Diffusivity measurements using holographic laser interferometry in a cubic lipid-water phase

CPL Chemistry and Physics of Lipids 84 (1996) l 12

ELSEVIER

CHEMISTRY AND

PHYSICS OF LIPIDS

Diffusivity measurements using holographic laser interferometry in a cubic lipid-water phase Charlotte Mattisson a,*, Tommy Nylander b,c, Anders Axelsson a, Guido Zacch? ~Chemical Engineering L Chemical Center, University of Lund, P.O. Box 124, S-221 O0 Lund, Sweden bFood Technology, Chemical Center, University of Lund, P.O. Box 124, S-221 O0 Lund, Sweden ~Physical Chemistrv 1, Chemical Center, University o/' Lund, P.O. Box 124, S-221 O0 Lund, Sweden

Received 18 September 1995; revised 14 June 1996; accepted 5 July 1996

Abstract

The diffusion of glucose in a cubic lipid-water phase was studied with holographic laser interferometry. The X-ray diffraction data showed that the structure of the cubic phase could be described as a primitive cubic lattice under the experimental conditions used. The present study shows that holographic laser interferometry is suitable for determining the concentration profile due to diffusion, even for such intriguing structures as cubic lipid-water phases. The concentration profile obtained revealed that the diffusion process can be described by Fick's Second Law, and the effective diffusion coefficient was obtained by fitting a theoretical concentration profile to the experimental data. The diffusion coefficient obtained for glucose was four times smaller than the corresponding coefficient in aqueous bulk solution. The effective diffusion coefficient in the cubic phase is greater than that predicted by models developed for polymer gels. Keywords: Monoolein; Glucose; Diffusion measurements; Cubic phase; Holographic laser interferometry: Concentration profile

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

The study o f diffusion processes in p o r o u s media is essential for the understanding o f processes occurring in nature, as well as in industrial applications. The transport o f molecules t h r o u g h a

* Corresponding author. Tel: + 46 46 2223618.

m e d i u m is principally determined by the microstructure o f the medium. This structure can be r a n d o m l y organised as in, for example, polymer gels, or highly organised such as the structures formed by some lipids in aqueous media. The latter are formed due to the self-assembly o f lipid molecules in water. D e p e n d i n g on the molecular shape, the temperature and water content, the lipids can be organised in planar bilayers (the

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2

C. Mattisson et al./Chemistry and Physics of Lipids 84 (1996) 1-12

lamellar phase) as well as in other phases, such as the hexagonal or the cubic phase. The most intriguing structure is the cubic lipid-water phase, in which a curved lipid bilayer is organised in such a way that a continuous water channel system is formed [1]. It is obvious that the transport of water-soluble compounds is facilitated in the cubic phase compared with the lamellar phase. The lipid monoolein forms a cubic lipid-water phase at room temperature. Monoolein has a very low solubility in water and the cubic phase, which is both viscous and transparent, can thus exist in equilibrium with an excess of water. This cubic phase has also been shown to accommodate a wide range of globular proteins [2]. Cubic phases with entrapped enzymes, eg. glucose oxidase, can be used to construct biosensors for the determination of glucose concentration [3]. The response of the biosensor was found to be controlled by the diffusion rate of glucose in the cubic phase. Clearly, it is of great interest to study the diffusion of various substances in the cubic phase, both to reveal the nature of the phase itself and to find new applications. The self-diffusion of water in the cubic phase has previously been studied by NMR [4]. In the present study, we have chosen glucose as a model biomolecule, since it is considered not to interact with the lipid bilayers within the concentration range employed in this study [5]. Furthermore, our earlier work, directed at the understanding of the action of enzymes, e.g. glucose oxidase entrapped in the cubic phase [3], required studies of substrate diffusion in the lipid matrix. In this paper, we present results from measurements of the macroscopic diffusivity by means of holographic laser interferometry, which, to our knowledge, has not previously been used to study mass transport in lipid-water systems. The buildup of a concentration profile in a gel or a liquid, due to diffusion, will result in a corresponding refractive index profile. In the case of the holographic technique an interference pattern is formed, which can be used to measure the concentration-length profile versus time. By means of Fick's law, the data obtained can be used to calculate diffusion coefficients. The method has previously been used to study diffusion in liquids

