DIFFUSION of LOW-MOLECULAR COMPOUNDS IN FOOD MODEL SYSTEMS

DIFFUSION OF LOW-MOLECULAR COMPOUNDS IN FOOD MODEL SYSTEMS

G. DIAZ, W. WOLF, A.E. KOSTAROPOULOS and W.E.L. SPIESS'

Federal Research Centre for Nutrition D-7500 Karlsruhe, Germany

Accepted for Publication July 14, 1993

ABSTRACT

The diffusion of NaCl and isopropanol was studied in a matrix of pure gel and one containing either carbohydrates, proteins or fat using the concentration-distance method. Concentrations of NaCl were measured by conductivity, those of isopropanol by gas chromatography. Diffusion experiments have shown that the temperature dependence of the diffusion coeflcient is in accordance with an Arrhenius approach.

Experimental andpredicted diffusion coefficients did not agree satisfactorily when only models of mere obstruction were considered. Including the effect of hydration by values obtained from pure gels, experimental difision coeflcients range throughout between the calculated values of mere obstruction and those obtained from a mathematical combination of obstruction and maximum hydration.

INTRODUCTION

In many sectors of industrial food processing mass transport processes such as absorption, evaporation and drying are important. Mass transport is accomplished by diffusion and convection. Although both mechanisms are usually involved in practice, diffusion is regarded in many cases as controlling parameter.

To describe and calculate diffusion processes, diffusion coefficients must be known (Rotstein 1987; Stahl and Loncin 1979), the experimental determination of which in real food is usually intricate. Wherever possible, model

'To whom correspondence should be sent: Prof. Dr.-Ing. W.E.L. Spiess. Institute of Process Engineering, Federal Research Centre for Nutrition, Engesserstr. 20, D-7500 Karlsruhe, Germany. Telephone number: 49 (07 21) 66 25-123.

Journal of Food Processing and Preservation 17 (1993) 437-454. All Rights Reserved. Copyright 1994 by Food di Nutrition Press, Inc.. Trumbull, Connecticut 437

438 G. DIAZ, W. WOLF, A.E. KOSTAROPOULOS and W.E.L. SPIESS

systems allowing transfer of the results to real food are advised. Model systems of solid gels have been found very suitable (Muhr and Blanshard 1982; Friedmann 1930; Friedmann and Kraemer 1930).

To get an idea of the changes in concentrations of ingredients occurring during storage of chilled moist food, studies on the transport of dissolved compounds in a food matrix consisting of carbohydrates (cellulose, starch), protein (casein) and fat were conducted by means of simple gel model systems. NaCl, whose diffusion in gels has been well documented in the literature (Gros and Riiegg 1987), was selected as diffusing substance. Furthermore, the diffusion of volatile aroma compounds such as ketones, esters, alcohols, etc., (Voilley and Le Meste 1985), exemplified by isopropanol, was investigated in various matrices with gel as carrier substance.

Theoretical Fundamentals

Mass transport by diffusion is due to differences in potentials (e.g., concentration gradient) caused by random molecular motion (Brownian movement). In analogy to Fourier’s theory of heat transfer by conduction, Fick developed mathematical equations of mass transport by diffusion in isotropic media. For various system geometries and different initial and boundary conditions several solutions to Fick’s differential equations have been reported in the literature (Cussler 1984; Crank 1975; Carslaw and Jaeger 1959; Jost 1952). There has been no universal theory allowing a sufficiently precise calculation of diffusion coefficients especially in multi-component systems such as food; diffusion coefficients hence are usually determined experimentally (Table 1). However, under certain conditions it may be possible indeed to predict or at least estimate diffusion coefficients on the basis of empirical relations, e.g., the equations of Stokes-Einstein (Cussler 1984). Wilke and Chang (1955), and one derived from Nernst-Planck’s theory of electrochemistry (Harned and Owen 1958). The usually strong temperature dependence of the diffusion process may be described by an approach according to Arrhenius (Rice and Selman 1984; Califano and Calvelo 1983; Tyrrel and Watkiss 1979; Kosfeld and Goffloo 1971; Menting 1970). Values typical of the activation energy in different diffusion processes are listed in Table 2. Experimental studies have confirmed that the activation energy increases linearly with increasing concentration (Gladden and Dole 1953; Fish 1958) and with increasing molecular weight (Longsworth 1954) of the diffusing compound.

