diffusion coefficient 1

A Method for the Determination of Diffusion Coefficients of Food Components in Low- and Intermediate Moisture Systems

W. NAESENS, G. BRESSELEERS, and P. TOBBACK

ABSTRACT A simple set-up to measure translational diffusion of components in low moisture systems is described. The tracer technique was tested on the diffusion of t4C-labeled tripalmitin and palmitic acid in a model system containing paraffin oil, microcrystalline cellulose and gum arabic. Different methods to evaluate the apparent diffusion coefficients from the experimental diffusion curve were compared. Good results were obtained using a curve-fitting procedure based on the sum of least squares. The technique appears to be suitable for measuring diffusion coefficients up to .10e9 cm2/sec. The procedure offers the possibility to quantify the mobility of chemical components in dried foodstuffs in order to elucidate the mechanisms and kinetics of reactions occurring during storage.

INTRODUCTION THE RELATIVE MOBILITY of chemical substances is one of the determining factors in chemical and enzymatic reactions. The displacement of reactants and reaction-products is important, not only in liquid and gaseous media but also in dry systems. With respect to dried foods, a relation between reactant mobility and chemical reaction has often been put forward (Labuza et al., 1970; Seow, 1975; Duck- worth et al., 1976). The diffusion of individual food components in dried systems has been suggested as one of the major factors restricting enzymic activity (Brockmann and Acker, 1978; Potthast, 1978).

Accurate knowledge of diffusion coefficients is a pre- requisite for a better understanding of the loosely used term “mobility” in order to approach the above mentioned subject from a more quantitative viewpoint. Duckworth (1962) has shown a relation between moisture content and diffusion of labeled glucose in dried potato and carrot tis- sues by means of autoradiography. In a companion study (Duckworth and Smith, 1963), various counting techniques have been used to measure the extent of diffusion of simple soluble constituents in both plant and animal products, but all the data obtained were qualitative. Later, Duckworth and co-workers have done extensive studies on the mobility of various low molecular weight proton-containing solutes in finely-devided samples of various colloidal test materials at different moisture levels using wide-line NMR (Duck- worth and Kelly, 1973; Seow, 1975; Duckworth et al., 1976).

Mobihty of components in dehydrated systems has also been of interest in the study of volatile retention in freeze-dried foods. Selective diffusion and microregion entrapment have been proposed as the basic underlaying mechanisms (Flink, 1975; Omatete and King, 1978). The microregion entrapment theory of Flink and Karel (1972) is more a qualitative description of mobility whereas the selective diffusion theory of Thijssen and Rulkens (1968) is a quantitative diffusion-based analysis. They derived diffusion coefficients for water and volatiles at different water concentrations from sorption- and desorption mea-

All authors are wifh the Laboratory of Food Preservation, Catholic University of Leuven, de Croylaan 42, 3030 Heverlee, Belgium.

1446-JOURNAL OF FOOD SCIENCE-Volume 46 (1981)

surements (Menting et al., 1970). Unfortunately, their techniques cannot be suitably used for nonvolatiles. In the same context, Chandrasekaran and King (1972) have described a diffusion technique which is only applicable to aqueous solutions.

Quantitative diffusion in foods or related systems has been predominantly measured in homogeneous liquids or liquefied heterogeneous media such as solutions or gels (Brown and Chitumbo, 1975; Busk and Labuza, 1979; Stahl and Loncin, 1979; and many others). When presented in dry systems, diffusion is mostly a matter of sorption or desorption of water or volatiles. The experimental set-up for diffusion measurements as described in these studies generally does not allow an adaption for measuring diffusion of nonvolatile components in dehydrated systems. This is also true for the extensive diffusion research carried out in the field of adsorption, synthetic polymers and heterogeneous catalysis. A useful method however has been described by Moore and Ferry (1962) for the quantitative determination of diffusion of 14C-labeled penetrants through polymer solutions. A thin layer of radioactive penetrant or a disk of polymer containing the radioactive penetrant was joined to a nonradioactive disk and the diffusion was followed by measuring the activity at the ini- tially inert surface by means of a Geiger tube.

While diffusion phenomena play an important role in relative dry systems and should be taken into consideration whenever quantitative analysis of the reactions occuring in such systems are made, methods for accurately determining diffusion coefficients seem to be lacking. In this study, a tracer method is proposed which allows the determination of the apparent translational diffusion in partially dried systems. Before handling the practical aspects of the set- up, some attention must be paid to the basic principles of diffusion.

