Buoyancy and surface-tension-driven convection in hanging-drop protein crystallizer

j. . . . . . . .

ELSEVIER

CRYSTAL GROWTH

Journal of Crystal Growth 165 (1996) 308-318

Buoyancy and surface-tension-driven convection in hanging-drop protein crystallizer R. Savino, R. Monti * Unit~ersit~ degli Studi di Napoli "Federico 11' '. Dipartimento di Scienza e lngegneria dello Spazio "Luigi G. Napolitano", P. le V. Tecchio 80, 1-80125 Napoli, Italy

Received 20 July 1995; accepted 20 November 1995

Abstract

This paper deals with natural and Marangoni convection in hanging (or sitting) drop protein crystallizers. In the pre-nucleation phase the drop is modelled as a mixture of water, precipitating agent and protein, bounded by an undeformable interface with a surface tension exhibiting a linear dependence on the concentrations; axial symmetry is assumed with respect to the drop axis. The post-nucleation phase is modelled assuming a given location of the crystal and appropriate boundary conditions for the concentrations of protein and precipitating agent in the neighbourhood of the crystal and at the drop surface. The final state of the pre-nucleation is used as the initial condition for the post-nucleation phase. The field equations, written in a suitable spherical co-ordinates system, are solved, with appropriate boundary and symmetry conditions, by a numerical algorithm based on finite-difference schemes. The study cases refer to the crystallization of lysozyme in a solution of sodium chloride in water, for two configurations, full-size and half-size geometries. The computations indicate that for these configurations solute transport is dominated by convection and that the convection velocities are one or even two orders of magnitude larger than the characteristic diffusion velocities. In the pre-nucleation phase solute Marangoni effects are negligible for the half-zone geometry but in the full-size geometry they are comparable to buoyancy driven flows. Calculations of buoyancy flows around a growing crystal show that in ground conditions non-uniform concentration gradients may have a detrimental effect on the growth kinetics.

1. I n t r o d u c t i o n

In recent years successful macromolecule crystallization experiments have been performed by investigators using a number of instruments of US and European design. In early vapour diffusion apparatus (VDA) experiments, protein crystals grew larger, displayed more uniform morphologies and yielded

Corresponding author. Fax: + 39 81 5932044.

diffraction data of higher resolution than equivalent crystals grown on earth [1]. More recently, experiments on protein crystallization were performed in the IML-I in the Cryostat [2] and on the IML2, in the European Space Agency (ESA) Advanced Protein Crystallization Facility (APCF) [3]. A group of US investigators has also carried out protein crystal growth experiments on the Russian MIR, using a number of different crystallization devices [4]. An interesting aspect of the results obtained is that when the same protein is crystallized by a variety of different techniques in microgravity,

0022-0248/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved PH S0022-0248(96)00151-0

309

R. Sat,ino, R. Monti/Journal of Cr3'stal Growth 165 (1996) 308-318

a range of crystalline samples could result, pointing out the need for multiple flight experiments and optimization. These experiments have provided persuasive evidence that growth in microgravity can produce protein crystals of larger size, better shape and higher quality when compared to those obtained on earth. On the other hand, although protein crystal growth experiments in microgravity have yielded a variety of encouraging and successful results, one of the basic problems remains the study of the fundamental mechanisms of the crystallization processes. Experimental results have not shown that protein crystals uniformly display improved properties when grown in microgravity. A reason for this result might be disagreement between protein and technique; another reason might be that crystallization conditions have not been optimized for growth in a microgravity environment. Recent developments in protein-crystal-growth fluid dynamics were obtained in Ref. [5], where the interest was focused on the role of convection by numerically modelling the time-dependent

diffusive-convective transport around a crystal in a two-dimensional growth cell bounded by solid walls. In this paper a numerical model of the hangingdrop fluid dynamics is proposed to study the relevance of solute convection in geometrical configurations with free surfaces. The drop is modelled as a mixture of water, precipitating agent and protein, bounded by an undeformable spherical surface, with constant transport properties and with density and surface tension exhibiting a linear dependence on the concentrations. The Boussinesq approximation for the density is considered and the effects of moving surface due to evaporation are supposed negligible (quasi-stationary evaporation). Axial symmetry is assumed with respect to the drop axis. Two geometrical configurations are considered: the half drop and the full drop (Fig. 1). The differences between the two geometries are that the contact boundary with the drop support is a circular disk of radius R, for the half sphere, and is limited to a small spherical cap, for the full size drop (0 < q~ < q~+). The sittingdrop problem is obtained from the same mathemati-

i

liquid-vapour interface ~

y f f

i

u

i

p r e c l l

i ~

q

x u

~ :symmetryaxis

(a)

J+evap~rat~2xn sytmnetryaxis

(b)

Fig.1.Geometryoftheproblemandcoordinatesystem:(a)halfdrop:(b)fulldrop.