Diffusion in Polymers

Diffusion in polymers is regarded as diffusion in a porous solid compound, whose pores are connected with each other and filled with the solvent. Among

DIFFUSION OF LOW-MOLECULAR COMPOUNDS 43 9

TABLE 1 . EXPERIMENTAL DIFFUSION COEFFICIENTS OF FOOD INGREDIENTS

IN WATER AT = 298 K (THE VALUES APPLY TO AN INFINITIVE DILUTION OF THE DIFFUSING SUBSTANCE)

Air I 2 9 I 2.00 I Cussler 1984 II n-Propanol 60 0.87 Cussler 1984

n-ProDanol 60 1.1 5 Johnson and Babb 1 9 5 6

IsoDroDanol I 6 0 1 1.10 I Landolt and Bornstein 1 9 6 9 11 NaCl 1 58.5 1 1.61 I Harned and Owen 1 9 5 8

NaCl I 58.5 I 1.53 I Landolt and Bornstein 1 9 6 9 11 Acetic acid 60 1.21 Cussler 1984

Acetic acid 60 1.16 Johnson and Babb 1 9 5 6

Coffein 21 2 0 .69 NRC 1929

Egg albumin 45000 0.078 Tanford 1961

Hemoglobin 67000 0.069 Gosting 1956

Glucose 180 0 .69 Chandrasekaran and King 1972

Saccharose 3 4 2 0.54 Chandrasekaran and King 1972

the components involved (polymer, diffusing substance and solvent) various interactions may take place, among which obstruction, hydrodynamic drag, changes in solvent properties and possible reactions between polymer and solvent, and polymer and diffusing substance are the most important. The obstruction effect is of practical significance as it can relatively easily be described in mathematical terms.

There have been no models so far to describe, in terms of quantity, the effect of hydrodynamic drag, i.e., the phenomenon of polymers in motion having a dragging effect on the more readily movable molecules (Friedman and Kraemer 1930). A mathematical model to describe changes in solvent properties is not available either.

The polymer molecules may produce changes in viscosity and structure of the solvent that, although they may influence diffusion (Osmers and Metzner 1972), are usually negligible. Reactions between polymer and solvent, and between polymer and diffusing substance can in principle be estimated by


TABLE 2. ACTIVATION ENERGIES OF DIFFUSION PROCESSES IN FOODS

AND FOOD MODEL SYSTEMS

I System

Reference

kJlrnole

Glucose in poly- acrylamid I 1 ;; I ; ;;; IBrown

and Chitumbo 1975

Ascorbic acid in glycerol

Ascorbic acid in water

Ascorbic acid in

water soluble substances in carrots during blanching

Thiamin in water

Thiamin in heat sterilized soinach

Giannakopoulos and Guil- 18.0 278 - 31 3 bert 1986

16.7 298 - 323 Schneeberger et al. 1975

44.9 343 - 370 Rice and Selman 1984

28.2 333 - 363 Selman et al. 1983

18.9 298 - 323 Schneeberger et al. 1975

65 283 - 300 Thornson 1982 I I I

Eyring’s theory (Ogston et al. 1973; Navari et al. 1971) and the free volume theory (Duda 1985); the latter is confined to some special cases, however, because of the difficulties arising in determining the parameters involved.