[6-10], and has also recently been applied to diffusivity measurements in gels [11-13]. The major advantages are that it is a direct in situ method and that the concentration profile, as well as the diffusion coefficient of a solute, can be determined without knowledge of the absolute concentration. Since the diffusion process is visualised, any disturbances during the measurements can easily be identified. Furthermore, any heterogeneities present in the gel structure causing changes in the diffusion process would be observed. The results of the diffusion measurements obtained using holographic interferometry are discussed in relation to self-diffusion measurements made using NMR. The structure of the lipid matrix studied was characterised with X-ray diffraction to determine the type of cubic phase, as well as the dimensions of the unit cell under the experimental conditions used in this study. Furthermore, referring to our earlier study on biosensors, using the cubic monoolein-aqueous phase as a matrix for glucose oxidase, we will also present X-ray data that indicates that the structure of the used cubic phase is retained even if buffer and glucose oxidase is introduced into the aqueous part of the cubic phase. 2. Materials and methods

2.1. Preparation of the cubic lipid-water phase Monoolein (1-monooleyl-glycerol, TS-ED 173), containing 98.1% monoglycerides, was kindly provided by Grindsted Products (Brabrand, Denmark), fl-D-Glucose (G 5250, lot no. 73H0139), glucose oxidase (GOD), type X-S from Aspergillus niger (E.C. 1.1.3.4), lyophilised, with a specific activity of about 100 150 U rag-1 lyophilisate and L-glucose (G 5500, lot no. 34H0959) were purchased from the Sigma (St. Louis, MO, USA). All chemicals used were of analytical grade. Water used was ion-exchanged, distilled and passed through a Milli-Q water purification system (Millipore). The glucose solutions, containing 3.5% glucose (w/w), were prepared from pure water or phosphate-buffered saline. The buffer contained 10 mM phosphate, 0.1 M KC1 and 1 mM EDTA, and the pH was adjusted to 7 by using 1 M KOH.

C. Mattisson et al. / Chemistry and Physics o f Lipids 84 (1996) 1 12

3

11 7

/

8

/5

13

Fig. I. Layout of the equipment used in holographic laser interferometry: (1) laser, (2) camera body used as a shutter. (3) polarisation filters, (4) beam-splitter, (5) mirrors, (6) spatial filters, (7) lenses, (8) diffuser, (9) the cell, (10) holographic plate, (11) screens, (12) CCD camera, (13) computer with image-analysis system.

The cubic phases were prepared by two methods. First, 0.62 g of monoolein was melted at 38°C in a vial, 0.38 g of the glucose solution was added and the mixture was equilibrated for 1 h at 38°C. The sealed vials were then allowed to equilibrate at 25°C for a minimum of 24 h (or until a completely transparent phase was obtained). The cubic phase was transferred into a 5 × 10 × 45 mm spectrophotometric cell and centrifuged at 2000 × g for 30 min at 25°C. The surface of the cubic phase was smoothed with a spatula and the cell was centrifuged for another 15 rain to obtain a plane surface. Alternatively the cubic phase was prepared directly in the spectrophotometric cell by adding 0.36 ml buffer to the cell, which was thermostated to 38°C. The monoolein (0.44 g), also thermostated to 38°C, was slowly added to the buffer. The mixture was equilibrated at 38°C for 1 h before it was centrifuged at 2000 × g for 30 min at 25°C. The cell was sealed and the cubic phase was allowed to equilibrate at 25°C for a minimum of 24 h (or until a completely transparent phase was obtained). The cubic phases, containing 2.5 mg G O D per ml buffer solution, were

prepared according to this method. All the cubic phases were stored in equal volumes of glucose solution of the same composition as that used for the preparation of the cubic phase at 25°C to ensure complete swelling of the cubic phase. The cubic phases used for the N M R self-diffusion measurements were prepared from monoolein, D20 and glucose, using the same molar ratio between D20, glucose and monoolein as in the holographic interferometry experiments, namely 40.5% D20 (w/w), containing 3.36% glucose (w/w), and 59.5% monoolein (w/w). The cubic phases were equilibrated in sealed vials prior to transfer into the N M R tubes, and centrifugation was carried out as described above until all air bubbles had been removed.

