Langmuir 2005, 21, 8155-8160
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Dynamics in Dialysis Process for Liquid Crystalline Gel Formation Masahiro Nobe and Toshiaki Dobashi Department of Biological and Chemical Engineering, Faculty of Engineering, Gunma University, Kiryu, Gunma 376-8515, Japan
Takao Yamamoto* Department of Physics, Faculty of Engineering, Gunma University, Kiryu, Gunma 376-8515, Japan Received May 10, 2005. In Final Form: June 21, 2005 The processes of gelation and liquid crystalline formation in the dialysis of Curdlan solution have been observed under crossed nicols, and the calcium concentration and pH of the inner solution were traced. The results showed that the gelation and the liquid crystalline formation occurred simultaneously to form liquid crystalline gel (LCG), but the birefringence of the LCG increased even after the gelation, suggesting further ordering of the Curdlan molecules. On the basis of the calcium ion diffusion, a simple theory for the time development of the thickness of the LCG layer was developed. The experimental and theoretical results agree very well until an amorphous gel (AG) ring appears. The whole process was expressed by a master curve by reducing time and distance data for different radius dialysis tubes by those at the final state; a scaling behavior with respect to the dialysis tube radius was found. The experimental analysis for the calcium concentrations and the pH indicates that forming Curdlan LCG with high ordering of Curdlan molecules consists of two steps: the diffusion of calcium ions inducing the ordering of Curdlan molecules and yielding cross-links simultaneously, and the local relaxation of the Curdlan molecules increasing the ordering degree further.
Introduction The unique structures of liquid crystalline gels (LCG) and liquid crystalline elastomers are of current interest both in colloid science and in technological industry.1-8 A variety of preparation schemes for these materials has been developed recently. Conventionally, liquid crystalline molecules are embedded in a gel matrix.1-6 In a previous paper,7,8 we showed that one of the polysaccharides, Curdlan, self-organizes to form an LCG with a refractive index gradient by Curdlan’s conformational change and by cross-linking through calcium ion binding in a dialysis process. In the process of the dialysis of Curdlan dissolved in aqueous sodium hydroxide into aqueous calcium chloride, the outflux of hydroxide anions changes the conformation of the Curdlan molecules from random coil to triple helix due to the pH change, and the influx of calcium cations cross-links the helical Curdlan molecules intermolecularly, resulting in a cylindrical LCG with a refractive index gradient and with an amorphous gel (AG) in alternating layers.7 It is expected that this method can be applied to other semi-flexible macromolecules if the * To whom correspondence should be addressed. E-mail:
[email protected]. Fax: +81-277-40-1026. (1) Kato, T. Science 2002, 295, 2414-2518. (2) Mizoshita, N.; Hanabusa, K.; Kato, T. Adv. Funct. Mater. 2003, 13, 313-317. (3) Suzuki, Y.; Mizoshita, N.; Hanabusa, K.; Kato, T. J. Mater. Chem. 2003, 13, 2870-2874. (4) Ivanova, R.; Lindman, B.; Alexandridis, P. J. Colloid Interface Sci. 2002, 252, 226-235. (5) Gebhard, E.; Zentel, R. Macromol. Chem. Phys. 2000, 201, 902910. (6) Meng, F.; Zhang, B.; Liu, L.; Zang, B. Polymer 2003, 44, 39353943. (7) Dobashi, T.; Nobe, M.; Yoshihara, H.; Yamamoto, T.; Konno, A. Langmuir 2004, 20, 6530-6534. (8) Dobashi, T.; Yoshihara, H.; Nobe, M.; Koike, M.; Yamamoto, T.; Konno, A. Langmuir 2005, 21, 2-4.
