organic compounds, etc. are diamagnetic. For diamagnetic material, wg has a negative value, typically wg 9 109 m3/kg [4] and a weak repulsive magnetic force acts on it. Fig. 1 shows a vertical superconducting magnet and a spatial distribution of magnetic field strength (H). When a container filled with water is set at the position A where a negative magnetic field gradient is applied on water, an upward magnetization force ðF m Þ is exerted on it. The vertical force exerting on a unit volume of water is 1 q 2 F ¼ F m þ F g ¼ r m0 wg H g ez ; ð3Þ 2 qz where g is the magnitude of gravitational acceleration ð.Þ. When a container is located at the position B, in a uniform field, F m is zero. When a container is located at the position C, F m is
133
downward. Thus, it is possible to reduce or increase vertical acceleration by applying a magnetic field gradient and generating a vertical magnetization force. Since 1997, the experiments of protein crystal growth under magnetic field gradients have been conducted [5,6]. Wakayama et al. [5] found that the number of lysozyme crystals segregated from the supersaturated solution was reduced when an upward magnetization force acted on the solution. When the solution was set in a uniform field, the crystal number did not change. Recently, Lin et al. [6] have found that the quality of FBP (fructose1,6-bisphosphatase) crystals segregated from the solution is improved compared with crystals grown in the absence of the field when the solution was set at the position A in Fig. 1. On the other hand, when the solution was set at the position C, the reverse tendency was observed. These phenomena indicate the effect of an upward magnetization force on the growth process of protein crystals. The decrease of vertical acceleration will affect buoyancy and damp the solute convection around a growing protein crystal. Furthermore, under a strong magnetic field, the Lorentz force is expected to exert on fluid motion because the aqueous solution of protein is lowconducting. In this paper, in order to clarify these magnetic effects quantitatively, we numerically simulate solute convection around a growing protein crystal, considering the influence of magnetization force and Lorentz force.
2. Mathematical model
Fig. 1. Experimental setup using a vertical superconducting magnet and the spatial distribution of magnetic field strength (H).
As shown in Fig. 2, we assume a cylindrical crystal (1 mm diameter and 1 mm height) growing from protein solutions (wg 9 109 m3/kg) in a cylindrical container (10 mm diameter and 10 mm in height). The crystal grows at the center of the bottom. Here we consider the axial symmetry system in cylindrical coordinates. The top surface of the solution is treated as a free surface. For other walls and the surface of the crystal, no slip flow conditions are given. The boundary conditions imposed on the concentration field are C ¼ Cl at the crystal surface, C ¼ C1 at the
134
J. Qi et al. / Journal of Crystal Growth 232 (2001) 132–137
magnetic field and flow. Due to the low electric conductivity of the solution, the electric potential and related effects, e.g. Joule heating can be negligible. Therefore, the basic equations used in our simulation are as follows: Equation of continuity: qr=qt þ r ðr*Þ ¼ 0:
ð6Þ
Equation of motion: q* r þ ð* rÞr* ¼ F g rp þ rnr2 * þ F m þ F L : qt ð7Þ Equation of dimensionless solute concentration: Fig. 2. (a) Model of numerical simulation and (b) computational grid.