2.2. Experimental procedure 2.2.1. Holographic laser &terferometry The experimental set-up used for the holographic laser interferometry measurements is shown in Fig. 1. The set-up is a modification of that described by Gustafsson et al. [13]. The laser

4

c. Mattisson et al. / Chemistry and Physics of Lipids 84 (1996) 1 12

beam from the He-Ne laser (5 mW, Melles Griot type 05-LHP-151) is split into an object beam and a reference beam. After passing through the spatial filter and the lens, the reference beam hits the holographic plate (Agfa 10E75) directly, while the object beam passes through a lens, a diffuser and the cell before hitting the plate. The cell containing the cubic phase was placed in a transparent Plexiglass holder in which temperature control was achieved by circulating thermostated water of temperature 25 + 0.2°C. The holographic plate was exposed to the laser light for 1/8 s. At the time of the exposure the solution above the cubic phase was of the same composition as the aqueous solution within the cubic phase. The holographic plate was developed and reinserted in exactly the same position as at the time of exposure. The holographic image and the real object were thus superimposed upon each other. The solution above the cubic phase was then replaced with pure water and the diffusion process was started. Care was taken not to move the cell out of position during the experiment. The diffusion process results in changes in the refractive index along the cell. The optical path of the object beam passing through the cell is thus changed, while the reconstructed beam arising from the hologram is unaffected. Thus, when the cell is observed through the holographic plate a number of interference fringes appear in the cubic phase as well as in the liquid above it (Fig. 2). The fringes move away from the cubic phase-aqueous phase interface. Seven experiments were performed and concentration profiles were measured after 3 and 4 h. This time was needed to get a good resolution of the fringes. The interference pattern was recorded with a video camera (Mintron CCD Camera MTV1802CB) connected to an image-analysis system (Overlay Frame Grabber, Model 768, Imaging Technology Inc. Bedford, MA, USA) and evaluated with Optimas 5.0 image analysing software (Optimas Corporation, Seattle, Washington USA). In some cases the interference fringes where inclined due to difficulties in preparing a smooth interface between the cubic and the aqueous phase. Thus the distances from the interface to each fringe, x, was measured in the

center of the cell. The mean of three measurements of each distance was used for the calculations. 2.2.2. X - r a y diffraction

The equilibrated samples were investigated with low angle X-ray diffraction using a DPT camera with Ni-filtered K7 radiation (2 = 1.542 /k) [14]. The cubic phase (a few mg) was placed in a sample holder constructed according to Hernqvist [15]. An exposure time of 24 h was used. 2.2.3. N M R

self-diffusion m e a s u r e m e n t s

The self-diffusion coefficients, D, were determined using the pulsed field gradient spin-echo technique PGSE [16] in the improved Fourier transform mode, as suggested by Stilbs [17]. The spectrometer, JEOL FX 100, was equipped with a home-built pulsed magnetic field gradient unit. The FT-PGSE measurements were performed at the controlled temperature of 2 5 _ I°C by varying the length of the gradient pulse (6) and with a constant gradient pulse interval A = 140 ms. The decay of the echo intensity (I) is given by the relation [16,17]:

©

Fig. 2. Interferencepattern from experiment no. 7 recorded after 4 h. The position of the interface is marked with an arrow and the scale is indicated with a bar.

C. Mattisson et al. / Chemistry and Physics of Lipids 84 (1996) I 12

Interface x=O

xi

5

aqueous phase is zero (6'2o) and is uniform in the water channel system of the cubic phase (G0). The effective diffusion coefficient for glucose in the cubic phase is Dlxff. In the aqueous phase the diffusion coefficient for glucose is D> Both phases are considered to be semi-infinite and the diffusion coefficients are regarded as constant. The diffusion process under the given conditions is assumed to be. described by Fick's second law, which in one dimension is given by ~C 8t

~2C D 8x 2

(2)

I

°°',%%.