Jwevaporation massflux

310

R. Savino, R. Monti/Journal of Co'stal Growth 165 (1996) 308-318

cal formulation as the hanging drop (Fig. l a) but with the sense of gravity reversed (in Fig. la ~p = -rr is the top of a sitting drop or the bottom of a hanging drop). The continuity, Navier-Stokes and mass species equations in the liquid phase together with the appropriate boundary and symmetry conditions are numerically integrated in a staggered grid in spherical co-ordinates system by a numerical algorithm based on finite-difference schemes. The radius of the drop is computed, at each time, by the mass flow rate of evaporation at the drop surface. Two different codes have been implemented for the pre-nucleation and the post-nucleation phases. The first phase is characterized by water evaporation from the surface, causing the concentrations of protein and precipitant to increase in the drop until appropriate values are reached and protein molecules begin to nucleate. The growing crystal phase is modelled assuming a given location of the crystal and appropriate boundary conditions for the concentrations of protein and precipitating agent in the neighbourhood of the crystal. The model is described in Section 2 and the numerical formulation in Section 3. The results are discussed in Section 4. Final comments and conclusions are given in Section 5.

tions of protein and precipitant (mass of the species divided by the total mass of the mixture) p = O o [ 1 - [3~,(cI -Cl0 )

-- [~c2(C2

-- C2o)]

.

(2)

The flow in the liquid phase is governed by the Boussinesq form of the continuity, Navier-Stokes and mass species equations that are written below in the spherical co-ordinates system shown in Fig. 1 (the geometry considered and the axial symmetry suggest to use the (r,q0) co-ordinates in order to easily impose the symmetry and boundary conditions) 1 a(r2u) 1 a( v sin(q))) re a-----r--+ -r sin(~p) - O, atp au

au

[ 1 au

0t

Or

r aqz

--+u--+v[

u 2 r""

= 1) V 2 H - -

r 2

v~

1 ap

r

Po Or

(3a)

) +----

2 a(c sin(~)) ) sin(~p) &p

Ap

+ --g

(3b)

• ir,

Po

at,

0v

--+u--+v at ar

( 1 ac'

u)

1

ap

----+-+--r &p r Po r &p

2 au =12 V2C+ r2 aq0

Ap

r 2 sin(~p)

+--g.i~, 9o

(3c) 2. Fluid-dynamic model

The drop has been modelled as an isothermal mixture of water, precipitant and protein bounded by an undeformable interface with a surface tension exhibiting a linear dependence on the concentrations.

ac 1 --+u at

ac 1

U 0C 1

+--Or r &p

1 a [ (r2 ) = D1177 ~7 1

o" = cr0[l--tY~.(c I --clo ) --%2(c2--c2o)],

(1)

where cl0 and c2o are the initial values of the precipitant and protein mass concentrations. The liquid is supposed to exhibit constant transport properties (viscosity, Ix and diffusion coefficients, Di); the effects of moving interface due to evaporation are considered negligible (quasi-stationary evaporation). For the density (mass per unit volume) a linear dependence is also assumed with the mass concentra-

+

r 2

aC 2

aC 2

--+u at

(3d)

sin(~p) U aC 2

+--ar r aq~

[lO

=D2 7 ~ r

r 2 --

1 +

r 2

sin(~p)

(3e)

R. Savino, R. Monti /Journal of Co,stal Growth 165 (1996) 308-318

where

+ r 2 sin(~p) 0q~ the subscripts (1) and (2) denote the precipitating agent and the protein, u and v are the radial and azimuthal velocity components, v is the kinematic viscosity, D i the diffusion coefficient of the component i. Eqs. (3a)-(3e) must be solved together with the following initial and boundary conditions: Initial c o n d i t i o n s ( t = O) b/ = U = 0 ;

C1 :

C1o ;

(4)

C2 : C2o.

