Capillaries in alginate gel as an example of dissipative structure formation

Chemical Physics ELSEVIER

Chemical Physics 208 (1996) 9-24

Capillaries in alginate gel as an example of dissipative structure formation J. Thumbs, H.-H. Kohler

*

Institute of Physical and Macromolecular Chemistry University of Regensburg, 93040 Regensburg, Germany

Received 6 November 1995

Abstract Capillaries of unifonn diameter (S-300 J-Lm) are fonned in a gel growing during diffusion of Me 2 + metal ions into a sol of Na alginate. The alginate chains are cross-linked by the metal ions. The transition from sol to gel is limited by diffusion and occurs in a propagating front. Our investigations provide evidence that the dissipative process of structure fonnation in this front is caused by friction between the contracting alginate chains and the surrounding solution, leading to a pattern of hydrodynamic flow similar to Rayleigh-Benard convection. This flow pattern is mapped on the growing gel and leads to a structure of parallel capillaries filled with aqueous solution.We present a reduced mathematical model describing the onset of a spontaneous structure fonnation. The model includes the hydrodynamics of the system fonnulated by a Navier-Stokes equation. In this equation the frictional force between the contracting alginate chains and the solution plays the role of an external force. A second basic equation describes the binding of alginate molecules to the gel front. The model shows that pattern fonnation occurs above a critical value of the contraction velocity. This critical value and the spacing of the resulting pattern depend on a number of parameters such as viscosity, chain density, thickness of the contraction zone, friction coefficient between contracting chains and surrounding solution, diffusion constant of alginate and concentration of the sol.

1. Introduction A capillary gel is formed by diffusion of Me2+ metal ions, e.g. Cu2+, into an aqueous solution of Na alginate [1,2]' Alginate is an anionic linear polysaccharide formed from units of guluronic and mannuronic acid [3]. The transition from sol to gel, which is a diffusion limited reaction, starts at the interface of the two solutions and occurs in a propagating front [4]. The overall process of gel formation can be written as: Na alginate

+ Cu 2 + ;;=!o Cu alginate + Na +

.

The gel is formed by Cu2+ ions cross-linking the alginate chain molecules, see Fig. la,lb [13-15]. The process of gel formation is accompanied by contraction of the alginate chains. In an appropriate concentration range parallel capillaries form within the gel. The capillaries are almost uniform in diameter (8-300 /-Lm, depending mainly on the alginate concentration) [1]. Sections perpendicular to the capillaries show that hexagonal patterns are formed, see Fig. 2. Literature does not provide any clear-cut physicochemical explanation of the process of • Corresponding author. 0301-0104/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved PII S030 1-0 I 04(96)00031·6

10

J. Thumbs. H.-H. Kohler / Chemical Physics 208 (1996) 9-24

a)

sol

1 b)

eu

2

+

gel sol Fig. I. (a) Solution of Cu2+ salt, e.g. CU(S04)' placed on top of a sol of Na alginate. (b) Sol/gel transition induced by diffusion of Cu2+ ions into the sol. Alginate chains are cross-linked by Cu 2 +. A capillary gel is fonned.

capillary formation. Rather vague concepts, such as demixing by small drops, are used [1,10-12]. From our investigations we conclude that capillary formation is not essentially due to buoyancy forces (in drops of alginate gel surrounded by a copper solution capillary formation exhibits radial symmetry) nor to electrical field forces (capillary formation is only weakly affected if potential gradients are short-circuited by the addition of sodium chloride). We have observed a periodic pattern of convective motion of the aqueous solution in the immediate neighbourhood of the gel formation front. This hydrodynamic flow could be made visible under a microscope by adding silica spheres (diameter about 300 nm). If the movement of the silica particles is photographed with a relatively long time of exposure, a picture of the velocity field v is obtained as shown in Fig. 3. Obviously, this convective motion is mapped on the growing gel. In the next section we will present a mathematical model of the process of capillary gel formation. Our basic assumption is that binding and cross-linking of the alginate chains by Cu 2 + ions causes rapid contraction of the chains, giving rise to frictional forces between chains and surrounding solution. We postulate that the force exerted on the solution, above a certain threshold, may lead to the spontaneous formation of a hydrodynamic

Fig. 2. Section of a capillary gel nonnal to the capillary axes. The lateral walls of the capillaries consist of Cu alginate gel. The capillaries are filled with alginate free aqueous solution. Mass fraction of alginate at the beginning of the experiment: 0.2%, diameter of capillaries: 100 ILm, concentration of CU(S04) solution: I mol/I.