mechanism and the dynamics of forming LCG are understood. The purpose of this paper is to observe precisely the gelation and liquid crystalline formation processes, as well as to measure the flow of calcium cations and hydroxide anions to understand the mechanism and the dynamics in the dialysis process. Experimental Section Curdlan was purchased from Wako Pure Chemical Co. Ltd. and used without further purification. The molecular weight of the Curdlan was determined7,9 by viscosity measurement to be 5.9 × 105. Reagent grade sodium hydroxide and calcium chloride and Milli-Q water were used for preparation of Curdlan solutions and extradialytic solutions. All of the sample preparations and measurements were performed at room temperature of around 25 °C, unless otherwise specified. A desired amount of Curdlan was dissolved in 0.3 M NaOH at 5 wt %. To trace the gelation and the liquid crystalline formation processes, we assembled a sample chamber as follows. Two circular glass plates with the radius R, the upper one of which had a small pit for pouring in the sample, were arranged face to face, connected with three supporting Teflon rods that were 1 mm in diameter and 5 mm long, and surrounded with a cellulose acetate dialysis membrane. After the sample was put into the cell, the cell was immersed in 1 L of aqueous calcium chloride at a concentration of 8 g/dL. The front lines of the gel layer and the liquid crystalline layer were easily determined from sharp increases in turbidity and birefringence, respectively. They were cylindrically symmetrical and moved to the center in the dialysis process. The distance between the front line and the dialysis tube was denoted as x. The relationship between the distance x and the dialysis time t was measured for the sample cells with the radius R ) 10, 15, and 20 mm. The measurements were performed at 20 °C for R ) 10 and 20 mm, and 25 °C for R ) 15 mm. To measure the calcium content as a function of dialysis time, 20 mL of 5 wt % Curdlan (9) Nakata, M.; Kawaguchi, T.; Kodama, Y.; Konno, A. Polymer 1998, 39, 1475-1481.
10.1021/la051246q CCC: $30.25 © 2005 American Chemical Society Published on Web 07/29/2005
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Figure 1. The turbidity τ (a) and the birefringence ∆n (b) observed at an outermost LCG layer near the dialysis tube. solution in 0.3 M aqueous sodium hydroxide was subdivided into seven 19.1 mm diameter dialysis tubes and put into 2400 mL of 8 g/dL of calcium chloride bath. The tubes were taken from the bath one by one at 30 min intervals. Then, the Curdlan solution in each tube was diluted 20 times with MilliQ water. The gel was washed five times in pure water, and 1 g of the gel was excised from the outermost part of the LCG layer. Each sample was dissolved in 10 mL of 0.01 M EDTA-4Na with gentle stirring of the solution for 2 h and then mixed with 10 mL of ammonium buffer at pH 10, and a few drops of eriochrome black T (EBT) were added as an indicator. The calcium content of the samples was determined by back-titration using 0.01 M MgCl2. The pH of the inner Curdlan solution was measured with a commercial pH meter (DKK TOA Co., HM-7J, Japan) as a function of dialysis time.
Results When the Curdlan solution in the dialysis tube was immersed into the extradialytic aqueous calcium chloride bath, a transparent gel was produced from the boundary with the bath, and the gel front line moved toward the center. Because the turbidity of the gel was larger than that of the inner Curdlan solution, we could determine clearly the boundary between the gel and the solution. The boundary between the liquid crystalline portion and the isotropic portion was also easily determined from the birefringence observed under crossed nicols. In the experiment, no difference in the front line of the gel and that of the liquid crystal was found, as is shown in Figure 1. Therefore, the gelation and the liquid crystalline formation occur simultaneously to form LCG, although the birefringence of the LCG layer steadily increased further even after the gelation. It is suggested that, in the liquid crystalline formation process, the orderedness of the Curdlan molecules increased locally after the gelation, as a result of the molecules being partially fixed by the cross-linking with calcium cations. In parallel with the latter process of further molecular ordering, a much more turbid layer was formed near the gel front line (inside the LCG layer). The increased turbidity indicates an inhomogeneous amorphous structure. As the time elapsed, the second LCG layer was formed near the gel front line (inside the turbid layer). The birefringence of the second LCG layer was far lower than that of the first LCG layer. It is suggested that the orderedness of the Curdlan molecules is high in the early stage of gelation, and then it becomes lower because of
Figure 2. Time course of LCG thickness for different radii of circular plate of 20 mm (circle), 15 mm (triangle), and 10 mm (square) (a) and that reduced by the values at the final state, that is, distance x divided by the radius R versus time t divided by the time required for complete gelation te (b). The arrow indicates the starting point of AG layer formation.