Boundary. c o n d i t i o n s ( t > O)

On the support 0 < q~ _< ,.p*,

r = R for the full drop: 0C 1

u = v = 0;

dr

- 0;

0C 2

- 0,

dr

~p = "rr/2,

r < R for the half drop:

u=v=0;

Oc1 --=0; Oq~

(5a)

Oc2 -0. 0q~

(5b)

u=0;

Ix

[()

R - v/R

0c l

= -~ ~ - ,

(6)

c1 Jw

(7)

Cw

Oc 2 dR c2 - p D 2 --~-r = P C2 --~t = - Jw - - .

(8)

Cw

On the symmetry axis (q~ = O; q~ = w) Ou --=0;

0q~

[~[(C2--

-

kp~/pc2]

=

C~ q ) / C ~ q - - ~ 0 ] =

- - P D 1 - O- n ' Oc2 -- pD2--~--n/(p

(lOa)

c 2 -- pCc2) ,

(10b)

3c I dR - P D I ~ - r = P C ' at

v=0;

The boundary condition u = 0 at r = R is a consequence of the hypothesis of quasi-stationary evaporation. Preliminary orders of magnitude analysis shows in fact that, with the assumed value for the evaporation flux, the ratio between the characteristic surface evaporation velocity and the characteristic diffusion velocity is of the order 10 3-10 4 so that the effects of moving interface due to evaporation can be neglected. In the post-nucleation phase we have considered a simple model consisting in a macroscopic single crystal located near the surface of the hanging drop in correspondence of the symmetry axis (region of maximum probability for nucleation). Since the growth velocity is supposed to be sufficiently slow (the ratio between crystal growth velocity and the diffusion velocity is estimated of the order of magnitude 10-2-10-3) the moving boundary can be safely ignored and no-slip boundary conditions for the velocity at the crystal interface can be assumed. Mass transport and interface kinetics are related using the boundary conditions for the changing concentrations on the crystal surface proposed in Ref. [5] on the basis of experimental results on the growth and kinetics of lysozyme crystals [7]. For the interface precipitant and protein concentrations we assumed pclVf[pC/p

On the liquid interface ( r = R; q~* < q~ < ~r) ~Ovr

311

aT --=0;

Oq~

OC1

--=0;

3qv

aC e

Oq~

--0,

where V f = 8 0 A / s is the face growth rate and k = 0.01 is the segregation coefficient, the kinetic coefficient [3 = 8.25 × 10 8 c m / s , ~o = 2.9 is the width of the zone in which no growth occurs, the superscript " e q " denotes the equilibrium protein solubility at the temperature considered, p~ and p ~ the protein mass density and the total mass density in the crystal. According to Lin et al. [5], we assumed p~ = 0.82 g / c m 3 and pC = 1.233 g / c m 3.

(9) where the water evaporation flux, proportional to the difference between the partial pressures of water on the drop and in the reservoir [6], is assumed equal to Jw = 2 X 10 -7 g / c m 2 s.

3. N u m e r i c a l m o d e l

The numerical solution of the field equations along with the appropriate boundary and symmetry

312

R. Savino, R. Monti /Journal q['Cr3,sml Growth 165 (1996) 308-318

conditions has been obtained using an explicit timedependent method in primitive variables, splitting at each time the computation of the velocity field in two steps. First, an approximate vector field (V *) corresponding to the correct vorticity of the field (but with ~7. V * v~ 0) is obtained from the momentum equation for Vp = 0. This vector is then corrected with the pressure field obtained solving with a SOR (successive over relaxation) iterative method the Poisson equation deriving from the divergence of the momentum, accounting for the continuity equation. The field equations have been discretized using a staggered grid and finite-difference methods. Central differencing scheme was used for the diffusive and convective terms. The length of the time steps was varied during the calculations in such a way to satisfy at each time the numerical stability both for the convective and diffusive terms of the field equations [8]. A uniform mesh was employed with (20 × 20) grid points for the numerical solutions of the prenucleation phase, whereas a finer grid (40 X 40) was used in the post-nucleation phase to accurately solve the field near the crystal. The radius of the drop is computed, at each time, by the integration of the rate of evaporation of the drop. When a new radius is determined at time (n + 1) a uniform mesh is generated assuming as boundary the new position of the surface. To compute the field solution at time (n + 1) the required solution at time (n) in the new grid points is determined by quadratic interpolation formulas using the previously computed values in the grid points used at time (n).