J. Thumbs, H.-H. Kohler / Chemical Physics 208 (1996) 9-24

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Fig. 3. Photographic picture of the torus-shaped convection cells in the neighbourhood of the gel fonnation front running horizontally through the center of the picture (section parallel to the capillary axes). The area above the frontline shows alginate sol, the area below capillaries embedded into Cu alginate gel. The front is moving upward in vertical direction.

pattern which is amplified by the structural organization of the gel. The hydrodynamic aspect of our postulate is similar to the situation in heat convection (Rayleigh-Benard phenomenon), where hydrodynamic flow, however, is driven by buoyancy instead of frictional forces [9,16,17]. The aim of our modelling approach is to analyze whether the above postulate is in agreement with physical reality. Seeing that the process of capillary formation is an extremely complex one, we strive for a minimal model containing only the most essential physicochemical features of the structurization process. To this end we will introduce simplifications whenever possible. Moreover, modelling will be limited to the onset of capillary formation. Thus, linearized forms of the basic differential equations describing the transition from the unstructured to the structured state can be used. Clearly, the conclusions drawn from such a reduced model, mainly based on the stability analysis of the unstructured state, will be of a half-quantitative or qualitative rather than of a quantitative nature. With regard to the contraction of the chains in the gel formation layer, we mention that alginate molecules have a rather extended conformation in water [5]. It is known that polymer chains, weakly grafted to a surface, stretch away from the surface for entropic reasons. With an increasing number of links between chain and surface the chains flatten to the surface [6,7]. This is what we call contraction of the chain. It is beyond the scope of this paper to give a detailed description of the propagation velocity of the so1/ gel front. Experiments show that the thickness of the gel layer roughly increases with t 1/ 2 suggesting that propagation of the gel front is limited in time by diffusion of the metal ions [4,8,9]. This time dependence is due to the three-phase character of the system and is quite different from the constant velocity behavior of 'ordinary' chemical waves in single phase solution systems [18,19]. Nevertheless, the time period for transition from the unstructured to the structured state of the gel is short compared with the overall time of gel growth. Hence, for the purpose of modelling this transition, we assume that the velocity of the gel formation front is approximately constant. At the present stage of modelling the precise value of this velocity need not be specified. As pointed out, we will introduce a number of fundamental simplifications. For instance, we will ignore the depletion of alginate molecules in front of the growing gel. (Such depletion necessarily occurs if the alginate concentration is higher in the gel than in the sol [4], cf. Fig. 10). Other simplifications (e.g. neglect of electro-diffusion and buoyancy) are suggested by experimental experience [1,2]. In the end, the mathematical model will include the hydrodynamics of the system, modelled by the Navier-Stokes equation, as well as a material balance of alginate in the gel formation layer, modelled as an ordinary differential equation which includes effects of the gel forming reaction, of diffusion and of convection. Hydrodynamics and reaction are coupled by the frictional force between the alginate chains and the surrounding liquid.

J. Thumbs. H.-H. Kohler / Chemical Physics 208 (1996) 9-24

12

2. Theory 2.1. Basic concepts We assume that the propagating gel front is moving with a constant velocity Vo = Voe z ' where e z is the unit vector in direction z. The model is based on a coordinate system moving with the gel front, see Fig. 4. A mass element moving with velocity v in the resting system is seen with velocity v * = v - Vo in the moving coordinate system. But the model will refer only to relative velocities so that Vo drops out. Therefore, for simplicity, all equations of the model will be written in v instead of v •. The gel front is covered by a layer of thickness Zo in which alginate chains, having come into contact with the gel front, are cross-linked by copper ions. Due to cross-linking they are contracted towards the gel front with a contraction velocity Vc = Vc e z' thus exerting a frictional force f on the surrounding solution. We assume that Vc is constant. The frictional force per volume, f, is written as f = a r( Vc - v), where v = v x ex + v y e y + v z e z is the velocity of the aqueous solution, r the mass fraction of the contracting chain molecules, and a a constant coefficient of friction. Inside the contraction zone (0:0;;; z:o;;; zo) r is assumed not to depend on z, outside (z> zo) it is zero. Note that the value of Vc is negative. 2.2. Basic equations In the following treatment we neglect inertial forces. Then the hydrodynamic behaviour of the aqueous solution is described by the Navier-Stokes equation

(1)

O=-gradp+1]Av+f,

where p is the pressure and 1] the viscosity of the aqueous solution, which is assumed to be constant. V and A have the usual meaning of nabla and Laplace operator. The right hand side of Eq. (0 describes the forces acting per volume element. One of these forces is the frictional force f exerted by the contracting alginate chains.

z

y

x

Fig. 4. Coordinate system of the model system. The front of the growing gel is located in the .xy plane at z = O. Thus the coordinate system is moving together with the gel front in direction z. Between gel (bottom) and sol (top) there is a layer of thickness Zo of chain molecules contracting towards the gel (gel formation layer).