container side wall, qC=qr ¼ 0 at the axial line ðr ¼ 0Þ, and qC=qz ¼ 0 at the bottom wall and the top surface. The density difference of the solution is assumed to be proportional to the difference in the protein concentration. Hence, the density of the solution can be expressed r1 rl C Cl r rl 1 þ rl C1 Cl ¼ rl ð1 þ GS fÞ;
ð4Þ
where f is the dimensionless concentration as ðC Cl Þ=ðC1 Cl Þ, and GS ¼ r1 rl =rl . The value for Cl will be somewhere between the saturation concentration and the supersaturated concentration ðC1 Þ. For a solution around a growing protein crystal in the presence of an external inhomogeneous magnetic field, the governing equations for flow and solute concentration are Navier–Stokes equations, including the gravity force F g ¼ r.; the magnetization force Fm given in Eq. (2) and the Lorentz force F L . The Lorentz force is given as follows F L ¼ m20 s*HH;
ð5Þ
where s is electric conductivity. The conductivity of 3% NaCl aqueous solution (4.13 O1m1) [7] is used in the simulations. In general, we should consider the effect of the electric potential that takes place in the presence of
qf=qt þ * rf ¼ DS r2 f;
ð8Þ
where DS is solute diffusion coefficient ( 1010m2/s). Estimated other coefficients are GS ¼ r1 rl =rl 0:01 and kinematic viscosity n 106 m2 =s [8]. These equations are solved by the finite volume method combined with the SIMPLEC algorithm [9]. The crystal growth is very slow so that the moving boundary is ignored here. A non-uniform staggered grid of 102 52 shown in Fig. 2 is used to get the good resolution in the region adjacent to the crystal interface where the greatest changes are taking place. Furthermore, we assume that a vertical gradient of H 2 is spatially uniform and that the horizontal magnetic field gradient is negligible.
3. Simulation results First, we numerically simulate convection in microgravity and normal gravity. Fig. 3(a) shows the concentration fields around a crystal at the center of the bottom growing in microgravity (0 g). It is characteristic that the omni-direction and isotropic transport of protein is obtained around the crystal for the diffusion-limited environment of 0 g. On the other hand, under normal gravity and in the absence of magnetic fields, the flow pattern changes drastically, as shown in Fig. 3(b). The solute convection is generated around the crystal due to the lower concentration of protein near the
J. Qi et al. / Journal of Crystal Growth 232 (2001) 132–137
Fig. 3. Velocity vectors and isoconcentration lines (f) for the crystal growing at the center of the bottom under (a) microgravity, (b) normal gravity, (c) a magnetic field gradient, m20 HðdH=dz)=685 T2/m and (d) m20 HðdH=dzÞ=1370 T2/m. The length of the arrow (!) above the figure represents the magnitude of the maximum velocity.
crystal. The lighter solution in the vicinity of the crystal rises and sinks near the wall of a container. The convection will produce steeper concentration gradient around the crystal and dominate over diffusive transport in microgravity. Thus, the convection causes high crystal growth rate and inhomogeneous composition in the crystal, which results in poorer crystal quality. Fig. 4(a) shows the spatial distribution of the vertical velocity along the central z-axis. In normal gravity (&), the maximum velocity, about 1.1 mm/s is observed 1.5 mm above the crystal. Fig. 4(b) shows the vertical velocity along radial direction (z ¼ 0:5 mm). In the vicinity of the crystal, the maximum local velocities, about 45 mm/s was observed. These local maximum velocities are
135
Fig. 4. Vertical velocity distribution (a) along the z-axis at r=0 mm, and (b) along radial direction at z=0.5 mm.
about 4% of the maximum global velocities that always occur above the crystal. Then, the numerical investigation under magnetic field gradients is performed. When a magnetic field gradient (m20 HðdH=dz)=685 T2/m, Fm 0:5 rg) is applied, the spatial distribution of magnetic strength (H) is shown in Fig. 5(a). When both an upward magnetization force and the Lorentz force exert on the solution, the convection is partially suppressed as shown in Fig. 3(c). The maximum velocity (*) is reduced to 0.58 mm/s, while the velocity in the vicinity of the crystal is 32 mm/s as shown in Figs. 4(a) and (b), respectively. In order to estimate the contribution from the Lorentz force, we also numerically simulate the convection, neglecting an upward magnetization force. The results are shown in Fig. 5(b) and the
136
J. Qi et al. / Journal of Crystal Growth 232 (2001) 132–137
Fig. 5. (a) Spatial distribution of magnetic strength (m0 H) when m20 HðdH=dzÞ=685 T2/m. The field strength at the center of the bottom (r ¼ 0, z ¼ 0 mm) is 10 T. (b) Velocity vectors and isoconcentration lines (f) when m20 HðdH=dzÞ=685 T2/m and only the Lorentz force acts on the solution.