increasing geometrical restriction,7 even after the reset by the AG formation. Figure 2a shows the time course of the thickness of the gel layer x. At the early stage of the first transparent (LCG) layer formation, x increases proportionally to the square root of time. After the turbid layer (AG) was formed, the growing rate of the gel became significantly slower, and the gelation appeared to stop once. Then gel production occurred at a higher rate, and the second transparent (LCG) layer was created. The time for the completion of gelation was dependent on the radius of the circular glass plate, but the entire behavior of gelation was the same regardless of the radius of the plate. Indeed, in terms of the time scaled by the elapsed time for forming the whole gel, te, the time development of the scaled distance x˜ ) x/R was in reasonable agreement for all of the measurements for plates having different radii, as shown in Figure 2b. Figure 3 shows the time course of the calcium concentration in the outermost LCG layer of the gel (a) and in the inner Curdlan solution (b). The calcium concentration in the LCG layer increased quickly to reach the critical value for forming LCG at the very initial stage and then increased slightly, proportionally to further time, as shown in Figure 3a. By extrapolating the values of Ca2+ to time being zero, we obtained the calcium content required to form gel, FG ) 2.06 × 10-4 mol/g. The linear increase of Ca2+ in the outermost LCG layer of the gel suggests that the curdlan gel captures a part of the calcium cations flowing through the gel, and the captured calcium cations may increase the cross-linking density of the gel. Since the birefringence also increased after the gelation, as discussed in Figure 1, molecular ordering coupled with cross-linking should have taken place after the gelation, too. However, we can neglect the influence of capturing calcium cations on the time development of the gel front line since the amount of the calcium cations captured during gel forming is not so large. The calcium content became a constant of 6.6 × 10-4 mol/g after 24 h dialysis. From the molar weight of the
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Figure 5. Upper view of dialysis system. Fs, F0, F′s, F′0, and F′(r) denote the calcium ion concentrations in the extradialytic solution, in the inner Curdlan solution, in the Curdlan gel at the boundaries with the extradialytic solution and with the inner Curdlan solution, and at the radius r, respectively. R is the radius of the dialysis tube, and x is the distance between the gel front line and the dialysis tube; dθ is the infinitesimal angle.
Theoretical Analysis
Figure 3. Time course of calcium content of outer LCG layer (a) and inner Curdlan solution (b). The arrow indicates the starting point of AG layer formation.
Figure 4. Time course of pH of inner Curdlan solution. The arrow indicates the starting point of AG layer formation.
glucose unit, 161, it is estimated that there exist two calcium ions per three glucose units in the gel at the final state. In contrast, the calcium concentration in the inner Curdlan solution was negligible until the AG was formed, where it increased sharply, as shown in Figure 3b. This suggests that almost all of the calcium ions are used in the LCG formation process, whereas parts of calcium ion flux pass through AG in the AG formation process. Figure 4 shows the time course of pH in the inner Curdlan solution. The pH reduced with time, with a slower rate in the LCG formation process, and with a faster rate in the AG formation process around t ∼ 150 min, and finally became a constant at 12.2 after the AG formation process. The larger decrease in pH in the AG formation process, compared with that in the first LCG formation process, indicates a larger amount of consumption of OHin the AG layer, suggesting a higher degree of helicity of Curdlan molecules. If we assume that this conformation results in the lower calcium content in the AG layer, the remaining calcium ions pass through the AG layer, which is consistent with the higher increase of calcium content in the inner Curdlan solution during the AG formation process. The final value of pH ∼ 12.2 is close to the critical pH required to dissolve Curdlan molecules.
Here, we present a simple theory for the LCG formation process. The process observed in the experiments could be analyzed by kinetic theories based on Fick’s law with an assumption that the gelation is induced by diffusion of calcium cations and proceeds cylindrical symmetrically. The illustration of the experimental system and the notation are given in Figure 5. Fs, F0, F′s, F′0, and FG denote the calcium ion concentrations in the extradialytic solution, in the inner Curdlan solution, in the Curdlan gel at the boundaries with the extradialytic solution and with the inner Curdlan solution, and at the critical concentration for forming the gel, respectively. To discuss the position dependence of the calcium ion concentration in the Curdlan gel, we denote the calcium ion concentration at the distance r from the center of the cylinder by F′(r), where R - x e r e R, and F′(R - x) ) F′0 and F′(R) ) F′s. µs(F), µ0(F), and µG(F) are the chemical potentials of calcium ions in the extradialytic solution, the inner Curdlan solution, and the gel, respectively. We consider the LCG formation process of a disklike gel with the radius R () the radius of the two circular glass plates), paying attention to the time development of the distance x from the dialysis membrane to the gel front line. Let us adopt the following assumptions with respect to the quantities, Fs, F0, F′s,F′0, and FG. (a) All of the calcium cations flowing into the inner Curdlan solution are used up to produce the LCG layer. (b) Although the structure of the LCG layer may vary with r and change with time t, the calcium concentration dependence of the LCG layer free energy is independent of r and t. (c) The LCG layer does not capture the calcium cations as a sink; all of the calcium cations flowing in the LCG layer can arrive at the inner Curdlan solution to realize a steady state. Mathematically, these assumptions require that all of the quantities, Fs, F0, F′s, F′0, and FG should be independent of time; they should be constants. At the boundaries between the gel and the extradialytic calcium chloride solution and between the gel and the inner Curdlan solution, the chemical potentials of calcium cations are balanced:
µ0(F0) ) µG(F′0) µs(Fs) ) µG(F′s)
(1)
The cylindrical symmetry of the system allows us to assume that the calcium cation flux is always along the radial direction. According to Fick’s law, the influx velocity
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of calcium cations into the Curdlan solution along the radial direction at the distance r is expressed as
∂µG(F′(r)) v)k ∂r
(2)
where k is the mobility of calcium cations and is assumed to be a constant in the present article. The flux density vector of the calcium cation is given by
bj ) -je br ∂µG(F′(r)) ∂r
j ) vF′(r) ) kF′(r)
(3)
where b er is the unit vector along the radial direction. At a steady state, we have
div bj ) -
1 ∂rj )0 r ∂r
(4)
centration, FG. For an infinitesimal angle, dθ, the amount of calcium cations flowing into the Curdlan solution during an infinitesimal time dt is expressed as
From the above equation, we obtain
j(r) )
Figure 6. Time course of LCG formation expressed by the reduced time ˜t and the function y˜ (x˜ ) of the reduced distance x˜ . The plots of the data lie on the straight line expressed by y˜ ) 1.62 × 10 -5 ˜t until the AG layer is formed. From the plot, we have K ) 1.62 × 10-5 cm2/s.