The solution to Eq. (2), under the conditions used in this study, gives the glucose concentration, C~, in the aqueous part of the cubic phase [19] as /

Go + Czo D/-K-G(x, t ) -

Fig. 3. Schematic illustration of the system studied consisting of a cubic phase (1) and an aqueous phase (2). The distance from the phase boundary is x. 6"2 and C 1 denote the concentration of glucose in the aqueous phase and in the aqueous part of the cubic phase, respectively. D 2 is the diffusion coefficient for glucose in the aqueous phase, and DL~f~ is the effective diffusion coefficient for glucose in the cubic phase.

-

" ' L / ~ /l.eff ~

k/ 2 1+ ~/Dl.cn" [D:

el

4

[D~

ok X / _ ~ - C:o~ O~,effertf-

x

\ 2 ~ /

(3) where G is the gradient strength, 7 is the gyromagnetic ratio of the observed nucleus (IH, in this case), and I0 is the signal intensity in the absence of a gradient pulse. The gradient strength was calibrated with a sample of D20 by observing the ~H-NMR signal for H20 and using the known value of D for HDO in D20, i.e. Do = 1.902 x 10 9 m2/s [18] at 25°C, under the same experimental conditions as those used for the samples.

and the glucose concentration in the aqueous phase, C2, as

3. Theory

The partition coefficient, k, is defined as the ratio between Cl and C2 at the final equilibrium between the aqueous part of the cubic phase (1) and the aqueous phase (2). The structure of the cubic phase is such that the partition coefficient is considered to be unity at final equilibrium. Eqs. (3) and (4) give the theoretical concentration-length profiles versus time. As diffusion progresses, the

f

ok + C2ok./-~D--22

C1

" ' / J ~ / 1,eft" •

C2(x, t) =

l+k

/_/)2 x/Dlxn

[62o-6,o 1 +1 + k ~

# Ixl en~'2--~zt )

(4)

V Dl,eff

The system studied consists of two phases, the cubic phase (1) and the aqueous phase (2), of equal volumes, with the interface at x = 0 (Fig. 3). As shown in the figure, x has been defined in such a way that it is positive in the cubic phase. Initially at t = 0, the glucose concentration in the

6

C. Mattisson et al. / Chemistry and Physics of Lipids 84 (1996) I 12

AQUEOUS PHASE

An FRINGE NO. (z) (NJ

x=O

~dlto t

(3-1/4+y/2) - -

(y+1/4)

_~

3

k

(3-1/2) - -

b CUBIC

k

PHASE

b

(2-1/2) - -

k b

b

k b

1

(1-1/2)

2b

--

k b

Fig. 4. Schematic picture of an interference pattern obtained from a diffusion experiment. The picture shows how to interpret the interference pattern to obtain the refractive index profile in the cubic phase. The same principles can be used for the aqueous phase.

concentration of glucose in the x direction will change in both phases. The refractive index along the cell will thus change and result in dark and light interference fringes in both phases. Fig. 4 shows a schematic interference pattern from a diffusion experiment. Each fringe corresponds to a certain change in refractive index, 2/b, where 2 is the wave length of the laser light and b the thickness of the cell. The fringes in the cubic phase are numbered 1, 2 ..... z ..... N as indicated in Fig. 4. The change in refractive index between fringe z and the unaffected part ( x = oo) of the cubic phase, Anz, can be expressed as Anz = (z -

0.5)

2

where N is the total number of whole fringes and y is the fraction of the fringe closest to the interface obtained by extrapolation. From the total number of fringes (N) and the distance from each fringe to the interface, x, an experimental concentration profile can be readily obtained [13,20]. However, it is necessary to know the relation between the refractive index and the concentration. This relation is found to be linear, with the proportionality constant K, under the experimental conditions used in our system, which at fringe z gives the following

(5)

The total change in refractive index between the interface ( x = 0 ) and the unaffected part (x = oo) of the cubic phase, Antot, can be expressed as

ACz

K(z -

A 0.5) ~

(z - 0.5)

+ (7)

c. Mattisson et al. / Chemistry and Physics of Lipids 84 (1996) 1 12

The diffusion coefficient is obtained by leastsquares fitting of the theoretical concentration profile to the experimental data.