maximum velocity, 1.08 mm/s, is nearly the same as 1.1 mm/s under normal gravity. The convection is slightly damped as shown by (*) in Fig. 4. When the conductivity is increased 5 times (21 O1 m1, 29% NaCl aqueous solution) [7], the maximum velocity is calculated to be 1.07 mm/s. These facts indicate that the contribution from the Lorentz force is negligible to damp natural convection of these NaCl aqueous solutions When an external magnetic field is imposed (m20 HðdH=dz)=1370 T2/m) and an upward magnetization force Fm rg, the solute convection is suppressed as shown in Fig. 3(d). The transport process recovers its diffusion limit as in microgravity (Fig. 3(a)). The residual velocity, 0.1 mm/s, is about 0.01% of that in normal gravity. As shown in Fig. 4, with increasing the value of m20 HðdH=dz) and an upward magnetization force, the convection will gradually decrease and nearly approach to the microgravity when m20 HðdH=dz) is about 1370 T2/m.
4. Discussion Generally, when the crystals grow in normal gravity, convection will dominate transport process of solute to the crystal interface. The results of our numerical simulations show that upward
magnetization forces can reduce the natural convection in the vicinity of the protein crystal caused by the concentration gradients. In order to damp natural convection during the process of protein crystal growth, the magnet which provides spatially uniform qH 2 =qz is necessary. The group of National Research Institute for Metals has done the research to construct this kind of magnet [11]. A magnetic field gradient of m20 HðdH=dzÞ 1370 T2/m (Fm Fg ) is necessary to damp convection completely. At present, it is difficult to obtain such value of magnetic field gradients by using a commercially available superconducting magnet. On the other hand, when m20 HðdH=dz)51370 T2/m (Fm 5Fg ), natural convection is proved to be partially suppressed. For example, when a magnetic field gradient of m20H(dH/dz)=685 T2/m is applied, the convection is reduced by about 50%. In such a case, the contribution from an upward magnetization force is much larger than that of the Lorentz force because of the low electric conductivity of the aqueous solution of protein. This reduction of natural convection may be related to the experimental results already carried out about protein crystal growth [5,6]. When an upward magnetization force is applied, the number of crystals segregated from the solution showed the tendency to decrease compared with that without magnetization force [5,6]. Furthermore, the improvement of crystal quality has also been recently found [6]. These phenomena were also observed in crystals formed in a spaceshuttle [1,2]: Of course, in the present method, we must consider the effects of a strong magnetic field on the process of protein crystal growth. In 1990, Kuroda [10] first found that the lysozyme crystals segregated from the solution were magnetically oriented when a magnetic field was applied. In our experiments [5,6], protein crystals was observed to be magnetically oriented. Small tetragonal lysozyme crystals suspending in the solution are magnetically aligned along the direction of a magnetic field when the following relation exists [12]: 1 2 N DE ¼ DKH 4kT; ð9Þ 2 8
J. Qi et al. / Journal of Crystal Growth 232 (2001) 132–137
where DK is the anisotropy of magnetic susceptibility per unit cell, N is the number of molecules in a crystal and k is Boltzmann constant. According to Eq. (9), most lysozyme crystals suspending in the solution will be magnetically oriented when N > 106 (crystal size 0.27 mm) under 10 T, while they will be orientated when N > 108 (1.2 mm) at 1 T. The magnetic orientation of small suspending crystals seems to even improve the crystal quality, judging from the following experimental results [6]: the crystal quality was improved to some extents when the solution was set at the position B in Fig. 1 where no magnetization force acts on the solution.