4. Results

and

discussion

Numerical experiments have been carried out for the lysozyme-NaC1 system, which is the most commonly employed reference model in protein crystallization studies so that its relevant physical properties are available in the literature, see, for example, Refs. [9-11]. The values used in the numerical calculations are given in Table 1. In particular the surface-tension derivatives with concentrations are taken from Ref. [11 ].

Table 1 Physical properties of the system (W,% indicate the percentage weights of solutes in the liquid mixture) Physical property Solution density (Po) Kinematic viscosity (v) Salt diffusivity (D I ) Protein diffusivity ( D 2 ) Salt solutal expansion coefficient ( [ 3 ) Protein solutal expansion coefficient ([3c~) Surface-tension derivative with salt concentration (%,) Surface-tension derivative with protein concentration (%~)

Value

l g/cm s 0.015 c m 2 / s 1.5xlO 5 cm~/s

10-~ cm2/s 5 × 10 -~ W t t

1.5xlO 4 W ~% 4 dyn/(cm W~ %) 1.5 dyn/(cm W, %)

In the pre-nucleation phase we have analysed the half- and the full-drop geometries (Figs. la and lb) and compared the "ideal" case of zero-g conditions with the ground (1 g) situation. For the half drop both the cases of hanging and sitting drops have been considered. The initial radius of the drops is R = 2 ram, corresponding to an initial volume of about 16 t*1 for the half drop and 32 t-tl for the full drop. The initial salt concentration is clo=0.029 and the initial lysozyme concentration is Q,, = 0.025. 4.1. Prenucleation phase in a half drop In Figs. 2 and 3 the concentration contours of NaC1 (Fig. 2) and lysozyme (Fig. 3) are shown after 48 h of evaporation for the half drop at zero-g (a), the hanging (b) and the sitting drop (c) at l g. In particular the salt concentration field is illustrated since the NaCI concentration difference is the main driver for buoyancy flow (the salt solutal expansion coefficient is one order of magnitude greater than the protein solutal expansion coefficient, see Table 1). The volume of the drop is about one half of the initial volume and the drop radius has been reduced to about 80% of its initial value. In the purely diffusive situation (zero-g, absence of any buoyancy convection), water evaporates from the drop surface, the surface concentrations of salt and protein become higher and a diffusion front moves from the surface towards the centre of the drop. At this time the diffusion front is completely developed and quasi-

R. Sa~'ino, R. Monti /Journal of Coastal Growth 165 (1996) 308-318

steady conditions have been reached. For this geometrical configuration (angle q~* between the symmetry axis of the drop and the liquid surface contacting the solid wall of the tip equal to 'rr/2) if diffusion conditions prevail the concentrations are homogeneous along the surfaces parallel to the drop surface and there are only radial variations (central symmetry). Under normal gravity conditions, both in the hanging drop and in the sitting drop, the density

(a)

313

Zero-g - Lysozyme concentration lines

(b) Hanging drop - Lysozyme concentration lines

(a)

Zero-g - NaCI concentration lines

y

Level CP C B A 9 8 7 6 5 4 3 2 1

0,048408 0.048377 0.048345 0.048314 0.048282 0.048251 0.048219 0.048188 0.048156 0.048125 0.048093 0.048062

(c) Sitting drop - Lysozyme concentration lines

(b) Hanging drop - NaCI concentration lines

Level C c B

0.055441 0.055435

A

0.055429

9 8 7 8

0.055423 0.055417 0.055411 0.055408

5

0.055400

4 3 2 1

0.055394 0.055388 0.055382 0.055377

Fig. 3. Iso-concentration contours of lysozyme at zero-g for the half drop (a), at 1 g for the hanging drop (b), and sitting drop (c), after 48 h.