4. Results 4.1. X - r a y diffraction

The results of the X-ray study are shown in Table 1. Indexing of the X-ray diffraction lines shows good agreement with a primitive cubic lattice. This was also previously found by Larsson [21] for cubic monoolein-water phases with high water content. The data show that the structure of the cubic phase can be described as a primitive cubic lattice, independent of the composition of the aqueous phase under the experimental conditions investigated. Only minor changes in the dimensions of the unit cell were observed. The presence of glucose leads to a decrease in the cubic unit cell size compared to that observed in pure water. The self-diffusion measurement gave a diffusion coefficient of the monoolein in the cubic phase of 1.4 x 10-i1 m2/s without glucose, and 1.3 x 10-11 m2/s with glucose. Together with the minor effects found in the X-ray study this suggests that glucose had almost no influence on the structure of the curved bilayer in the cubic phase in the glucose concentration range applied. This is consistent with observations made on planar bilayers in the lamellar phase of phosphatidylcholine [5]. When the cubic water-glucose phase was exposed to an excess of water, the cubic phase swelled due to change in glucose concentration. When buffer was used instead of water, almost no effect of glucose was observed. A cubic phase prepared from 38% by weight of buffered solution was less affected when exposed to an excess of buffer than the corresponding samples prepared from pure water. The presence of G O D did not affect the structure of the cubic phase. 4.2. Diffusion measurements with laser holographic interfi;rometry

As shown in Fig. 2, a broadening of the interfacial zone occurs when the cubic phase containing

7

glucose is exposed to the pure aqueous solution. This zone, which extends to less than 10% of the depth of the entire cubic phase, makes it difficult to determine the exact position of the interface. The results of the X-ray diffraction experiments, where the effect of maximal swelling was investigated, indicate that the effect is minor under the prevailing experimental conditions. However, to compensate for the swelling we developed a method of determining the position of the interface. Swelling appears only at the interface, leaving the interference pattern in the rest of the cubic phase unaffected. Swelling leads to an increase in the size of the water channels in the cubic phase and consequently the interfacial zone is not expected to give rise to any additional mass transfer resistance due to the swelling. Thus, no mass will be accumulated at the interface, which is one condition for using the correction model applied. If the interface is wrongly assigned to the upper border of the interracial zone the fitting procedure results in a diffusion coefficient for glucose of 2.4 x 10 10 m2/s. To correct for swelling of the interfacial zone, the variable x in Eq. (3) was replaced with x = xinit- Ax . . . . . where xioit is the distance from the upper border of the interfacial zone and Ax ..... is the correction for the width of the interfacial zone. The value of Ax ..... was adjusted until the best fit was obtained, that is minimising the object function Z 2. The experimental concentration profiles obtained with and without application of the correction model, after 4 h for experiment no. 7 are shown in Fig. 5, together with theoretical fits according to Fick's law. About 75% of the decrease in glucose concentration occurs within 2.5 mm from the interface (Fig. 5). The X-ray data shows that the unit cell dimensions are 4% less in the presence of glucose compared to a cubic phase containing only water. This corresponds to a volume increase of 12.5% and if we assume that maximum swelling occurs in this 2.5 mm zone then the corresponding height amounts to about 0.3 mm. This rough estimate is comparable with Ax .... in Table 2. In some cases, a larger adjustment had to be made, probably due to difficulties in preparing a smooth interface between the cubic and the aqueous phase. One advantage of the holographic technique is that

Water W a t e r + 3 . 5 % glucose W a t e r + 3 . 5 % glucose b Buffer c Buffeff + 3.5% glucose Buffeff Buffeff + 0.25% G O D Bufferc + 3.5% glucose BufferC+3.5% glucose+0.25% G O D

38.0 38.0 38.0 38.0 38.0 45.5 a 48.5 a 45.2 a 45.1 a

68.7 64.2 67.5 66.1 66.7 68.1 68.1 67.2 67.2

97.1 90.8 95.5 93.5 94.4 96.2 96.2 95.1 95.1

54.4 52.9 55.9 53.4 54.2 55.1 54.7 55.4 55.0

d (/~)

d (A)

a¢~lc (A)

(111)

(110)

(hkl)

94.2 91.6 96.9 92.5 93.9 95.4 94.7 96.0 95.3

ac~l~ (A) 38.8 37.7 39.6 38.2 38.6 39.5 39.5 39.4 39.4

d (A)

(211)

~Equilibrated with excess of aqueous solution. bThe cubic phase containing glucose is equilibrated with the same volume water before X-ray analysis. CThe buffer contained 10 m M potassium p h o s p h a t e + 0 . 1 m M K C I + 1 m M E D T A and p H was 7.