In the case of obstruction, the effect of which can be described by an effective diffusion coefficient D,, (Geankoplis 1972), diffusion is solely inhibited by the presence of the polymer matrix, which is impermeable to the diffusing substance (Hendrickx et af. 1987; Slade et af. 1966). Fricke (1924) developed an equation to describe diffusion in a suspension of randomly arranged and impermeable spheroids:

DQ I + * a

where Do is the diffusion coefficient in the pure solvent, $ the volume fraction of the polymer, and a a form factor. The form factor depends on the particle shape only, and not on its size. For spherical particles and aggregates of a! =

DIFFUSION OF LOW-MOLECULAR COMPOUNDS 441

2, Eq. (1) changes into Maxwell's equation for diffusion (Maxwell 1873). For very flat oblate spheroids a tends towards zero, and for prolate spheroids or needle shaped particles a tends towards 312.

When the polymer phase consists of several spherical macromolecules, a modification of Fricke's model applies (Barrer 1968; Eucken 1932):

1 -ru A

In view of similar observations, Polder relation:

(2) ai

and van Santen (1946) developed the

with /3 as form factor; the relation may be extended to

to include several particle shapes.

(1 955) developed the following equation from the stochastic point of view: To describe the diffusion of ions in ion exchangers, Mackie and Meares

Other models to describe the obstruction effect in mathematical terms were reported by Hamilton and Crosser (1962), Bruggeman (Crank and Park 1968), Rayleigh (1892), Cheng and Vachon (1969), and Prager (1960).

The diffusion coefficients for diffusion in a polymer of needle-shaped molecules calculated acceding to the various models are shown in Fig. 1 . It becomes evident that for small volume fractions ($ < 0.1) all models - except for the Mackie-Meares model - are in correspondence and that diffusion is independent of the geometry of the polymer particles (Slade er al . 1966).

Another parameter influencing obstruction in aqueous systems is the state of hydration of polymer molecules; it characterizes the quantity of water absorbed by the macromolecules. The absorbed water is not available for mass


1 .o

0.8

0.6 no \

0 0.4 f

0.2

0.0 0.0 0.2 0.4 0.6 0.8 1 .o

Volume fraction ( ~ p )

FIG. 1. DETERMINATION OF THE EFFECTIVE DIFFUSION COEFFICIENT

POLYMER IN THE TOTAL SYSTEM BY MEANS OF DIFFERENT OBSTRUCTION MODELS (Do DIFFUSION COEFFICIENT IN THE PURE SOLVENT)

AS A FUNCTION OF THE VOLUME FRACTION OF A NEEDLE-SHAPED

transport in the aqueous phase. Hydration has been defined as the amount of water transported by a unit mass of polymer molecules, when the molecules are moving, e.g., diffusing, in a solution (Wang 1954). The state of hydration, which depends also on the shape of the macromolecules, has been described by Wang (1954) as:

1 H 'Q = cp ( - + - 1

Q P Q w

where cp is the polymer concentration (g water-free polymer per ml of total matrix), H is the hydration of the polymer (g water per g water-free polymer) and pp and pw are the density of the water free polymer and of the water, respectively.

Equation (6) was used in diffusion tests to determine the hydration of proteins and gels (Langdon and Thomas 1971; Hendrickx ef al. 1987; Wang 1954). Most polymers are capable of forming gels; the gels may be homogeneous or heterogeneous, depending on the kind of solvent and on the interaction between solvent and gel forming substance.


MATERIAL AND METHODS

Diffusion coefficients were determined according to the concentration-distance method (Schneider et al. 1976). Under the initial and boundary conditions of

t = O

c = c " , X C O I -l

I

with t time, cH and cL, the higher and lower initial concentrations, x the coordinate relating to space, the solution of Fick's differential equations is (Naesens et al. 1981; Crank 1975)

Because of the different chemical structures, agar gels (Merck No. 1615) and carrageen gels (Serva No. 16 245) were included in the studies (Glicksman 1987; Rees 1972; Rees et al. 1969).

The gel forming substances were filled into two round flasks (250 ml each) with reflux condenser and mixed with demineralized water. The mixtures were heated in an oil bath at 12OC for 6 h and poured into glass tubes (length: 145 mm, inner diameter: 12 mm) to allow solidification. Two gel cylinders of equal diameter and different concentrations of the diffusing compounds (NaCI, Merck No. 6404, and isopropyl alcohol, Merck No. 9634) were brought into contact and separated again after a certain time. The cylinders were cut into thin slices (ca. 1.5 mm). The slices were weighed and the concentration of the diffusing substance in the slices was determined. The diffusion coefficient was determined by fitting Eq. (7) to the concentration profiles (SAS Software Package, 1982).