J. Thumbs. H.-H. Kohler / Chemical Physics 208 (1996) 9-24

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Due to incompressibility, the continuity equation reads . av x av y avz dwv=-+-+-=O. ax ay az

(2)

Capillary formation implies time-dependent and lateral changes of the alginate chain mass fraction, r, within the contraction zone. These changes are modelled by 2

ar

- = -ko(r- rIO) + D at

2

( -a2r + -a2r ) ax ay

v z ( x,y,ZO)K, 0 ~ Z ~ zoo

(3)

The first term on the right side describes diffusive exchange of alginate chains between contraction zone and sol, where ko is a diffusion rate constant and rIO is the mean density of the gel, the second term describes diffusion of alginate in lateral directions (x,y) with diffusion constant D. The relation between ko and D will be discussed later. The third term covers binding of alginate ions transported to the contraction zone by convection. vz(x,y,zo) is the z component of the velocity v of the aqueous solution at z = zo; K is a coefficient of convective transport and will increase with the alginate concentration in the sol. Let us assume, for the moment, that Vc is zero. Then f in Eq. (1) is a pure damping force. Therefore, at large t, the system approaches a stable stationary state given by v = 0, f= 0, grad Po = 0 and r = r o, where r 0 = rIO in the contraction zone and r 0 = 0 outside this zone. If Vc is not zero, but small and constant, a similar stable stationary state will develop with v = 0, r = r o, f= grad Po = urovc' However, for Vc sufficiently large, the effect of f will be mainly accelerating. Thus the one-dimensional state might become unstable and pattern formation might occur. To analyze the stability behaviour of the one-dimensional stationary state, we will consider small variations of velocity and alginate density relative to this state, 8v = v and 8r = r - roo 2.3. Stability analysis 2.3.1. Linearized equations These variations v and 8r can be described by the linearized equations following from Eqs.(l) and (3). The linearized form of Eq. (1) is (4)

0= -grad8p+llLl.v+8f,

where and 8 f = 0 for z > zo. Accordingly, Eq. (3) yields 2

a8r

(a 8r

at

8x

2

a 8r)

- = -k D 8r+D --2 +--2

8y

-vzCx,y,Zo)K.

(5)

Variations of the z components of v, v z are introduced as plane waves of wavevector k. k is parallel to the xy plane with components kx and kyo Thus: V

z

= W( z)ei(kxx+kYY)e hl ,

(6)

where W( z) is the z-dependent amplitude and ei(kxx+ k,Y) the complex representation of cos(k x x + k Y y). h is introduced as a complex parameter with real part (T and imaginary part w: h = (T + iw. Thus e hl is a complex representation of e"l cos(wt). Accordingly, variations of the alginate density r are written as 8r = R( z)ei(kxx+k'Y)e hl .

Inside the contraction zone (0 ~ z ~ zo) R( z) is constant and is denoted by R [. Outside R( z) is zero.

(7)

f. Thumbs, H.-H. Kohler / Chemical Physics 208 (1996) 9-24

14

Stability of the one-dimensional stationary solution requires (J" < O. Pattern formation will occur if the absolute value of the contraction velocity IVel reaches a critical value, lugitl, with (J" = 0 (stability border). The phenomenological behaviour of the system suggests that the critical pattern is stationary in time (w = 0). Similar to the case of heat convection [8], stationarity of the critical pattern can be also deduced directly from the basic equations of the model (Eqs. (4) and (5) plus boundary conditions). Thus the critical pattern obeys av aSr - = = 0 or h = O.

at

(8)

at

2.3.2. Stationary solution Spatially periodic convective patterns require rotrot f=l= O. Applying rotrot to the Z component of Eq. (4) and using both Eq. (8) and diu v = 0, we obtain 2 a2Sr a sr) dro auz T)ll.2 u,=-aue ( --2 +--2 +aroll.u,+a--. (9) ax ay dz az Due to rotgrad Sp = 0, pressure variations &p drop out. The right hand side of (9) results from rotrot Sf (see Appendix A). The last term in (9) is only nonzero for z = zo, where ro(Z) steps from rIO to O. With Eqs.(6), (7) and e = + Eq. (9) transforms into 2 d2 d aro) auee a dro dW(z) 2)( k k 2 - - W(z)=--R(z)+-. (10) 2 ( dz 2 dz T) T) T) dz dz

k; k;,

Inside the zone of contraction R( z) is constant and equals R I' With the substitution s( z) = W( z)/ R I we get 2 d2 d auee a dro ds(z) - k 2)( _ _ k 2 _aro) _ s(z) = -R(z) ---+ ----. (11) 2 ( dz 2 dz T) T) RI T) dz dz Let us denote s by s I inside this zone. Because of R( z) = R I and r 0 = rIo = const, Eq. (11) simplifies to (dd:2 -e)(dd:2 -k2-

a~o )SI(Z)= au~e, 0.:;; z.:;;zo'