Composition of aqueous solution % (w/w)

Aqueous content (w/w)%

95.0 92.2 96.9 93.5 94.5 96.7 96.7 96.6 96.6

a~L~ (/~)

31.9 30.6 32.4 31.6 31.6 32.2 32.2 32.3 32.1

d (/~)

(221)

95.6 91.8 97.1 94.7 94.7 96.5 96.5 96.9 96.2

a¢~1~ (A) 95.5 91.6 96.6 93.5 94.4 96.2 96.0 96.1 95.8

ac~l~ (mean, A)

Table 1 Indexing according to space group no. 224 (Pn3m) of the X-ray diffraction lines recorded at 25°C from cubic monoolein-aqueous phases of different compositions

7"

C. Mattisson et al. / Chemistry and Physics of Lipids 84 (1996) 1 12

9

1.30 0.8 ¸

1.85

A, • o

q

Fit obtained without ", • c o r r e c t i o n

0.6

N

o" q

2.40 0.4-

2.95

0.2

Fit obtained \ • with correction .....

0 0

i

2 Distance

3

•

4

3.5~0 6

from interface, x (mm)

Fig. 5. Concentration profiles in the cubic phase, from experiment no. 7, after 4 h with and without correction for swelling. The concentration is given as the ratio between AC(x) and ACtot. AC(x) is the difference between the concentration at the unaffected part of the cubic phase, x = oc, and at distance x from the interface. ACtot is the difference between the concentration at the unaffected part of the cubic phase, x = or, and the concentration at the interface, x = 0. The absolute concentration C~ is also inserted. The solid lines represent the best theoretical fit, according to Eq. (3), to the experimental data. As illustrated in the figure a considerable improvement of the fitting was achieved with the correction for the interfacial zone. In fact the value of Z 2 decreased with almost an order of magnitude (from 0.09 to 0.01).

any disturbances deriving from misalignment or drift in the set-up will lead to deterioration of the interference pattern and will thereby easily be discovered. The good agreement between the fitted and experimental profiles indicates that the diffusion of glucose in the cubic phase is Fickian in nature under the experimental conditions used. The diffusion coefficients resulting from the fitting procedure, with the above described correction model, are summarised in Table 2. The arithmetic mean value of the effective diffusion coefficients obtained, Dlxff, is 1.7 x 10-~o m2/s with a standard deviation of 0.3 x 10 10 m2/s. A comparison between this value and literature data for the bulk diffusion coefficient D z = 6 . 9 x 10 - I ° m2/s [22], gives a ratio of D~.eff/D2= 0.25 rather than 0.34 which would have been obtained if the interface had been assigned to the upper border. The fringes in the aqueous phase can be used in a similar way to calculate the diffusion coefficient of glucose in aqueous solution. These calculations gave a value of O 2 = 7.1 x 10 -1° m Z / s for one of

the experiments, the experiment illustrated in Fig. 2 using D~,eff= 1.7 x 10 ~0 m2/s. The obtained value is close to the literature value of the bulk diffusion coefficient of glucose, demonstrating the validity of the evaluation procedure. The results obtained from N M R self-diffusion measurements of water gave a diffusion coefficient of 3.6 × 10 ~o m2/s in the cubic phase, in both the presence and absence of glucose. This gives a ratio of D/Do=O.19 which is close to the value obtained for glucose in the holographic laser interferometry measurements. Due to the relatively low glucose concentration in the sample, the JHN M R signal from glucose is very weak and is partly overlapped by the signal from monoolein, making self-diffusion measurements of glucose in the cubic phase under the present conditions very tricky. Recently we managed to optimise conditions and measured the diffusion coefficient of glucose in the cubic phase as 1.2 × 10 ~o m2/s [23]. Bearing in mind the structure of the cubic phase and the dimensions of the water channels

C. Mattisson et al. / Chemistry and Physics of Lipids 84 (1996) 1 12

10

Table 2 Diffusion coefficients obtained from experiments on the diffusion of glucose ~ out from the cubic phase (62% (w/w) monoolein and 38% (w/w) water solution containing 3.5°/,, (w/w) glucose), into a solution of pure water Experiment no.