To study the diffusion of NaCl, different concentrations of sodium chloride were added to the gel mixtures before heating. The sodium chloride concentrations were determined by conductivity measurement (AOAC method, digital conductometer, Knick). To study the diffusion of isopropanol, different concentrations of isopropanol were added under stirring to the gels that had been allowed to cool down to ca. 50C after heating in the oilbath. Isopropanol concentrations in the gel slices were determined by head-space gas chromatography (Perkin-Elmer Sigma 3B gas chromatograph with Ucon-LB 550X column and ionizing flame detector).


Diffusion Experiments in Gels Plus Components

In another experimental series different concentrations of additional components, besides the diffusing substances, were added to the agar gels; components added were microcrystalline cellulose (Avicel pH 10 l), starch (Merck No. 152), edible semolina, casein (Riedel de Haen, No. 39 102) and hardened vegetable fat (30-32 m.p., Chemische Fabrik Gruenau GmbH). The concentration of agar in the matrix was 3% throughout. The components were added before the heating process, if possible. Casein was added after heating. The hardened plant fat was melted with lecithine (Serva, No. 57 556, 1 % fat) at 80C first and then added to the gel mixture before the heating process.

RESULTS AND DISCUSSION

Diffusion of NaCl

Figure 2 shows the results of the NaCl diffusion experiments in pure agar gels as a function of agar concentration and temperature. Measured points are mean values of three to eight repeated measurements. In the concentration ranges tested, diffusion coefficients are independent of the gel concentration, but not of the temperature, i.e., the diffusion coefficient increased with increasing temperatures.

The temperature dependence of the experimental diffusion data can be adequately described by the Arrhenius equation. Calculated activation energies for the diffusion coefficients at different gel concentrations are listed in Table 3. The calculated apparent activation energies, including that of 17.9 kJ/mole, which was extrapolated for an agar concentration of 0% , are typical of diffusion processes.

Activation energy and gel concentration have been found to correlate linearly, i.e., diffusion is inhibited by increasing gel concentrations. The inhibitive effect is so faint, however, that it escaped significant experimental detection.

Diffusion of Isopropanol in Agar Gels

Isopropanol diffusion coefficients were studied as a function of agar concentration and temperature in analogy to the experiments using NaCl. The results presented in Fig. 3 show a clear relation between gel concentration and diffusion coefficient, which decreases with increasing agar concentration. Comparable to NaCl diffusion in agar gel, activation energies calculated by the Arrhenius equation have been found to be clearly correlated to the agar concentration (Table 3).

DIFFUSION OF LOW-MOLECULAR COMPOUNDS

v) 1 "E 1.5 0 \ u) g 1.0- x

% P

0.5

445

-

-

2.5 ' v ... V 313K

1 A l I I

1 v

4 ' I

A , 305 K

I &

I I 298 K

1

I A l 4

_ _ _ ~ . .I . o n

4 1

291 K 0 n I I - I

0 I

i ~ 277,5 K ' i

I 0.0 ' 1 I I

0 2 4 6 8

Concentration / g Agar/100 g Gel FIG. 2. INFLUENCE OF AGAR CONCENTRATION AND TEMPERATURE ON THE EFFECTIVE DIFFUSION COEFFICIENT OF NaCl IN AGAR GEL

TABLE 3. ACTIVATION ENERGY E, OF THE DIFFUSION OF

NaCl AND ISOPROPANOL IN AGAR GELS OF DIFFERENT AGAR CONCENTRATIONS

g Agar/100 g Gel

2

3 I 20.37 I 19.1 11 4 1 21.3 I 19.3 11

Suitability of the Obstruction Models to Calculate Diffusion Coefficients

Graphs as presented in Fig. 4 are obtained when obstructive diffusion is described by experimental diffusion coefficients, i.e., when Deff/D0 is plotted against the volume ratio of the polymer matrix. Effective diffusion coefficients