(c) Sittin drop - NaCI concentration lines

tt lftttt Fig. 2. Iso-concentration contours of NaCl at zero-g for the half drop (a), at 1 g for the hanging drop (b), and sitting drop (c), after 48 h.

gradients are sources of natural convection, with typical buoyant plumes rising along the symmetry axis and flowing down along the drop surfaces (Fig. 4). Comparing the streamlines of Fig. 4a (hanging drop) with those of Fig. 4b (sitting drop) we recognize that the stream function has a negative sign in the first case and a positive sign in the second case, as consequence of the fact that for the hanging drop the flow along the symmetry axis is from the surface to the solid support, whereas it is opposite for the

R. SaL,ino, R. Monti/Journal of Crystal Growth 165 (1996) 308-318

314

sitting drop. The maximum calculated flow velocities are about Vmax = 5 p~m/s for the hanging drop and 15 l,z m / s for the sitting drop, i.e. one or two orders of magnitude larger than the characteristic diffusive velocities. Indeed, the characteristic diffusive velocities (Di/R) are of the order of magnitude of 0.5 p~m/s for sodium chloride diffusion and 0.05 ~tm/s for lysozyme diffusion. As a consequence solute transport is dominated by convection that alters the concentration fields with respect to the diffusive case (Figs. 2 and 3) and these differences are more pronounced for lysozyme. Whether these perturbations are relevant depends on the crystal growth mechanisms. In particular, due to the convective circulation cell, fluid with lower concentrations flows from the bottom to the top, and fluid with higher concentrations flows down along the drop surface, so that the surface concentrations become higher on the bottom and lower near the top.

a)

5.544E-2 ~

*~

5.542E-2 -

o

5.540E-2 -

n

g

drop

Hanging drop

z

5.538E-2

]

I

rr/2

2rd3

~o

i

I

5rd6

7~

(b) ~ o

4.836E-2 --

...=

4.826E-2 -

tting drop

Hangi?gd9 (a)

(b)

Hanging drop - Streamlines

Level C B

0.00002 -0.0001

A

-0.0002

g

-0.0004

8

0.0005

7

0.0008

8

-0.0008

5

0.0009

4

-0.0010

3

-0.0012

2

-0.0013

1

-0.0014

Level

Sitting drop - Streamlines

PSI

PSi

C

0.0023

B

0.0021

A

0.0019

9

0.0017

8

0.0015

7

0.0013 0.0011 0.0009 0.0007 0.0005 0.0003 0.0001

Fig. 4. Streamlines at 1 g for the hanging (b), and sitting drop (c), after 48 h.

Zero-g

8 t,;: o @

4.816E-2 -

N

o

4.806E-2

rr/2

27z/3

5rr./6

r~

Fig. 5. Profiles of surface concentration of NaCI (a), and lysozyme (b), at z e r o - g for the half drop and at l g for the hanging and sitting drop, after 48 h.

This non-uniformity in the surface compositions is illustrated in Fig. 5, showing the surface concentration profiles of NaCI (a) and lysozyme (b); ~ = w / 2 denotes conditions at the solid support (top of a hanging drop or bottom of a sitting drop) and ~ = w at the symmetry axis (see Fig. 1). In the purely diffusive case (zero-g) the concentrations are uniform along the surface, so that Marangoni effects should be completely absent. In the sitting drop the concentrations are higher for q~ = w / 2 , i.e. near the solid wall; the opposite takes place for the hanging drop. If Marangoni effects are taken into account no relevant differences are observed. This can be ex-

R. Sat~ino, R. Monti / Journal of Co'stal Growth 165 (1996) 308-318

plained by the relatively small surface concentration differences in this geometrical configuration. 4.2. Prenucleation phase in a full drop

In zero-g it would be possible to choose any shape of the drop without the risk of the drop "falling". The case of "almost" full drop, anchored to a support of size small with respect to the drop radius (see Fig. l b), has been considered to compare the results with the half-drop case and to make assessment on the microgravity relevance of the hanging-drop crystallization. Figs. 6 - 8 show the results of numerical computations in the case of a full drop, after 48 h of evaporation. The angle between the symmetry axis and the liquid surface contacting the solid support is q~* = 30 °. In this case, even in the ideal diffusive situation (Figs. 6a and 7a) central symmetry no longer holds and the concentration contours show a distortion with respect to the sphero-symmetrical configuration ( " o n i o n " configuration). The effect of the geometry on the diffusive field causes the surface concentration to be lower in the region close to the top wall than the surface concentrations at the bottom of the drop. If Marangoni effects are taken into account surface-tension-driven flows arise (Fig. 8). The fluid in the drop is driven by the surface-tension gradient, flowing from the bottom of the surface (where the concentrations are higher and the surface