5. Conclusions The key points of the present study are as follows: 1. Magnetization force caused by a magnetic field gradient is a body force and can cause buoyancy. 2. We numerically simulate the solute convection around a growing protein crystal when an upward magnetization force acts on the solution. The effect of the Lorentz force is also taken into account. 3. When a magnetic field gradient m20HðdH=dz)=685 T2/m (Fm 0.5rg) is applied, the maximum velocity is reduced by about 50%, while the maximum velocity in the vicinity of the crystal is reduced by about 24%. 4. Due to the low electric conductivity of the solution, the contribution of the Lorentz force is proved to be negligible when m20HðdH=dz)=685 T2/m
137
5. When m20 HðdH=dzÞ=1370 T2/m (Fm rg), convection is suppressed completely. The transport process shows its diffusion limit like in microgravity. 6. The convection is damped with increasing the value of m20 HðdH=dzÞ and an upward magnetization force. 7. Our results indicate a new method of controlling natural convection on Earth.
References [1] L.J. DeLucas et al., Science 246 (1989) 651. [2] A. Mcpherson, Trends Biotechnol. 15 (1997) 197. [3] McGraw-Hill Encyclopedia of Science and Technology, 7th Edition, Vol. 10, McGraw-Hill, New York, 1992, p. 315. [4] Kagakubinran (Ed.), Chemical Society of Japan (Table of Chemical and Physical Data), 4th Edition, Maruzen, Tokyo, 1993, p. II445. [5] N.I. Wakayama, M. Ataka, H. Abe, J. Crystal Growth 178 (1997) 653. [6] S.X. Lin, M. Zhou, A. Azzi, G.J. Xu, N.I. Wakayama, M. Ataka, Biophys. Res. Commun. 275 (2000) 274. [7] Kagakubinran (Ed.) Chemical Society of Japan (Table of Chemical and Physical Data), 2nd Edition, Maruzen, Tokyo, 1975, p. II 1184. [8] N. Ramachandran, Ch.R. Baugher, R.J. Naumann, Microgravity Sci. Technol. VIII/3 (1995) 170. [9] J.P. Van Doormaal, G.D. Raithby, Numer. Heat Transfer 7 (1984) 147. [10] R. Kuroda, Master Thesis, Department of Applied Physics, Faculty of Engineering, University of Tokyo, 1990. [11] T. Kiyoshi, O. Ozaki, H. Morita, H. Nakayama, H. Jin, H. Wada, N.I. Wakayama, M. Ataka, Proceedings of 1998 Applied Superconductivity Conference, Palm Desert, CA, USA. [12] M. Ataka, E. Katoh, N.I. Wakayama, J. Crystal Growth 173 (1997) 592.
Magnetic suppression of convection in protein crystal growth processes Jianwei Qia,b, Nobuko I. Wakayamaa,b,*, Mitsuo Atakab,c a
Institute for Advanced Industrial Science and Technology (AIST), 1-1 Higashi, Tsukuba, Ibaraki 305-8565, Japan b Core Research in Evolutional Science and Technology (CREST), JST, Japan c AIST, Kansai, 1-8-31 Midorigaoka, Ikeda, Osaka 563-8579, Japan
Abstract Magnetization force caused by a magnetic field gradient ðFm Þ is a body force and can cause buoyancy. We numerically simulate the natural convection arising from the depletion of protein concentration around a growing protein crystal when an upward magnetization force acts on the solution. The numerical predictions reveal that an upward magnetization force can damp convection. When a magnetic field gradient m20 HðdH=dzÞ ¼ 685 T2/m is applied, the maximum flow velocity is reduced by about 50% and the velocity in the vicinity of the crystal is reduced by about 24%. Due to the low electric conductivity of the solution, the contribution of the Lorentz force is negligible. When m20 HðdH=dzÞ ¼ 1370 T2 =m, the upward magnetization force (Fm rg) damps convection completely. Our study shows a new method of controlling convection in the process of protein crystal formation. # 2001 Elsevier Science B.V. All rights reserved. PACS: 44.25; 47.62; 81.10.Dn; 87.14.Ee Keywords: A1. Biocrystallization; A1. Convection; A1. Magnetic fields; A2. Growth from solutions; A2. Microgravity conditions; B1. Proteins
1. Introduction Producing protein crystals of adequate size and quality is often the ‘‘bottleneck’’ for three-dimensional X-ray structure analysis of protein molecules. Recent crystallization experiments conducted on board of a spaceshuttle have indicated that crystals grown in microgravity environment may be larger, display more uniform *Corresponding author. Institute for Advanced Industrial Science and Technology, 1-1 Higashi, Tsukuba, Ibaraki 3058565, Japan. Tel.: +81-298-61-4519; fax: +298-61-4487. E-mail address: [email protected] (N.I. Wakayama).