446 G . DIAZ, W. WOLF, A.E. KOSTAROPOULOS and W.E.L. SPIESS

1.5

u) \ 1.0 “E 0 \ m 0 r

0.5 E n

0.0 0 2 4 6 8

Concentration / g Agar/100 9 Gel FIG. 3. INFLUENCE OF AGAR CONCENTRATION AND TEMPERATIJE ON THE

EFFECTIVE DIFFUSION COEFFICIENT OF ISOPROPANOL IN AGAR GEL

predicted according to Fricke and Mackie-Meares are shown as well (noninter- Npted lines). Form factors for the different matrices were taken from Hendrickx e? a1.(1987), Muhr and Blanshard (1982). and Slade ef al. (1966), details of the geometrical structure from Lienhard (1991) (Table 4).

Accordingly, calculated and experimental diffusion coefficients agree satisfactorily only for the diffusion of NaCl in agar gel according to Mackie and Meares. In the remaining cases, experimental and calculated results differ strongly from each other. Possible explanations are:

- -

The form factor is not correct. Diffusing molecules cannot diffuse through pores smaller than the molecule diameter; such pores apparently respond like the solid polymer matrix (theory of the excluding volume). Diffusion is influenced not only by hindrance through the matrix, but also by, e.g.. hydration, electrochemical affinity of the polymer matrix, and interactions between diffusing substance and solvent.

-

Assuming that, besides mere obstruction, hydration of the gel is involved as well, hydration values of pure gels can be calculated by combining Eq. (6) with Eq. (2) and the experimental diffusion coefficients. Diffusion coefficients calculated according to Fricke in consideration of hydration are shown in Fig.


Fricke

% t Hydr.: 0.31

Mackie and Meares I

I Fricke t Hydr.: 2.68

Fricke t Hydr.: 5 9

Fricke t Hydr.: 11.9 ".J I

0.00 0.01 0.02 0.03 0.04 0.05 0.06

Volume fraction ( c p )

FIG. 4. EFFECTIVE DIFFUSION COEFFICIENTS OF NaCl AND ISOPROPANOL IN A HOMOGENEOUS MATRIX

The curves in full lines were calculated according to Fricke. and Mackie

combination of the hydration effect and Fricke model. and Meares obstruction models. The curves in broken lines were calculated by

4 as well (broken lines). The fluctuation of hydration values for agar gel (Table 5 ) is remarkable if one assumes that hydration of the matrix should be independent of the diffusing compound. Literature data range from 0.7 g H,O/g dry agar (Derbyshire and Duff 1974; Woessner and Snowden 1970) to 3.4 g H,O/g dry agar (Langdon and Thomas 1971). The differences in literature hydration data are partly due to different methods employed to determine "bound water".

Our diffusion experiments using NaCl in carrageen yielded 11.9 g H20/g dry matrix. Hendricks et al. (1987) report 5.9 g H,O/g dry carrageen from diffusion tests using glucose. When the present experiments are analyzed on the basis of Hendrickx's value, which is more realistic, then it becomes obvious that obstruction and hydration are not the only mechanisms affecting diffusion; some other diffusion hindering effects must be involved as well.


TABLE 4. GEOMETRY AND FORM FACTORS OF VARIOUS FOOD COMPONENTS (MATRICES)

Matrix I Geometry Form factor II Polder (a) Fricke (a)

I I I

Carbohydrates (Agar, Needle shaped particles, carrageen, MCC, starch, randomly distributed 513 312 semolina) (prolate spheroids)

Protein (Casein) Aggregates of spheres 312 2 I Fat Dispersed spheres 312 2

TABLE 5 . CALCULATED HYDRATION VALUES

(DIFFUSION IN HOMOGENEOUS GELS)

Diffusion of NaCl in Inhomogeneous Matrices

Results of diffusion experiments using NaCl and isopropanol in agar gels plus carbohydrates, proteins and fat are presented in Fig. 5-7.