Basic diffusion theory The one-dimensional solution of Fick’s second law for

an extended source of infinite extent with initial condi- tions

c = c, x<o for t = 0 (1) c=o x>o >

and for a constant diffusion coefficient is given by Crank (1956):

c (x,t) =F X

1 - erf 2 (Dt)r 12 > (2)

where C(x,t) is the concentration of the diffusing substance at distance x and time t, D is the diffusion coefficient, Co is the initial concentration and erf is the well-known error function of which elaborate tables are.available. The diffusion coefficient can be estimated at any point of the experimental diffusion profile by means of Eq(2).

A relative simple method to estimate the diffusion coefficient is stated by Jost (1957). The equation of the tangent at the inflection point of the sigmoid c_urve is given by

CO CO

C=-2(nDt)r/2X+?-

The intercept with the x-axis (xc) is thus given by

x0 = (rrDt)’ /2 (4) from which D can readily be calculated.

A more general solution, which is also valid for concentration-dependent diffusion coefficients, is given by Crank (1956):

Dc=co = - 2t 1 dx

dC I

-- o Cl x dC

and CO xdC=O 0

(5)

(6)

with Cr any value of C between 0 and Co. The procedure is to plot the concentration-distance curve for a known time, to locate plane x=0 by use of Eq (6), and then to evaluate D at various concentrations C, from Eq (5). The latter results from the tangent of the diffusion curve at C=C, (dC/dx) and from the area under the curve from C=O to C=Cr (JxdC). A method of improving the accuracy of the calculations near the extremes of the concentration range is given by HalI (1953).

EXPERIMENTAL Preparation of the diffusion model system

Tripalmitin or palmitic acid was chosen as diffusants in a medium containing 20 times more liquid araffin oil in which it is completely miscible at 50°C. 45 mg 1 % C-labeled (~1 PCi) tripalmitin and 900 mg paraffin oil were dissolved in 2 ml chloroform and emulsified with 20 ml of a 5% (w/v) aqueous solution of gum arabic (Merck 4282) with an Ultra-Turrax Mixer Emulsifier (Janke and Kunkel K.G., Type TP 18/10, Staufen, W. Germany) for 2 min at maximum speed. The emulsion was then well mixed with 15g microcrystalline cellulose (Avicel, Merck 2230) and 60 ml bidis- tilled water by means of the Ultra-Turrax for 3 min at maximum speed. When palmitic acid was used instead of tripalmitin, 60 ml 0.2M Tris-HCl pH 8.0 was used instead of 60 ml water. Similar suspensions were prepared in the absence of the diffusants.

The suspensions were poured into plastic petri-dishes to a depth of about 3 mm, quickly frozen with liquid air and freeze- dried at a pressure of less than 0.1 Torr in such a way that the maximum temperature never exceeded 25’C (Secfroid, Lyolab C 3023, Lausanne, Suisse). The freeze-dried suspensions were then stored at 5O’C above standard saturated salt solutions (Greenspan, 1977) for about 7 days. MgCl, (a, = 0.31) and NaCl (a, = 0.74)

MEASUREMENT OF DIFFUSION IN DRY SYSTEMS.. .

were chosen. After equilibration, the water activity (aw,) of the samples was measured with a calibrated electrical hygrometer (Nova Sina Ag, SMT-B, Zurich, Suisse) at the same temperature.

Diffusion set-up The diffusion experiments were performed in thick glass-wall

cylinders of 1.2 cm inner diameter and about 7 cm length. Each cylinder was graduated at 1 mm accuracy (Fig. 1). A glass plunger fitted precisely in the cylinder. Approximately 0.75-lg of the model system without diffusant was weighed and brought rapidly into the cylinder which was stoppered at one end. The dry mixture was compressed so that lg of system was distributed over exactly 2 cm. Attention was paid to the upper face of the compressed system, so that it was perfectly flat and orthogonal to the length-axis of the cylinder because this plane was considered later as the initial (x=0)-plane. An equal amount of the diffusant-containing model system was deposited on the first layer in the same manner and the cylinder was stoppered at the other end. The cylinder was stored at the desired condition of relative humidity at 50°C.

Concentrationdistance curve After a certain time, the diffusion system was analyzed in order

to obtain the concentration-distance curve. The stoppers at both ends were removed and the plunger was gently placed in the glass cylinder on the side of the diffusant-containing model system. The plunger was moved forward by a screw action so that the model system was gradually forced out of the cylinder. Slices of about 0.2-0.3 mm thickness were scraped off with a razor blade and the powder of each slice was collected in a pre-weighed 20 ml plastic scintillation vial. The weight of the powder was determined and the activity was measured with a liquid scintillation counter (Beckmann, LS 9000) using 1 ml water and 7 ml Hydroluma (Lu- mat, 1077) as scintillation medium. No correction for quenching was found to be necessary and the background was substracted. The activity of the last slices expelled from the system, expressed as cpm/g, was set equal to Co and the relative concentration of the diffusant in the preceding slices was calculated as C/Co. The diffusion distance could be easily calculated from the weights of the successive slices and the compression factor.