( 12)

Outside the zone of contraction both R( z) and ro are zero. Denoting S in this region by S2' Eq. (11) simplifies to d2 ( dz 2 -

e )2 S2(Z)=0,

(13)

z>zo·

A particular solution of the inhomogeneous Eq. (12) is found by setting all derivatives of SI(Z) to zero:

aUe Sl,inh

=

T)

k2

( 14)

+ arlO

Using sl,hom = e AZ to solve the homogeneous part of Eq. (12), one obtains A4 _ 2k2A2 - A2

a~lo + e( e + a~lO)

= 0,

This characteristic equation has four real roots: All =k, AI2 = -k,

(15)

J. Thumbs, H.-H. Kohler / Chemical Physics 208 (1996) 9-24

>"'4 = -

_I

15

0: rIO

Vk 2 + ---:;;- .

Therefore, the general solution of Eq. (12) is

s,( z) = A"e kz

+ A'2ekz + A\3 exp(Je + 0: r lO /11

z)

+ A'4 exp( -

ue Je + 0: r lO /11 z) + TJk 20:+o:r , lO ( 16)

with constant coefficients A jr The homogeneous differential equation (13), accordingly, has a characteristic equation >,,4 _ 2>,,2k 2

+e

=

0,

(17)

with roots >"2' = >"22 = k and >"23 = >"24 = -k. Therefore the general solution of Eq. (13) is S2 ( Z) --A 2,e kz+zA 22 e kZ+A 23 e -kz+zA 24 e -kz . kz But e diverges for z -+ 00. Therefore, A2\ = A22 = 0, and

( 18)

(19)

S2(Z) =A23e-kz+zA24e-kz.

The coefficients A,i' A'2' A\3' A'4' A 23 , A24 in Eqs. (16) and (19) are defined by two boundary conditions at z = 0 and four transition conditions at z = zo.The boundary conditions are

s,(O)

= 0,

(20)

ds,(O)

---=0 (21) dz ' where Eq. (21) is obtained from Eq. (2) with the "sticking condition" u/x,y,O) = u/x,y,O) = O. The transition conditions are (22)

S2( zo) = s,( zo)' ds 2( zo) dz

ds,( zo)

d 2 S2( zo)

dz 2 d s,( zo)

dz 2

dz 2

d 3s 2( zo) dz 3

d 3s,( zo) dz 3

(23) (24) 0: ds,( zo) 11

dz

riO'

(25)

Eq. (25) is obtained by integrating Eq. (11) between Zo - e and Zo + e with e -+ O. In this integration all terms of Eq_ (I 1) being at least stepwise steady in Zo drop out. According to Eqs_ (22)-(24), s( z) is steady up to the second derivative. Consequently, the only term to be considered on the left hand side of Eq.(11) is d 4s/dz 4. Thus

zo+e d4s 0: ds( zo) jzo+e drlO --dz 4 j zo-e --dz=dz 11 dz zo-e dz '

(26)

which leads to Eq. (25). With Eqs. (16) and (19), Eqs. (20)-(25) give the following system of algebraic equations:

M·A=B,

(27)

J. Thumbs. H.-H. Kohler / Chemical Physics 208 (1996) 9-24

16

with a ve -b l rIO

All

All

0

AI3

,

A=

B=

AI4

a ve -l b rIO

A23

0 0 0

.414

e'O 'oe'o '5e'0 'o( '5 - a )e'o

0 0

I

-'0'0

'0

M=

= B(~o,a,ve/rlO)'

e-

- 'oe - '0 '5e -'0 -'0('5- a)e-'o

0 0

b

-b

eb be b

e- b

- e-'o

- e-'o

-be- b

'oe - '0 - '5e-'0 ,~e - '0

ao- l)e-'O - 'oC'o - 2)e-'0 '5ao - 3)e- (0

_b 2 e- b

b 2e b b(b 2

-

a)e b

- b(b 2

-

a)e- b

= M('o.a).

Here the abbreviations

have been used. Since B is proportional to ve, A, and hence SI(Z) and Sl(Z), are proportional to ve' More precisely, the value of s( zo) = S I( zo) = Sl( zo) following from the above equations can be written as ve s(zo)=-fo(~o,a), (28) rIO

where fo is a (positive) function of

~o

and a alone.

2.3.3. The critical conditions A stationary solution s( z) must obey not only Eq. (9) but also Eq. (5). With Eq_ (6), Eq. (7), and Eq. (5) yields S( zo)

=

ko K

(1 + Dkko

a8rfat = 0,

2 ).