As in homogeneous matrices before, mere obstruction effects and obstruction effects intensified by gel hydration were again calculated by means of the form factors listed in Table 4. Maximum hydration values were 0.31 for diffusion of NaCI, and 2.68 for isopropanol in agar gel (mean value of 3 measurements); these values had been obtained from the diffusion experiments in pure gels (Table 5). In Fig. 5 experimental and calculated results of diffusion experiments in carbohydrate containing matrices are presented. Experimental results lie between the two curves of predicted diffusion coefficients, the upper


1 .a

0.8

0" 0.6 \ t

0

0.4

0.2

NaCI-MCC + NaCI-Starch -

A Isopropanol-MCC 0 NaCI-Semolina

I I

Fricke + Hydr.. 0.31 - - - - - _ _ Zricke + Hydr.: 2.68

0.00 0.02 0.04 0.06 0.08 0.10

Volume fraction of carbohydrates added FIG. 5. EFFECTIVE DIFFUSION COEFFICIENTS O F NaCl AND ISOPROPANOL IN

INHOMOGENEOUS AGAR-CARBOHYDRATE MATRICES (AGAR CONCENTRATION 3 %)

curve representing a mere obstruction model and the lower combining obstruction and hydration of 2.68 g H,O/g dry matrix.

Figure 6 shows experimental and calculated (mere obstruction, and obstruction plus hydration) diffusion coefficients for NaCl and isopropanol in agar gels plus casein. Experimental and calculated values are in good agreement if the model takes into account obstruction and hydration of 2.68 g H,O/g dry matrix of the gel. A possible hydration of casein was not taken into account. Results of diffusion in agar gel plus fat are presented in Fig. 7. Again, experimental values range in between the two calculated curves, the upper one illustrating mere obstruction, the lower a combined maximum hydration-obstruction effect.

CONCLUSION

The experimental results have shown that the diffusion of NaCl and isopropanol representative of low molecular compounds in pure polymer matrices as well as in inhomogeneous polymer matrices may be calculated using obstruction models and taking the hydration of the polymer into account. From the experimental results it may be further concluded that hydration, interpreted

450

1 .o

0.8

0" 0.6 \ t

0

0.4

0 p 0 6 \

f P

G. DIAZ, W. WOLF, A.E. KOSTAROPOULOS and W.E.L. SPIESS

NaCl A lsopropanol

A lsopropanol

- -

ce

ce + Hydr.: 0.31

(e t Hydr.: 2.68

0.2 0.00 0.02 0.04 0.06

Volume fraction of casein added FIG. 6. EFFECTIVE DIFFUSION COEFFICIENTS OF NaCl AND ISOPROPANOL IN

INHOMOGENEOUS AGAR-PROTEIN MATRICES (AGAR CONCENTRATION 3 %)

1.01 I

I --+@ Fricke t Hydr.: 0.31

Fricke t Hydr.: 2.68 - - - - - - - - - 4

0.00 0.02 0.04 0.06 0.08 0.2

Volume fraction of fat added FIG. 7. EFFECTIVE DIFFUSION COEFFICIENTS OF NaCl AND ISOPROPANOL IN

INHOMOGENEOUS AGAR-FAT MATRICES (AGAR CONCENTRATION 3 46)

DIFFUSION OF LOW-MOLECULAR COMPOUNDS 45 1

as "bound water", of the matrix is not independent of the diffusing compound. So for the pure agar matrix hydration values of 0.31 in the case of diffusing NaCl, and of 2.68 in the case of diffusing isopropanol were determined. Using these values for predicting the diffusion coefficient of NaCl in inhomogeneous matrices (agar gels plus carbohydrates or proteins or fat added) good agreement with the experimentally determined diffusion coefficients was obtained in the case of added carbohydrates and fat. When casein was added the diffusion coefficient of NaCl in the inhomogeneous agar-matrix was not predictable at the desired accuracy, probably due to interactions between casein and NaCl. In which way the diffusion process of NaCl was influenced by these possible interactions has not been studied within the framework of this project. Neither was NaCl diffusion studied in inhomogeneous carrageen gels, and so it remains open whether diffusion coefficients in these matrices are predictable.