RESULTS & DISCUSSION THE MODEL SYSTEM presented here stands for a typical example of a powdered, heterogeneous and porous substance. In such systems, it is convenient to define D as an overall or apparent diffusion coefficient (Stahl and Loncin, 1979). As shown in Figure 2, the tripalinitin system gives a symmetric profile. This suggests that the displacement of

Fig. 1 -Perspective view of a diffusion cylinder and plunger. The arrow indicates the initial (x=0)-plane; N. L. stands for the nonlabeled layer and L. for the labeled layer of the system.

Volume 46 (1981)-JOURNAL OF FOOD SCIENCE- 1447

tripalmitin is only due to Brownian motion. Little is known about the physical state of the oil in the model system. Probably the emulsion is reconstituted upon rehydration as described by Lladser and Arancibia (1972) and by Gejl- Hansen and Flink (1978). We have also found by scanning electron microscopy that at least part of the oil is present as oil globules. An asymmetric and much less developed diffusion profile is obtained with the palmitic acid system (Fig. 2). This was even true in the absence of the Tris-HCl buffer. It is well-known that diffusion of ,ionic compounds is complicated by interactions with the matrix (Duckworth and Smith, 1963; Karel, 1976). The Ca++ present in the model system may be one of the interacting factors. The diffusion coefficient of palmitic acid cannot be calculated from the simple equations as given above and is not further discussed.

The tripalmitin system in Figure 2 shows that our diffusion technique yields a fairly good experimental curve. Similar symmetric curves were obtained after 90 days and at a,=0.33. Apparent diffusion coefficients from the experimental curves were calculated using the described methods (Table 1). Eq (4) offers very rapidly a good estimate of D. Eq (4) can also be easily applied to get an idea of the required diffusion time when D is approximately known. The method using Eq (5) and (6) was found to be rather inaccurate and time consuming. It was necessary to calculate D at more than one point of the experi-

DISTANCE x (cm)

Fig. 2-Diffusion curves after 45 days of tripalmitin l - l and palmitic acid o - o in the model system containing paraffin oil, microcrystalline cellulose, and gum arabic at an a, of 0.71 and 50°C. A Tris-HCI buffer pH 8.0 was used in the preparation of the palmitic acid system.

Table I-Tripalmitin diffusion coefficients at 5@C in the model system at two water activities as calculated by three different methods

D x 10’ (crn%~-~)

ha Time (days) Eq (4) Eq (5) and (6jb

Least squares

0.33 45 4.2 4.2 4.0 90 4.7 4.5 4.4

0.71 45 7.4 7.8 7.6 90 7.4 7.8 7.7

a As measured by means of the Sina hygrometer b Mean value

1448-JOURNAL OF FOOD SCIENCE-Volume 46 (1987)

mental curve in order to obtain a reproducible value. Solutions using Eq (2) are also quite acceptable. The value of D becomes also more accurate if several experimental points are used. Curve fitting based on the method of the least squares can be usefully applied for this purpose. According to this method, a number of diffusion coefficients in the range of the expected value of D are selected. The best fit diffusion coefficient minimizes the sum (S) of the squared deviations between the experimental concentrations (C,) and the calculated concentrations (C,) at the different locations:

s = c (C, - Cc)2 (7) The C,‘s are calculated from Eq (4) and the plane corres- ponding to x=0 can be determined by Eq (6). The procedure can be implemented on a computer which, if necessary, also plots the diffusion curve. A reliable diffusion coefficient is rather easily obtained in this way. Even with this method, small differences between the D values of similar diffusion experiments may still exist. This is illus- trated in Table 1 where the diffusion coefficients at two diffusion times are compared. The differences seem to diminish as the diffusion profile becomes more developed. Moore and Ferry (1962) reported a precision of D within about 5% in their diffusion methods. When diffusion was slow, their precision was largely dependent on the accuracy with which the thickness of the disks, varying between as little as 0.15 and 0.4 cm, was known. Also in our set-up, the deviations of the experimental compression factor from the theoretical value of 2 cm/g may represent a source of error. However, it must be stressed that adding the second layer in the diffusion set-up did not cause the initial layer to compress.