(29)

On equating Eq. (28) with Eq. (29), one gets

Ivel=

1 e(I+Dk2), fo(~o,a) ko

where the parameter

(30)

e is defined by

rlOko

e=--

(31)

K

If we assume that the diffusion constant of alginate molecules in the contraction zone is independent of direction, ko can be approximated by ko = 2D/z6, as is shown in Appendix B. Then Eq. (30) becomes

IVel =

(

1

fo ~o,a

) e(1 + ~~~).

(32)

J. Thumbs, H.-H. Kohler / Chemical Physics 208 (1996) 9-24

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1,2xl0-4,-----,--------------,

8,Oxl0"

'in

§. 6,Oxl0"

2!: 4,Oxl0" 2,Oxl0" ....... vccnt

0,0 0,0

1,Oxl0'

2,Oxl0'

3,oxl0'

4,Oxl0'

5,Oxl0'

k [11m)

Fig. 5. IVel as calculated from Eq. (33) for typical parameter values: Zo = 10- 5 m (cf. Fig. (10», Cl = 5 X 1011 kg m - 3 s- 1 (according to Ref. [20] the value of Cl in a fully cross-linked gel is about 5 X 10 13 kg m - 3 S - I; since cross-linking is just starting in the contraction zone, we use a smaller value), rIO = 5X 10- 3 and 11 = 0.05 kg m- I s- 1 (typical values of our alginate sols), kD = 10- 2 S-I (with Eq. (44) corresponding to a diffusion constant D of alginate in the contraction zone of 0.5 X 10 - 12m2 s - I), K = 10 2 m - 1 (corresponding to convective exchange between sol and contraction zone with a free alginate mass fraction of 0.001). These values give a = 5 and 9=5XIO- 7 m S-I. In this case the minimum of the curve appears at k~rit=1.20XI05 m- I or ~grit=1.20, the corresponding characteristic wavelength is Acrit = 52 fLm, and, according to Eq. (32), the critical velocity of contraction is vg it = - 6.5 fLm s - I.

Assume that the values of zo, u, rIO' kD' K, T) are given. Then, as discussed above, the critical contraction velocity v~rit is given by the minimum value of IVc I to yield a stationary pattern, i.e., to satisfy Eq. (32). Hence Ivgitl = IVclmin = min

6{1 + Hgrit2) 6{1+1.,2)) ° = 6 min (1+1.'2) ° =

( fo('o,a) 2

2

fo('o,a)

fo{ ,grit ,a)

6

(33)

where the substitution

is used.In (33) the minimum is taken with respect to the free parameter value of '0 is ,grit. The critical pattern has a characteristic wavelength

'0

or, equivalently, to k. The according (34)

An example is shown in Fig. 5. From Eq. (33) it is obvious that ,grit is a function of the reduced friction coefficient a alone so that v~rit is a function of a and 6. It is useful to introduce the reduced critical velocity of contraction v

. Crtt

( a)

Ivgit( a)1 1 6 = go (rCrit )' 0,,0 ,a

(35)

which is a function of a alone. A plot of vcrit(a) is shown in Fig. 6. It is seen that, for a < I, vcrit(a) can be approximated by v Crit (a) "'" 49.8/a. (The number of 49.8 is the analogue of the Rayleigh number in heat convection.) In the original variables, this can be written as

Iv~ritl""'49.8

T)k2D • UZOK

(36)

2.3.4. Size of the critical pattern According to Eq. (34) the wave length of the critical pattern, }.crit, is inversely proportional to ,grit}, where, as pointed out in connection with Eq. (33), ,grit is a function of a. The latter dependence, obtained by minimizing

J. Thumbs, H.-H. Kohler / Chemical Physics 208 (1996) 9-24

18

1~0,---------------------------------,

----vcrit

E>

100

10

1+---~~~~--~~~~~--~~~~

0.D1

0.1

10

a Fig. 6. v cril versus the reduced friction coefficient a on a double logarithmic scale. The dotted line is the approximation v eril

""

49.8/ a.

1.5

1,0

~ 0.5

0,0 0

2

3

4

5

a Fig. 7. ,~ril as a function of the reduced friction coefficient a.

gol(~o,a) with respect to ~o' is shown in Fig. 7. As is seen, ~grit is nearly constant and roughly equals 1.2. Eq.

t1T

(34) thus givesA.crit "'" zo.This leads to the important conclusion that A.crit is approximately proportional to the thickness of the contraction zone. 2.4. The hydrodynamic pattern

Under critical conditions, v z and r form a pattern of plane waves of wavevector k (for simplicity the superscript crit is omitted). k is parallel to the xy plane. According to the definition of s( z) the z-dependent z., 1,0 0,8 ~ 0,6

:;: :;:

0,4 0,2

2,OX10"

4,OX10"

8,OX1a-s

z[m)

Fig. 8. Dependence of W( z). the amplitude of the velocity v z • for the parameter values used in Fig. 5.