REFERENCES

BARRER, R.M. 1968. Diffusion and permeation in heterogeneous media. In Diffusion in Polymers, (J. Crank and G.S. Park, eds.) pp. 165-217, Academic Press, London.

BROWN, W. and CHITUMBO, K. 1975. Solute diffusion in hydrated polymer networks. Part 2. Polyacrylamide, hydroxyethylcellulose and cellulose gels. J. Chem. SOC., Faraday 1 , 71, 12-21.

CALIFANO, A.N. and CALVELO, A. 1983. Heat and mass transfer during the warm water blanching of potatoes. J. Food Sci. 48, 220-225.

CARSLAW, H.S. and JAEGER, J.C. 1959. Conduction of Heat in Solids, 2nd Ed., Clarendon Press, Oxford.

CHANDRASEKARAN, S.K. and KING, C.J. 1972. Volatiles retention during drying of food liquids. AIChE J. 18, 520-526.

CHENG, S.C. and VACHON, R.l. 1969. The prediction of the thermal conductivity of two and three phase solid heterogeneous mixtures. Int. J. Heat Mass Transfer 12, 249-264.

CRANK, J. 1975. Zle Mathematics of Diffusion, 2nd Ed., Clarendon Press, Oxford.

CRANK, J. and PARK, G.S. 1968. Diffusion in Polymers, Academic Press, New York.

CUSSLER, E.L. 1984. Diffusion. Mass Transfer in Fluid System, Cambridge University Press, Cambridge.

DERBYSHIRE, W. and DUFF, l.D. 1974. Gels and gelling process. NMR of agarose gels. Faraday Discuss. Chem. SOC. 57, 243-254.

DUDA, J.L. 1985. Molecular diffusion in polymeric systems. Pure Appl. Chem. 57, 1681-1690.


EUCKEN, A. 1932. Die Wheleitfhigkeit keramischer feuerfester Stoffe. VDI-Forschungsheft 353. VDI-Verlag, Berlin.

FISH, B.P. 1958. Diffusion and thermodynamics of water in potato starch gel. In Fundamental Aspects of the Dehydration of Foodstuffs, (SOC. Chem. Ind., ed.) pp. 143-157, Metchim & Son, London.

FRICKE, H. 1924. A mathematical treatment of the electric conductivity and capacity of disperse systems. Phys. Rev. 24, 575-587.

FRIEDMAN, L. 1930. Structure of agar gels from studies of diffusion. J. Amer. Chem. SOC. 52, 1311-1314.

FRIEDMAN, L. and KRAEMER, E.O. 1930. The structure of gelatin gels from studies of diffusion. J. Amer. Chem. SOC. 52, 1295-1304.

GEANKOPLIS, C.J. 1972. Transport Processes and Unit Operations, 2nd Ed., Allyn and Bacon, London.

GIANNAKOPOULOS, A. and GUILBERT, S. 1986. Determination of sorbic acid diffusivity in model food gels. J. Food Technol. 21, 339-353.

GLADDEN, J.K. and DOLE, M. 1953. Diffusion in supersaturated solutions. 11. Glucose solutions. J. Amer. Chem. SOC. 75, 3900-3904.

GLICKSMAN, M. 1987. Utilization of seaweed hydrocolloids in the food industry. Hydrobiologia 151/152, 31-47.

GOSTING, L. J. 1956. Measurement and interpretation of diffusion coefficients of proteins. In Advances in Protein Chem., Vol. XI, (M.L. Anson, K. Bailey and J.T. Edsall, eds.) pp. 430-557, Academic Press, New York.