morphologies, and yield diffraction data of significantly higher resolutions than the best crystals of these grown on earth [1,2]. Since an obvious difference between the space- and earth-based experiments is the magnitude of gravity and buoyancy force, the role of solute convection around a growing crystal has been of great interest. In this paper, we will report a new method of damping natural convection around a growing protein crystal by applying a magnetic field. Generally, magnetization force acting on a unit volume of material is [3] 1 1 F m ¼ m0 wrH 2 ¼ m0 rwg rH 2 ; ð1Þ 2 2
0022-0248/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 1 ) 0 1 1 4 7 - 2
J. Qi et al. / Journal of Crystal Growth 232 (2001) 132–137
where m0 is the permeability of vacuum (4 107 Henry/m), w is the volume magnetic susceptibility, H is the magnitude of magnetic field strength (H), r is the density and wg is the mass magnetic susceptibility. Magnetization force along a vertical axis z is 1 q F m ¼ m0 rwg H 2 ez ; ð2Þ 2 qz where ez is the unit vector along z-axis. As shown in Eqs. (1) and (2), it is characteristic that magnetization force is a body force and is proportional to density (r) . Most materials such as water, protein crystals, organic compounds, etc. are diamagnetic. For diamagnetic material, wg has a negative value, typically wg 9 109 m3/kg [4] and a weak repulsive magnetic force acts on it. Fig. 1 shows a vertical superconducting magnet and a spatial distribution of magnetic field strength (H). When a container filled with water is set at the position A where a negative magnetic field gradient is applied on water, an upward magnetization force ðF m Þ is exerted on it. The vertical force exerting on a unit volume of water is 1 q 2 F ¼ F m þ F g ¼ r m0 wg H g ez ; ð3Þ 2 qz where g is the magnitude of gravitational acceleration ð.Þ. When a container is located at the position B, in a uniform field, F m is zero. When a container is located at the position C, F m is
133
downward. Thus, it is possible to reduce or increase vertical acceleration by applying a magnetic field gradient and generating a vertical magnetization force. Since 1997, the experiments of protein crystal growth under magnetic field gradients have been conducted [5,6]. Wakayama et al. [5] found that the number of lysozyme crystals segregated from the supersaturated solution was reduced when an upward magnetization force acted on the solution. When the solution was set in a uniform field, the crystal number did not change. Recently, Lin et al. [6] have found that the quality of FBP (fructose1,6-bisphosphatase) crystals segregated from the solution is improved compared with crystals grown in the absence of the field when the solution was set at the position A in Fig. 1. On the other hand, when the solution was set at the position C, the reverse tendency was observed. These phenomena indicate the effect of an upward magnetization force on the growth process of protein crystals. The decrease of vertical acceleration will affect buoyancy and damp the solute convection around a growing protein crystal. Furthermore, under a strong magnetic field, the Lorentz force is expected to exert on fluid motion because the aqueous solution of protein is lowconducting. In this paper, in order to clarify these magnetic effects quantitatively, we numerically simulate solute convection around a growing protein crystal, considering the influence of magnetization force and Lorentz force.
2. Mathematical model
Fig. 1. Experimental setup using a vertical superconducting magnet and the spatial distribution of magnetic field strength (H).