J. Thumbs. H.-H. Kohler / Chemical Physics 208 (1996) 9-24

19

x Fig. 9. Streamlines in the xz plane calculated from Eq. (37) with the parameter values used in Fig. 5. The driving force is provided by the layer of contracting chain molecules schematically shown at the bottom with a periodic fluctuation in density.

amplitude of v z , W(z), is given by W(z)=s(z) R I • s(z) is obtained from Eqs. (16) and (19), with A i •j following from Eq. (27). The dependence of W on z is shown in Fig. 8. Without loss of generality, we may assume that k is pointing in direction x. The velocity field v then displays a pattern of parallel rolls with alternating sense of rotation and axes pointing into direction y. The axis to axis distance of the rolls is d = (A/2) = (Tilk). As shown in Appendix C, a streamline passing through a point with x= ±(Ti/2k) and arbitrary z=zs is given by 1

. s( zs)

x(z)= ±-arcsm-(-)-, k s z

Ti

Ti

2k

2k

--~x~-.

(37)

A flow pattern following from this equation is shown in Fig. 9. The according alginate density pattern is the early stage of a gel structure of parallel walls. A solution of the differential equations fonning a hexagonal pattern of torus-shaped convection cells (which can be considered as a first indication to cylindrical capillaries) can be easily constructed by superposing three systems of parallel rolls including angles of 60°. The center to center distance of the convection cells then is d = .fi A.

3. Discussion According to our model capillary fonnation takes place in the neighbourhood of the gel fonnation front. Alginate chains coming from the sol are continuously linked to the existing gel. Subsequently, these chains undergo inter- and intramolecular cross-linking. The process of cross-linking is equivalent to contraction of the chains and leads to an orientation predominantly parallel to the gel body, as schematically shown in Fig. 9. Contraction is enhanced by an increasing supply of cross-linkers, i.e. by increasing copper concentration. The contracting chains exert a frictional force on lhe surrounding solution directed towards the gel phase. Since, according to experimental observations, capillary fonnation cannot be attributed to buoyancy nor, most probably, to electrical forces, we interpret capillary fonnation as a result of this frictional force. According to this interpretation, the appearance of capillaries is immediately related to the formation of a pattern of hydrodynamic flow. Alginate molecules (or ralher chains of these molecules, see below) would accumulate in regions where hydrodynamic flow is directed from the sol to the gel body. Reaching the gel body these molecules are fixed and, being contracted, fonn the capillary walls. The counter-flow, coming from the interior of the capillaries, carries copper ions and free alginate molecules into the sol. As this flow turns back, the fluid

in thr nclet,t,,,rh(x~ of the gel front (section P~nrallel10 the IF. 10. I n l ~ r f ~ n ' n cmlc'roccclplc r pictun. of rontnrllon ,onr tirplc~lon capillary ;~.crc).Thr < h a p o f the ~nlrrfrrrncefrlngrs rhowq thr spatla1 partillon of [he optlral tlencltv and provides quantitative information :1ho111thr mas< drnslly. Stn~gtitfnnprs lncllcatr constant dcnqttv. tlrvintlons to left lndlcatr decrcaslnt! density. SLlfllne from the hulk sol (lop) the ~ I p l n N cticnelly IS w r n 111 tlrcrcasc in a ronr of strong clcplrtion, ~t distance of a b u t 10 Wm from the sol/ecl lmnt the tlrns~tystarts lo Incrraqc lowanic the prl, mils meion corrrrp,nclr to the rontnction Tone. For ccrmpan" with a normal microcro~lc is plrturr, thr ~ntrrfrrcncrfrinpn arc h d r t l nut on thc npht. Thc d~rtancrlvctwcen two adjacent fnngrs In the UI'Frr Part of lhc 10 p m .