Zero-g - NaCL concentration lines

315

tension lower) towards the syringe tip (where the concentrations are lower and the surface tension higher). In particular the streamlines of Fig. 8 indicate that Marangoni flows are comparable (if not greater than) buoyancy-driven flows. The maximum computed bulk velocities are about V = 50 p , m / s for buoyancy-driven convection, whereas the maximum surface speed, for Marangoni flow, is one order of magnitude greater. This means that for the full-drop geometry, for the assumed values of ~c, surface-tension effects are more important than buoyancy effects. The implication for the design of crystal growth experiments is two-fold. On the one hand, if the dependence of the surface tension on the concentrations is really the one reported in Ref. [11] and there are not the contaminants or surfactants lowering the surface-tension derivatives, microgravity crystal growth experiments with full-drop configurations should not give, in principle, better crystals compared to those grown in earth laboratories. On the other hand, this conclusion should be carefully checked both with fluid-dynamic experiments aimed at investigating the real size of solutal Marangoni effects in lysozyme water mixtures and with accurate numerical computations of the growing crystal growth phase, taking into account surfacetension-driven flows. The first phase of the process terminates when

Hanging drop - NaCI concentration lines

Zero-g with Marangoni effect - NaCl concentration lines ~evel C

(a)

(b)

C B

O 05557 O 05555

A 9 8 7 6 5 4 3 2 1

005553 OO5551 0 05549 005547 005545 005545 0 05541 005539 0 05537 0 0553E

(c)

Fig. 6. Iso-concentrationcontours of NaCI for the full drop at zero-g (a) and l g (b) without Marangoni effect, and at zero-g with Marangoni effect (c), after 48 h.

316 Zero

R. Sat,ino, R. Monti /Journal of Co,stal Growth 165 (1996) 308-318 g

-

Lysozyme concentration lines

Hanging drop Lysozyme concentration lines

Zero-g with Marangoni

-

effect Lysozyme concentration lines

LeJel CP C B A 9

s 7

5 4 3 2 1

0C4Sa [048727 ~ 048654 0 oaase! o 048509 0048436 0,348363 504829C ,:1.048218 o 048145 C O48072 0048

(b)

(a)

j(c)

Fig. 7. lso-concentration contours of lysozyme for the full drop at zero-g (a), and I g (b) without Marangoni effect, and at zero-g with Marangoni effect (c), after 48 h.

conditions for nucleation prevail at some locations inside the drop. The contours of "isoprobability" for the crystal nucleation can be computed on the basis of the couples of values for the concentrations of precipitating agent and protein [6]. Generally the nucleation occurs when the solubility limit is trespassed and the concentrations of protein and salt are close enough to the precipitation conditions, that could be represented by suitable supersaturation conditions. In this case the NaC1 concentration is so uniform throughout the drop that the protein saturation concentration is independent of position, so that

Zero-g with Marangoni effect - Streamlines

the isoprobability contours are essentially rescaled versions of the isoconcentration lines of lysozyme. 4.3. P o s t - n u c l e a t i o n p h a s e

In order to analyse the buoyancy effects in the crystallization phase, numerical experimentations have been performed for the model consisting of a hanging drop with a single crystal located on the bottom, in the region of maximum probability of nucleation. A fine mesh with 40 X 40 grid points was used in order to solve with a good accuracy the

Hanging drop - Streamlines Level PSI C B

0.059285 0054324

A

0.049383

9

0.044442

8 7

Level PSI C

-0,000405

B A

0000903 0001402

0039501

9 8

-0,001900 0 002398

0.034560

7

-0 002896

6

0.029619

6

-0003394

5

0.024678

5

0003893

4

0.019737

4

-0.004391

3

0.014796

2 1

0.009855 0.004914

3 2

0.004889 -0005387

1

-0005886

(a)

(b)

Fig. 8. Streamlinesfor the full drop at zero-g with Marangoni effect (a), and 1g without Marangoni effect (b), after 48 h.