C r
(5)
where the parameter C is independent of r and is determined by the boundary condition given by eq 1. Equation 5 is rewritten as
∂µG(F′(r)) C ) kF′(r) ∂r r
(6)
j(R)Rdθdt
The necessary amount of calcium cations for forming a new gel of width dx is expressed as
(FG - F0)(R - x)dxdθ
j(R)Rdθdt ) (FG - F0)(R - x)dxdθ (7)
∫F′F′ µG(F)dF] ) C lnR R- x s
Note that F′(R - x) ) F′0 and F′(R) ) F′s. Equation 8 gives the expression of the constant C. Introducing the calcium cation concentration dependence part, fG(F′), of the free energy per unit volume for the Curdlan gel layer, we have µG(F′) ) ∂fG(F′)/∂F′ and s
0
µG(F)dF ) fG(F′s) - fG(F′0)
(9)
(15)
k[F′sµG(F′s) - F′0µG(F′0) - fG(F′s) + fG(F′0)] R ln R-x
where
K)
k[F′sµG(F′s) - F′0µG(F′0) - fG(F′s) + fG(F′0)] FG - F0
(16)
Introducing a scalded time, ˜t ) t/R2, and a scaled distance, x˜ ) x/R, we have universal expression independent of the gel radius
dx˜ 1 )K 1 dt˜ (1 - x˜ )ln 1 - x˜
Therefore, the constant C is expressed as
C)
1 dx )K dt R (R - x)ln R-x
0
(8)
∫F′F′
(14)
From eq 11 and this one, hence, the time development equation of x is given by
Integrating both sides of the above equation from r ) R - x to r ) R, we have
k[F′sµG(F′s) - F′0µG(F′0) -
(13)
If the thickness of the gel grows up by dx during dt, using eqs 12 and 13, we have
Since ∂µG/∂r ) (∂µG/∂F′)(∂F′/∂r), we have
∂µG(F′) ∂F′ C ) kF′(r) ∂F′ ∂r r
(12)
(17)
The solution of eq 17 is easily obtained and given by
(10)
Hence, the flux density j is given by
k[F′sµG(F′s) - F′0µG(F′0) - fG(F′s) + fG(F′0)] 1 j(r) ) R r ln R-x (11) We assume that the Curdlan solution gels when the concentration of calcium cations exceeds a critical con-
1 1 1 (1 - x˜ )2ln(1 - x˜ ) - x˜ 2 + x˜ ) Kt˜ 2 4 2
(18)
At the early stage of the gelation process, assuming that x˜ 2 , 1, approximately we have
x˜ ) x2Kt˜
(19)
Figure 6 shows the experimental data plotted in terms of the scaled time and the function y˜ ) y˜ (x˜ ) of the scaled distance defined by
Dialysis Process for Liquid Crystalline Gel Formation
1 1 1 y˜ (x˜ ) ) (1 - x˜ )2ln(1 - x˜ ) - x˜ 2 + x˜ 2 4 2
(20)
All the data before the AG formation lie on a straight line. This shows that the relation (eq 18) is valid until the turbid AG layer is formed. We obtained K ) 1.62 × 10-5 cm2/s from the slope of the line. After the formation of the turbid AG layer, the plots of the data deviate from the straight line predicted by the theory. It is suggested that the later stage of gelation cannot be explained only by the simple diffusion theory. The gelation speed during the AG layer formation is slower than that predicted by the diffusion theory. Thus, the rate-determining process of the AG layer formation is not the diffusion process of calcium cations, but is probably the nucleation process of the AG structure. In Figure 6, even in the later stage, all the data for the gel radii R ) 10 and 20 mm seem to lie on a single curve, which deviates from the straight line predicted by the theory during the AG layer formation. On the other hand, the data for R ) 15 mm agree with those for R ) 10 and 20 mm in the initial linear stage, but deviate from those after the AG formation characterized by a low slope. These slight deviations might be attributed to the difference of the measured temperature of 20 °C for R ) 10 and 20 mm and 25 °C for R ) 15 mm; that is, initialization of AG formation or nucleation of AG domains could be very sensitive to the temperature. The fair agreement for R ) 10 and 20 mm suggests that in the whole time the relation between the time and the position of the gel front is expressed by the scaling form