As shown in Fig. 2, we assume a cylindrical crystal (1 mm diameter and 1 mm height) growing from protein solutions (wg 9 109 m3/kg) in a cylindrical container (10 mm diameter and 10 mm in height). The crystal grows at the center of the bottom. Here we consider the axial symmetry system in cylindrical coordinates. The top surface of the solution is treated as a free surface. For other walls and the surface of the crystal, no slip flow conditions are given. The boundary conditions imposed on the concentration field are C ¼ Cl at the crystal surface, C ¼ C1 at the
134
J. Qi et al. / Journal of Crystal Growth 232 (2001) 132–137
magnetic field and flow. Due to the low electric conductivity of the solution, the electric potential and related effects, e.g. Joule heating can be negligible. Therefore, the basic equations used in our simulation are as follows: Equation of continuity: qr=qt þ r ðr*Þ ¼ 0:
ð6Þ
Equation of motion: q* r þ ð* rÞr* ¼ F g rp þ rnr2 * þ F m þ F L : qt ð7Þ Equation of dimensionless solute concentration: Fig. 2. (a) Model of numerical simulation and (b) computational grid.
container side wall, qC=qr ¼ 0 at the axial line ðr ¼ 0Þ, and qC=qz ¼ 0 at the bottom wall and the top surface. The density difference of the solution is assumed to be proportional to the difference in the protein concentration. Hence, the density of the solution can be expressed r1 rl C Cl r rl 1 þ rl C1 Cl ¼ rl ð1 þ GS fÞ;
ð4Þ
where f is the dimensionless concentration as ðC Cl Þ=ðC1 Cl Þ, and GS ¼ r1 rl =rl . The value for Cl will be somewhere between the saturation concentration and the supersaturated concentration ðC1 Þ. For a solution around a growing protein crystal in the presence of an external inhomogeneous magnetic field, the governing equations for flow and solute concentration are Navier–Stokes equations, including the gravity force F g ¼ r.; the magnetization force Fm given in Eq. (2) and the Lorentz force F L . The Lorentz force is given as follows F L ¼ m20 s*HH;
ð5Þ
where s is electric conductivity. The conductivity of 3% NaCl aqueous solution (4.13 O1m1) [7] is used in the simulations. In general, we should consider the effect of the electric potential that takes place in the presence of
qf=qt þ * rf ¼ DS r2 f;
ð8Þ
where DS is solute diffusion coefficient ( 1010m2/s). Estimated other coefficients are GS ¼ r1 rl =rl 0:01 and kinematic viscosity n 106 m2 =s [8]. These equations are solved by the finite volume method combined with the SIMPLEC algorithm [9]. The crystal growth is very slow so that the moving boundary is ignored here. A non-uniform staggered grid of 102 52 shown in Fig. 2 is used to get the good resolution in the region adjacent to the crystal interface where the greatest changes are taking place. Furthermore, we assume that a vertical gradient of H 2 is spatially uniform and that the horizontal magnetic field gradient is negligible.
3. Simulation results First, we numerically simulate convection in microgravity and normal gravity. Fig. 3(a) shows the concentration fields around a crystal at the center of the bottom growing in microgravity (0 g). It is characteristic that the omni-direction and isotropic transport of protein is obtained around the crystal for the diffusion-limited environment of 0 g. On the other hand, under normal gravity and in the absence of magnetic fields, the flow pattern changes drastically, as shown in Fig. 3(b). The solute convection is generated around the crystal due to the lower concentration of protein near the
J. Qi et al. / Journal of Crystal Growth 232 (2001) 132–137
Fig. 3. Velocity vectors and isoconcentration lines (f) for the crystal growing at the center of the bottom under (a) microgravity, (b) normal gravity, (c) a magnetic field gradient, m20 HðdH=dz)=685 T2/m and (d) m20 HðdH=dzÞ=1370 T2/m. The length of the arrow (!) above the figure represents the magnitude of the maximum velocity.