Adjusted interface zone (3 h/4 h, mm)

D I x 10 ~° (mZ/s) after 3 h

1 2 3 4 5 6 7 Mean _+ S.D.

0.2 0.1/0.5 0.3/0.5 -/0.4 0.1/0.2 0.8/0.9 0.6/0.8

1.9 2.1 b 1.8d 2.2 1.4 1.7

D t × 10 I° (m2/s) after 4 h

1.9c 1.3 1.6 2.2 1.5 1.4 1.7 _+ 0.3

aD2 = 6.9 × 10 to m2/s, CI ° = 3.5%, and C2o = 0%. bValue recorded after 208 min. Walue recorded after 227 min. °Value recorded after 149 min.

discussed below, we consider both glucose and water fully mobile in the water channels in the cubic phase. The N M R self diffusion measurements show that the ratio of the measured diffusion coefficients in the cubic phase and the bulk solution value are similar for both water and glucose (0.19 and 0.18, respectively). The relatively low glucose concentration combined with it's fairly high mobility in the cubic phase as compared for example with that in a lamellar phase, makes swelling or shrinking of the cubic phase due to osmotic pressure effects less likely.

5. Conclusions

To our knowledge, apart from our self-diffusion measurements discussed above, no values for the diffusion coefficient of glucose in the cubic monoolein-aqueous phase have as yet been reported. The present study shows that holographic interferometry is suitable for determining the concentration profile resulting from diffusion, even for such intriguing structures as cubic lipid-water phases. The cubic phase studied is bicontinuous and based on curved non-intersecting lipid bilayers which are organised to form two non-connected continuous systems of water channels. If an interface is placed in the gap between the methyl end groups of th e lipid it will form a plane that can be described as an infinite periodic minimal surface (IPMS) [1,24]. The primitive cubic

lattice (Pn3m) which describes the cubic phase in the present study corresponds to a diamond type of IPMS. The average channel radius, (R)D, for such a surface is ( R ) D ---- 0.39 x aca|c, where acalc is the size of the unit cell [25]. Using the value of acalc = 96 A (Table l) gives a channel diameter of 75 A. From this value we have to subtract the thickness of two monolayers of monoolein. The bilayer thickness of monoolein can be estimated as 32 A from the X-ray data of Lindblom et al. [4] giving an average diameter of the water channels of 43 A, thus free transport of small water-soluble molecules through the water channels is possible. Indeed even larger molecules such as proteins, can move through the water channels [26]. However, it should be borne in mind that in practice, the cubic lattice structure will exist in domains, as shown in electron microscope images of the phase [27]. The N M R self-diffusion data are based on measurements over a very short time, and thus represent comparatively short diffusion paths. Values can be anticipated to originate from diffusion within single domains. Holographic laser interferometry gives an overall diffusion coefficient, often called the effective diffusion coefficient, ie. the transport across the boundaries between the domains is also included. The ratio between DLeff and D2 obtained from the H20 and glucose [23] self-diffusion measurement is similar to that obtained for glucose by holographic interferometry even though the two values were obtained by different methods based on alternative types of