t)
R2 g(x/R) K
(21)
where g(z) is the nondimensional scaling function behaving
1 1 1 g(z) ≈ (1 - z)2 ln(1 - z) - z2 + z 2 4 2
g(1) 2 R K
(23)
(24)
the diffusion coefficient, DCa, of calcium cations in the Curdlan gel is defined. Note that the coefficient generally depends on the calcium concentration; DCa ) DCa(F′). Using the cylindrical symmetry, we rewrite it as
∂F′ j ) DCa ∂r
From eq 3, the relation ∂µG/∂r ) (∂µG/∂F′)(∂F′/∂r), and the above, we have
∂µG(F′) ∂F′
DCa(F′) ) kF′
(25)
(26)
Instead of analyzing the above “precisely defined” diffusion coefficient, let us pay attention to the mean diffusion coefficient defined by
D h Ca ≡
∫F′F′
1 F′s - F′0
s
0
DCa(F)dF
(27)
It is rewritten as
D h Ca ) )
The relationship between the completion time obtained experimentally and the radius is shown in Figure 7. The proportional relation te ∝ R2 is verified experimentally. The proportional coefficient is given by g(1)/K ≈ 1.55 × 104 s/cm2. Therefore, we have g(1) ≈ 0.251. The proportional behavior te ∝ R2 shown in Figure 7 and the scaling behavior with respect to the completion time te shown in Figure 2b strongly support the scaling expression of eq 21. Let the diffusion coefficient of calcium cations transported into the inner Curdlan solution be analyzed. On the basis of the relation between the flux density and the concentration gradient of the calcium cations
bj ) -DCa grad F′
Figure 7. Relationship between the square of the radius of the circular plate R2 and the complete time te. The plot shows the proportional relation te ∝ R2.
(22)
in the small z region. By the scaling function g, the completion time te is expressed as
te )
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∫F′F′
1 F′s - F′0
s
0
∂µG(F) dF ∂F
kF
∫
F′s k [F′ µ (F′ ) - F′0µG(F′0) - F′ µG(F)dF] ) 0 F′s - F′0 s G s FG - F0 FG K ≈ K (28) F′s - F′0 F′s
Using the dilute limit approximation F′s ≈ Fs, we have
D h Ca ≈
FG K Fs
(29)
The diffusion coefficient is estimated from the above expression and, we have D h Ca ≈ 5.0 × 10-6 cm2/s. The value is around 1/4 of the diffusion coefficient of the calcium cation in pure water at 20 °C obtained by assuming the Einstein-Stokes equation of 1.88 × 10-5cm2/s, which could be attributed to the effect of an excess amount of friction resulting from the Curdlan network of the gel. In conclusion, the gelation and the liquid crystalline formation occur simultaneously to form liquid crystalline gel (LCG), and further ordering of the Curdlan molecules occurs even after the gelation. LCG/AG formation is driven by the diffusion of calcium cations in the dialysis process. A simple theory based on the calcium cation diffusion explains the time development of the thickness of the LCG layer until an amorphous gel (AG) ring appears very well. The whole thickening process was expressed by a master curve by reducing time and distance data for different radius dialysis tubes by those at the final state; a scaling behavior with respect to the dialysis tube radius was found.
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Acknowledgment. We are grateful to Professor Akira Konno in Senri Kinran University for his continuous encouragement. This work was partly supported by Grantin-Aid for Science Research from The Ministry of Educa-
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tion, Culture, Sports, Science and Technology in Japan (Grant No. 16540366). LA051246Q