is enriched hv atlditional alginatc molecules. Therefore. a linkage of alginate molecules by c o p p r ions to long chains may occur prior to the suhscquent fixation and contraction of this chain to the gel body. So c~mpi~ri~tively thick zones of contraction (L,) may he formed. Recent observations of the gel formation front during capillary growth hy microscope interferometry revealed that the value L,-, is approximately 10 h m (see Fig. 10). As discussed in the previous chapter, the hydrodynamic flow may organize in torus shaped convection cells, leading to a structure of straight, parallel capillaries (see Fig. I I a.1 I b). From the stability analysis of the mathematical model we have found that pattern formation requires a critical (minimum) absolute value of the contraction velocity, o:.".' Obviously, the absolute value of the velocity of contraction will increase with increasing copper concentration. Therefore. if all other parameters are fixed, a minimum value of the copper concentration is required to get segregation. Similarly, according to Eq. (36). if all other parameters are fixetl, a minimum value of the alginate concentration is necessary to obtain a sufficiently high value of K . Increasing values of q respectively k, (if all other parameters are fixed) lead to an increase of thc critical threshold ul."'. In any case. to guarantee a sufficiently high value of the contraction velocity, the rate constant for cross-linking of the gel forming molecules should be high and the chains should be rather flexible. On the search of alternative molecules. natural or synthetic. for the formation of capillary gels, these aspect shoultl be taken into account. In our calculations we have coupled k D (rate constant of normal diffusion) and D (diffusion constant of lateral diffusion) by the approximation k, = 2 0 / $ (cf. Eq. (32)). Without such coupling, analysis of the model shows that l/kUi'. i.e. the size of the pattern. increases with increasing values of D ( a and kept constant). Thus, as expected. lateral diffusion opposes segregation. Likewise, if the value of k, is decreased at constant D, lateral diffusion becomes more dominant and ACrl' increases,finally tending to infinity. If, on the other hand, k, is increased. lateral diffusion becomes less dominant and A'"" tends to a minimum value which is not zero but still has the order of magnitude of z,. The capillary structure of alginate gel is an interesting example of a chemically fixed dissipative structure. AS a model. the alginate capillary gel is of interest for capillary and pore formation in biological tissue. Capillarv gels are used in biotechnology for the fixation of cells and microorganisms and in pharmacy for the purpose of

:,

J. Thumbs. H.-H. Kohler / Chemical Physics 208 (1996) 9-24

21

a)

Fig. II. (a) In three-dimensional space the streamlines obtained by superposition of parallel rolls (shown by arrows) cover the surface of a torus. The picture schematically shows the torus corresponding to a pair of medium size streamlines in Fig. 9. (b) Schematical representation of torus cells and capillary structure: spatially organized tori form a hexagonal flow pattern which moves upwards with the front of freshly formed gel. Each torus leaves behind a 'tube' filled with aqueous solution poor of alginate. The tube is surrounded by a wall consisting of gel. Thus a structure of parallel capillaries is formed. The direction of Cu2+ diffusion is from bottom to top.

drug delivery. Attempts are made to use these gels as a porous support for thin membranes in the fields of ultra and hyperfiltration. In this contribution we have described the formation of a capillary gel in a halfquantitative way. Starting from the experimental observation of convective patterns, the modelling approach provides insight concerning the physicochemical nature of capillary formation. In future work, a more complete and more quantitative model should be developed including a detailed description of the mechanisms of cross-linking and contraction and of the role of diffusion of metal ions and alginate (especially alginate depletion). Such a model should be also able to describe the time-dependent propagation of the solj gel front and the coupling between propagation and pattern formation, which has been neglected in the present work.

Acknowledgements Financial support by the Deutsche Forschungsgemeinschaft (DFG) is gratefully acknowledged.

22

J. Thumbs. H.-H. Kohler / Chemical Physics 208 (1996) 9-24

Appendix A Operation rotrot applied to 8/ The z component of rotrot a is given by a (aa x (rotrot a) z = ax

-az- -

2 aa,) a (aa, aa y ) (a a, ax' - ay ay' = - ax;

-az-

+

2 a a,) ai

+

a (aa x az ax

+

aa y ) ay .

(38)

For 0 ~ z ~ Zo we have 8/= ex8rv e e, - arov. Applying rotrot to the first term, the z component becomes 2 a28r a 8r) (rotrot(ex8rv c e z )L= -ave ( - - 2 + - - 2 . (39) ax ay

( 40) Since ro does not depend on x and y, this can be written as 2 av)) (rotrot(exrov)L= -ex ( ro ( a2 v + a u) + -a ( ro-' ax ay az az

----+ ----+

2 2 a2 V z a Vz a Vz ) =-ex ( r ( --+--+-o az 2 ayZ az 2

aro av ) az az '

+ - -z

(41 )

where we have used div v = 0 in the second term. Combining this with Eq. (39), we obtain

( rotrot8f) z = -ex(vc (aZBr axz

Z

+ a 8r) -r ay2

0

.6.

v dro au z ) z dz az '

(42)

which is the right hand side of Eq. (9).