crystal. The lighter solution in the vicinity of the crystal rises and sinks near the wall of a container. The convection will produce steeper concentration gradient around the crystal and dominate over diffusive transport in microgravity. Thus, the convection causes high crystal growth rate and inhomogeneous composition in the crystal, which results in poorer crystal quality. Fig. 4(a) shows the spatial distribution of the vertical velocity along the central z-axis. In normal gravity (&), the maximum velocity, about 1.1 mm/s is observed 1.5 mm above the crystal. Fig. 4(b) shows the vertical velocity along radial direction (z ¼ 0:5 mm). In the vicinity of the crystal, the maximum local velocities, about 45 mm/s was observed. These local maximum velocities are
135
Fig. 4. Vertical velocity distribution (a) along the z-axis at r=0 mm, and (b) along radial direction at z=0.5 mm.
about 4% of the maximum global velocities that always occur above the crystal. Then, the numerical investigation under magnetic field gradients is performed. When a magnetic field gradient (m20 HðdH=dz)=685 T2/m, Fm 0:5 rg) is applied, the spatial distribution of magnetic strength (H) is shown in Fig. 5(a). When both an upward magnetization force and the Lorentz force exert on the solution, the convection is partially suppressed as shown in Fig. 3(c). The maximum velocity (*) is reduced to 0.58 mm/s, while the velocity in the vicinity of the crystal is 32 mm/s as shown in Figs. 4(a) and (b), respectively. In order to estimate the contribution from the Lorentz force, we also numerically simulate the convection, neglecting an upward magnetization force. The results are shown in Fig. 5(b) and the
136
J. Qi et al. / Journal of Crystal Growth 232 (2001) 132–137
Fig. 5. (a) Spatial distribution of magnetic strength (m0 H) when m20 HðdH=dzÞ=685 T2/m. The field strength at the center of the bottom (r ¼ 0, z ¼ 0 mm) is 10 T. (b) Velocity vectors and isoconcentration lines (f) when m20 HðdH=dzÞ=685 T2/m and only the Lorentz force acts on the solution.
maximum velocity, 1.08 mm/s, is nearly the same as 1.1 mm/s under normal gravity. The convection is slightly damped as shown by (*) in Fig. 4. When the conductivity is increased 5 times (21 O1 m1, 29% NaCl aqueous solution) [7], the maximum velocity is calculated to be 1.07 mm/s. These facts indicate that the contribution from the Lorentz force is negligible to damp natural convection of these NaCl aqueous solutions When an external magnetic field is imposed (m20 HðdH=dz)=1370 T2/m) and an upward magnetization force Fm rg, the solute convection is suppressed as shown in Fig. 3(d). The transport process recovers its diffusion limit as in microgravity (Fig. 3(a)). The residual velocity, 0.1 mm/s, is about 0.01% of that in normal gravity. As shown in Fig. 4, with increasing the value of m20 HðdH=dz) and an upward magnetization force, the convection will gradually decrease and nearly approach to the microgravity when m20 HðdH=dz) is about 1370 T2/m.
4. Discussion Generally, when the crystals grow in normal gravity, convection will dominate transport process of solute to the crystal interface. The results of our numerical simulations show that upward
magnetization forces can reduce the natural convection in the vicinity of the protein crystal caused by the concentration gradients. In order to damp natural convection during the process of protein crystal growth, the magnet which provides spatially uniform qH 2 =qz is necessary. The group of National Research Institute for Metals has done the research to construct this kind of magnet [11]. A magnetic field gradient of m20 HðdH=dzÞ 1370 T2/m (Fm Fg ) is necessary to damp convection completely. At present, it is difficult to obtain such value of magnetic field gradients by using a commercially available superconducting magnet. On the other hand, when m20 HðdH=dz)51370 T2/m (Fm 5Fg ), natural convection is proved to be partially suppressed. For example, when a magnetic field gradient of m20H(dH/dz)=685 T2/m is applied, the convection is reduced by about 50%. In such a case, the contribution from an upward magnetization