R. Sacino, R. Monti/Journal of Crystal Growth 165 (1996)308-318

fluid-dynamic field in the neighbourhood of the crystal. Since the post-nucleation behaviour is just a continuation of the pre-nucleation phase, the final concentrations and flow fields of the pre-nucleation have been assumed as initial conditions for the postnucleation phase. In Fig. 9 the concentration contours of lysozyme are illustrated some minutes after the nucleation, in zero-g (Fig. 9a) and l g (Fig. 9b). In zero-g the protein concentration at the crystal interface decreases with time, causing a depletion zone with high concentration gradients where protein diffusion mass transfer takes place. In normal gravity conditions this depletion region is distorted by buoyancy-driven convection (Fig. 10a and 10b): the lighter fluid

zero-g Postnucleation - Lysozyme Concentration lines Level CP F

0.045

E

0.043928

D C B

0.042857 0.041785 0.040714

A

0.039642

9 8 7

0.038571 0,0375 0.036428

8 5

0.035357 0.034285

4 3 2 1

0.033214 0.032142 0.031071 0.03

one-g Postnucleation - Lysozyme Concentration lines Level CP F E

0.045 0.043928

D C B A

0.042857 0.041785

9 8 7

0.038571 0.0375 0.036428

6 5 4

0.035357 0.034285 0.033214

3 2 1

0.032142 0.031071 0.03

0.040714 0.039642

Fig. 9. Iso-concentration contours of lysozyme 6 min after the crystal nucleation, at "zero-g" and " 1 g " (without Marangoni effect).

317

Postnucleation - Streamlines Level PSI

11 \ \ 1 ~\ \\ \\ X

\

N

\

\ \\ \

\\

\ \

\ %` \ "%

D

-4,438E-5

C

8

-0,000173 -0.000301

A 9

-0.000430 -0.000559

8

-0.000688

7

-0.000817

6

-O.000946

5 4

-0.001074 -0.001203

3 2

*0.001332 -0.001461

1

-0.001590

.." "%` "-- "" " ""

""

"~

Fig. 10. Streamlines and vector plots near the crystal 6 min after the crystal nucleation, at

"1 g'"

(without Marangoni effect).

surrounding the crystal, with lower protein concentration, rises from the upper face of the crystal along the symmetry axis, whereas the heavier fluid, with higher concentration, flows down along the drop surface towards the lateral face of the crystal. Mass transfer is enhanced by convection and therefore the crystal growth rate increases, but the non-uniformity in the interface concentration gradients around the growing crystal might have a detrimental effect on the growth kinetics. The maximum computed velocities are on the free surface and not in the vicinity of the crystal, as in the case of a crystal growth crystallizer with solid boundaries (Rosenberger [12]). Since the protein concentration at the surface nearest to the crystal is lower than in the surrounding areas, Marangoni effects could play an important role in this phase. We will analyse these effects in future simulations.

318

R. Sat,ino, R. Monti/Journal of Crystal Growth 165 (1996) 308-318

5. Conclusions

Acknowledgements

Numerical solutions of the hanging-drop fluid dynamics have been obtained for a half drop and for a full-size drop, at different conditions corresponding to zero-g and 1 g, with and without Marangoni effect. The results concerning the hanging-drop configuration indicate that on ground, solute transport is dominated by buoyancy convection and that the convection velocities are one or two orders larger than the characteristic diffusion velocities. For the full-drop configuration, for the range of the assumed values of the surface-tension derivatives with concentrations, Marangoni effects are comparable if not more important than buoyancy effects. If this would be true, microgravity crystal growth experiments with full-drop configurations should not give, in principle, better crystals compared to those grown in earth laboratories. However, these conclusions must be carefully checked with measurements of the surface tension and with fluid-dynamic experiments aimed at investigating the real size of solutal Marangoni effects in lysozyme water mixtures. Moreover accurate numerical computations of the crystal growth mechanisms in the presence of surface-tension effects must give a definite answer to the question. In this paper the calculations for the post-nucleation phase have been restricted to the cases of purely diffusion (zero-g) conditions and buoyancy ( l g ) convection, showing that in ground conditions mass transfer is enhanced by convection and therefore affects the crystal growth, causing non-uniform concentration gradients around the growing crystal that may have a detrimental effect on the growth kinetics.

The authors acknowledge the constructive comments of the Referees. The computations were carried out on Convex C38 Computer with the support of the Italian Aerospace Research Center (CIRA) and with the partial support of the Italian Space Agency (ASI).

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