C. Mattisson et al./ Chemistry and Physics of Lipids 84 (1996) 1 -12

diffusion. This indicates that the boundaries between the domains do not give rise to any further mass transfer resistance. In the simplest model of a porous medium, consisting of disconnected parallel water channels oriented perpendicular to the interface, the ratio between Dl.en-and D 2 will be the same as the volume ratio between the water and the matrix. It is assumed that the limiting factor is obstruction and that diffusion only takes place in the water channels. This gives rise to ratio for the cubic phase of 0.36. However, the cubic phase consists of two systems of continuous water channels, which implies an increased tortuous diffusional path. Thus, a ratio of less than 0.36 is expected, which is in agreement with the results presented. It is interesting to note that the effective diffusion coefficient in the cubic phase is larger than expected from calculations using correlations developed for common polymer gels such as calcium alginate, polyacrylate or agarose. In these kinds of calculations the amount of polymer present in the gel obstructing the diffusive flow is used to predict the magnitude of the effective diffusion coefficient relative to the aqueous diffusion coefficient. Examples of such calculations are stochastic approaches such as the random pore [28] and Mackie Mears equations [29], which when applied to our system give ratios between D i,eft and D2 of 0.13 and 0.02, respectively. The low ratio obtained from the Mackie-Mears equation originates from the assumption that the polymer phase is finely dispersed. From these calculations and our experimental data it can be concluded that the cubic phase studied here is less obstructive in nature to a diffusive flow than polymer gels, as would be expected since the cubic phase is a very organised and relatively open structure. Thus, a model in which the obstructing phase consists of regularly shaped objects must be used. Using Maxwell's equation, initially derived for electric conduction in dilute dispersions of spheres but also applicable to diffusion, gives a Dl.efl./D 2 ratio of 0.29 [30]. The results from the X-ray diffraction measurements showed an increase in the size of the unit cell in the cubic phase when the glucose concentration was decreased. This swelling makes identification of the interface difficult. It is, however, possible to compensate for this so as to determine the position

I1

of the interface. Furthermore, the X-ray data indicates that both the structure and the dimensions of the water channels of the cubic phase is retained even if buffer and glucose oxidase are introduced into the aqueous part of the cubic phase.

6. Notation acalc

b

Cj

size of the unit cell (A) cell thickness (m) concentration of glucose in aqueous phase (w/w %) concentration of glucose in the aqueous medium of the cubic phase

(w/w %) C jo

initial concentration of glucose in aqueous phase (w/w 0/0) initial concentration of glucose in the aqueous medium of the cubic phase

(w/w %) D2 DI .eft

D

Oo

diffusion coefficient for glucose in the aqueous phase (m2/s) the effective diffusion coefficient for glucose in the cubic phase (m2/s) self-diffusion coefficient for H20 in the cubic phase measured by NMR (m2/s) self-diffusion coefficient for H 2 0 in D 2 0 (m2/s)

d

spacing between the crystal planes

(A) G I

/o k K N (R)D l x Xmit

Y

gradient strength (T/m) echo intensity echo intensity in the absence of a gradient pulse partition coefficient proportionality constant ACz/'Anz = AC, ot/Anto t (w/w %) total number of interference fringes average channel radius (/k) time (s) distance from phase boundary (m) distance from the upper border of the interfacial zone (m) fraction of the interference fringe closest to the interface fringe number

12

C. Mattisson et al. / Chemistry and Physics of Lipids 84 (1996) 1-12

Greek letters 3 gradient pulse interval (s) ACtot total change in concentration between the interface and the unaffected part of the cubic phase ACz change in concentration between fringe z and the unaffected part of the cubic phase Anto t total change in refractive index between the interface and the unaffected part of the cubic phase An, change in refractive index between fringe z and the unaffected part of the cubic phase AX. . . . correction for the width of the interfacial zone (m) cf length of the gradient pulse (s) 2 wavelength in air for the laser light

(m) gyromagnetic ratio of the observed nucleus

Functions erf(q) = (2/7r 1/2) .Iq e _q2 dq

z=l ~x~ A Ctot/]exp

~xA Ctot/calc/]

Acknowledgements The authors are indebted to Prof. Kfire Larsson and Dr. Stephen Hyde for valuable discussions. Prof. Maura Monduzzi and Annalisa Caria are greatly acknowledged for assistance and discussions concerning the N M R measurements. Charlotte Mattisson acknowledges the financial support of The Swedish Research Council for Engineering Sciences. Tommy Nylander acknowledges the financial support of The Swedish Council for Forestry and Agricultural Research.

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