Appendix B Approximation for kD Based on the assumption of isotropic diffusion of alginate molecules in the contraction zone with diffusion constant D, the discontinuous z-dependence of the alginate mass fraction r(z) = Pa/ps (where Pa is the mass of solved alginate per volume and Ps the density of the sol with Ps :::= const) can be replaced by a linear function r' (z) having the same integral over z. Accordingly, 8r' (z) is defined to decrease linearly from 8r' (0) = 28r to 8r' (zo) = O. Thus, the alginate mass diffusion flow (per area) in direction normal to the sol! gel front, j~ass' can be approximated by a8p;

j:;'ass

= -D--

az

a8r'

= -Dps--

az

28r =Dps--'

(43)

Zo

The temporal change of mass fraction r due to this flow is - j~ass(1/zoXl/ps)' In Eq. (5) this quantity is written as -k D 8r. With Eq. (43) for j~ass' comparison gives 2D kd= - z . Zo

J. Thumbs, H.-H. Kohler / Chemical Physics 208 (1996) 9-24

23

Appendix C Calculation of streamlines in the xz plane We consider the stationary case, where k is parallel to the x-axis, i. e. k = k x; k y = O. The velocity v then can be written as v( x,z)

vx ( x,z)e x

=

+ v z ( x,z)e z •

Treating the velocity as a real quantity, we have (see Eq. (6) with h

=

0):

v z ( x,z) = W( z) cos( kx),

( 45)

Assume, without loss of generality vx(O,z) = O. From the equation of continuity we have (see Eq. (2»

av x

av,

= + - ' = O.

div v = -

ax

az

or, with Eq. (45),

avx

-

ax

dW( z) =

-

dz

cos(kx).

Integration gives 1 dW( z)

vx(x,z)=-"k

dz

( 46)

sin(kx).

A path element along a streamline is given by dx = dx

Vx

dt and dz

= V z dt. Thus with Eqs. (45) and (46):

[dW(z)/dz]sin(kx)

Vx

( 47)

kW ( z) cos( kx )

dz

Integration between x s ' x and zs' z yields

In

sine kx) sin(kx.)

or choosing

Xs

W ( z)

= -In---

= ±('IT/2k): 1

x( z)

=

( 48)

W(zs)'

. W( zs)

± "k arcsm W( z) ,

'IT

'IT

--~x~-

2k

~

'" 2k'

( 49)

Because of W(z) = s(z)R 1 Eq. (49) can be also written as x( z)

1

=

. s( zs)

± -arcsm-(-) . k

s z

This solution is periodic in x with period

('IT /

k).

References [I] H. Thiele, Histolyse und Histogenese, Gewebe und ionotrope Gele, Prinzip einer Strukturbildung (Akademische Verlagsgese\lschaft, Frankfurt, 1967). [2] R.J. Schuberth, Dissertation, Regensburg (1992). [3] O. Smidsr0
24 (5] (6] (7] (8] (9] (10] (I J] (12] (13] (14] (15] (16] (17] (18]

J. Thumbs, H.·H. Kohler / Chemical Physics 208 (1996) 9-24

O. Smidsr0d, Carbohydr. Research 13 (1970) 359. E.A. Di Marzio, in: Physics of polymer surfaces and interfaces, ed. I.C. Sanchez (Butterworth-Heinemann, Boston 1992) p. 73. S. Misra and S. Varanasi, Macromolecules 24 (199)) 322. K. Potter, B.J. Balcom, A. Carpenter and L.D. Hall, Carbohydr. Research 257 (1994) 117. T.K. Sherwood and RL. Pigford, Adsorption and extraction (McGraw-Hili, New York, 1952). H.H. Kohler and J. Thumbs, Chern. Ing. Tech. 67 (1995) 489. J. Hartmann, B. Philipp and H.J. Purz, Acta Polymerica 38 (1987) 277. T. Heinze. D. Klemm, F. Loth and B. Philipp, Acta Polymerica 41 (1990) 259. R.M. Hassan, S.A. EI Shatoury, M.A. Mousa and A. Hassan, Eur. Polym. J. 24 (1988) 1173. Z.-Y. Wang, Q.-Z. Zhang, M. Konno and S. Saito, Chern. Phys. Letters 186 (1991) 463. P.-G. de Gennes, Scaling concepts in polymer physics (Cornell University Press, Ithaca, 1979). L.D Landau and E.M. Lifschitz, Lehrbuch der theoretischen Physik, Band IV. Hydrodynamik (Akademie Verlag, Berlin, 1991). S. Chandrasekhar, Hydrodynamic and hydromagnetic stability (Clarendon Press, Oxford, 1968). R.J. Field, in: Oscillations and travelling waves in chemical systems, eds. RJ. Field and M. Burger (Wiley-Interscience, New York 1985) p. 55. (19] P. Gray and S.K. Scott, Chemical oscillations and instabilities (Clarendon Press, Oxford, 1990). (20] D. Hariharan and N.A. Peppas, J. Membrane Sci. 78 (1993) L