force is much larger than that of the Lorentz force because of the low electric conductivity of the aqueous solution of protein. This reduction of natural convection may be related to the experimental results already carried out about protein crystal growth [5,6]. When an upward magnetization force is applied, the number of crystals segregated from the solution showed the tendency to decrease compared with that without magnetization force [5,6]. Furthermore, the improvement of crystal quality has also been recently found [6]. These phenomena were also observed in crystals formed in a spaceshuttle [1,2]: Of course, in the present method, we must consider the effects of a strong magnetic field on the process of protein crystal growth. In 1990, Kuroda [10] first found that the lysozyme crystals segregated from the solution were magnetically oriented when a magnetic field was applied. In our experiments [5,6], protein crystals was observed to be magnetically oriented. Small tetragonal lysozyme crystals suspending in the solution are magnetically aligned along the direction of a magnetic field when the following relation exists [12]: 1 2 N DE ¼ DKH 4kT; ð9Þ 2 8
J. Qi et al. / Journal of Crystal Growth 232 (2001) 132–137
where DK is the anisotropy of magnetic susceptibility per unit cell, N is the number of molecules in a crystal and k is Boltzmann constant. According to Eq. (9), most lysozyme crystals suspending in the solution will be magnetically oriented when N > 106 (crystal size 0.27 mm) under 10 T, while they will be orientated when N > 108 (1.2 mm) at 1 T. The magnetic orientation of small suspending crystals seems to even improve the crystal quality, judging from the following experimental results [6]: the crystal quality was improved to some extents when the solution was set at the position B in Fig. 1 where no magnetization force acts on the solution.
5. Conclusions The key points of the present study are as follows: 1. Magnetization force caused by a magnetic field gradient is a body force and can cause buoyancy. 2. We numerically simulate the solute convection around a growing protein crystal when an upward magnetization force acts on the solution. The effect of the Lorentz force is also taken into account. 3. When a magnetic field gradient m20HðdH=dz)=685 T2/m (Fm 0.5rg) is applied, the maximum velocity is reduced by about 50%, while the maximum velocity in the vicinity of the crystal is reduced by about 24%. 4. Due to the low electric conductivity of the solution, the contribution of the Lorentz force is proved to be negligible when m20HðdH=dz)=685 T2/m
137
5. When m20 HðdH=dzÞ=1370 T2/m (Fm rg), convection is suppressed completely. The transport process shows its diffusion limit like in microgravity. 6. The convection is damped with increasing the value of m20 HðdH=dzÞ and an upward magnetization force. 7. Our results indicate a new method of controlling natural convection on Earth.
References [1] L.J. DeLucas et al., Science 246 (1989) 651. [2] A. Mcpherson, Trends Biotechnol. 15 (1997) 197. [3] McGraw-Hill Encyclopedia of Science and Technology, 7th Edition, Vol. 10, McGraw-Hill, New York, 1992, p. 315. [4] Kagakubinran (Ed.), Chemical Society of Japan (Table of Chemical and Physical Data), 4th Edition, Maruzen, Tokyo, 1993, p. II445. [5] N.I. Wakayama, M. Ataka, H. Abe, J. Crystal Growth 178 (1997) 653. [6] S.X. Lin, M. Zhou, A. Azzi, G.J. Xu, N.I. Wakayama, M. Ataka, Biophys. Res. Commun. 275 (2000) 274. [7] Kagakubinran (Ed.) Chemical Society of Japan (Table of Chemical and Physical Data), 2nd Edition, Maruzen, Tokyo, 1975, p. II 1184. [8] N. Ramachandran, Ch.R. Baugher, R.J. Naumann, Microgravity Sci. Technol. VIII/3 (1995) 170. [9] J.P. Van Doormaal, G.D. Raithby, Numer. Heat Transfer 7 (1984) 147. [10] R. Kuroda, Master Thesis, Department of Applied Physics, Faculty of Engineering, University of Tokyo, 1990. [11] T. Kiyoshi, O. Ozaki, H. Morita, H. Nakayama, H. Jin, H. Wada, N.I. Wakayama, M. Ataka, Proceedings of 1998 Applied Superconductivity Conference, Palm Desert, CA, USA. [12] M. Ataka, E. Katoh, N.I. Wakayama, J. Crystal Growth 173 (1997) 592.