The diffusion of tripalmitin is low compared with the diffusion coefficients for nonionic solutes and ions in liquefied food related systems which range from lop5 to 5 x lop8 cm2sec-1 (Karel, 1976; Busk and Labuza, 1979). On the other hand, it is well known that diffusion coefficients are strongly dependent on water concentration. A possible decrease over seven orders of magnitude from very high moisture to nearly dry material has been reported (Bomben et al., 1973). Few diffusion data in the low moisture range are available. Fish (1958) accurately measured the diffusion coefficient of water by adsorption and desorption in starch gel and scalded potato at moisture contents as low as 0.8% (dry basis). The diffusion coefficient was found to decrease very markedly with decreasing moisture content, especially below 30% moisture: 3.6 x 1O-8 cm2sec-l at 14.1%, 1.5 x 10-9cm2sec-1 at 6.3%, and 1.05 x lo-i0cm2sec-’ at 0.8% for starch gel. Menting et al. (1970) measured the diffusion of water and aceton in malto-dextrin at moisture contents above 10% (wet basis) at 21.5’C. The D of aceton was 1.5 x lo-l1 cm2sec-l at 10.2% (awZO.6) and 3.0 x 10-9cm2sec-1 at 20.1% (awr0.85). Diffusion coefficients of water were one to two orders of magnitude higher but followed the same pattern. The authors also showed that there is an inverse linear relationship between log D and the molecular diameter of the diffusant in this moisture range. Compared to these results, the diffusion of the large molecule tripalmitin (Table 1) in our model system at a,=0.33 (4.5% H20 on dry basis) and aw=0.71 (7.5% H,O) is remarkably high, even when corrected for the difference in diffusion temperature. Activation energy for diffusion is below 10 kcal/ mol (Fish, 1958; Menting et al., 1970; Stahl and Loncin, 1979), SO one can easily calculate that the diffusion coefficient of aceton in the malto-dextrin system at 10.2% water should not exceed 5.5 x lo-“cm2sec-’ at 5O’C. The relative low extent of binding between tripalmitin and the carbohydrate may be responsible for the higher tripalmitin mobility. Moreover, the negligible solubility of tripalmitin

MEASUREMENT OF DIFFUSION IN DRY SYSTEMS. . .

in water makes it reasonable that it is not molecularly transported like water-soluble diffusants. It is probable that the diffusion of tripalmitin is poorly dependent on the water in the system. On the contrary it is likely that its mobility is highly influenced by the paraffin oil present in the model system in which it is completely miscible at 5O’C. It has been mentioned that any liquid ingredient other than water might play a role in the process of mobili- zation (Duckworth et al., 1976; Brockmann and Acker, 1978). Probably, tripalmitin is transported as oil globules. This view is also supported by the relative small difference between the diffusion coefficients of tripalmitin at the two water activities tested (Table 1). The higher diffusion coefficient at higher a, is probably due to the greater avail- ability of the water, giving rise to a greater Brownian motion and a lowering of the viscosity of the diffusion medium.

The tracer diffusion technique here described offers the advantage that, apart from classical counting instru- ments, no special apparatus is required. When calculating D using the least squares procedure, a x, of about 0.2 cm is already sufficient to determine a reliable value of D. This means that it is possible to measure diffusion coefficients up to about 2 x 10-9cm2sec-1 within 2 months. It is obvious that with a diffusion coefficient in the range of 1 Om5 to 1 O- %m2 see-’ , a smaller diffusion time or thicker slices may be taken in order to have the same reasonable accuracy. With diffusion coefficients at the upper end of this range it may be easier to use a direct scanning-method which is less time consuming and nondestructive.

The measurement of the translational diffusion of components in this way is more quantitative than the above- mentioned NMR technique of Duckworth and his group, which gives only a semi-quantitative idea of the molecular mobility of the components and of the solvent properties of the restricted water. Moreover, the increase in the NMR signal does not occur at a relative low hydration level, although there is molecular mobility in that region (Karel, 1975; Duckworth et al., 1976).

CONCLUSION THE SENSITIVE TRACER technique combined with the curve-fitting procedure based on the sum of least squares seems to be useful to measure the apparent diffusion of food components in systems at low- and intermediate moisture level. Although the diffusion of a rather complicated system is described in this paper, the technique is surely applicable in general. The procedure may be used to quantify the loosely used term “mobility” of food components in dry systems, especially those components in dry systems which give rise to a diffusion profile obeying Fick’s law.

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Karel, M. 1975. Physico-chemical modification of the state of water in foods. A speculative survey. In “Water Relations of Foods ” Ed. Duckworth, R.B., p. 639. Academic Press, London.

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Menting, L.C., Hoogstad, B.. and Thijssen, H.A.C. 1970. Diffusion coefficients of water and organic volatiles in carbohydrate-water systems. J. Food Technol. 5: 111.

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Omatete, 0.0. and King; C.J. 1978. Volatiles retention during re- humidification of freeze dried food models. J. Food Technol. 13: 265.

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M S received 10/30/80;revised 314181; accepted 3/10/81.

This work was supported by the Fund for Joint Basic Research of Belgium. Author W. Naesens is working as Research Assistant with the National Fund for Scientific Research of Belgium.

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diffusion coefficient 1

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