GROS, J.B. and RUEGG, M. 1987. Determination of the apparent diffusion coefficient of sodium chloride in model foods and cheese. In Physical Properties of Foods - 2, (R. Jowitt, F. Escher, M. Kent, B. McKenna and M. Roques, eds.) pp. 71-108, Elsevier Applied Science, London.

HAMILTON, R.L. and CROSSER, O.K. 1962. Thermal conductivity of heterogeneous two-component systems. Ind. Eng. Chem. Fundam. 1,

HARNED, H.S. and OWEN, B.B. 1958. The Physical Chemistry of Electrolyte Solutions, 3rd Ed., Reinhold Publishing, Corp., New York.

HENDRICKX, M., OOMS, C., ENGELS, C., VAN POTTELBERGH, E. and TOBBACK, P. 1987. Obstruction effect of carrageenan and gelatin on the diffusion of glucose. J. Food Sci. 52, 11 13-1 114.

JOHNSON, P.A. and BABB, A.L. 1956. Liquid diffusion of non-electrolytes. Chem. Rev. 56, 387-453.

JOST, W. 1952. Diffusion in Solids, Liquids and Gases, Academic Press, New York.

KOSFELD, R. and GOFFLOO, K. 1971. Selbstdiffusion kleiner Molekule in polymerer Umgebung. Kolloid-Zeitsch. Zeitsch. Polym. 247, 801-810.

187- 191.


LANDOLT and BORNSTEIN. 1969. Zuhlenwerte und Funktionen aus Physik, Chemie, Astronomie, Geophysik und Technik., 6th Ed. Part 5. Springer Verlag, Berlin.

LANGDON, A. and THOMAS, H. 1971, Self-diffusion studies of gel hydration and the obstruction effect. J. Phys. Chem. 75, 1821-1826.

LIENHARDT, M. 1991. Diffusion d’isopropanol dans des gels d’agar presentant des inclusions proteiques, lipidiques ou polysaccharidiques. Diploma-work: Ensbana, Universite de Bourgogne, Dijon.

LONGSWORTH, L.G. 1954. Temperature dependence of diffusion in aqueous solution. J. Phys. Chem. 58, 770-773.

MACKIE, J.S. and MEARES, P. 1955. The diffusion of electrolytes in a cation-exchange resin membrane. 1. Theor. Proc. R. SOC. 232A, 498-509.

MAXWELL, C . 1873. Treatise on Electriciry and Magnetism. Vol. 1, Oxford University Press, London.

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, 111-126.

MUHR, A.H. and BLANSHARD, J.M.V. 1982. Diffusion in gels. Polymer.

NAESENS, W., BRESSELEERS, G. and TOBBACK, P. 1981. A method for the determination of diffusion coefficients of food components in low and intermediate moisture systems. J. Food Sci. 46, 1446-1449.

NAVARI, R.H., GRAINER, J.L. and HALL, K.R. 1971. A predictive theory for diffusion in polymer and protein solutions. AIChE J. 17, 1028-1036.

NRC. 1929. lnternational Critical Tables. Vol. 5, Natl. Res. Council, McGraw-Hill, New York.

OGSTON, A.G., PRESTON, B.N. and WELL, J.D. 1973. On the transport of compact particles through solutions of chain-polymers. Proc. R. SOC. Lond. A. 333, 279-316.

OSMERS, H.R. and METZNER, A.B. 1972. Diffusion in dilute polymeric solutions. Ind. Eng. Chem. Fundam. IZ, 161-169.

POLDER, D. and VAN SANTEN, J.H. 1946. The effective permeability of mixtures of solids. Physica 12, 257-271.

PRAGER, S. 1960. Diffusion in inhomogeneous media. J. Chem. Phys. 33,

RAYLEIGH, C. 1892. On the influence of obstacles arranged in rectangular order upon the properties of a medium. Phil. Mag. 34, 481-502.

REES, D.A. 1972. Polysaccharide gels. A molecular view. Chem. Ind. 16, 630-636.

REES, D.A., STEELE, I.W. and WILLIAMSON, F.B. 1969. Conformational analysis of polysaccharides. 111. The relation between stereochemistry and

23 (SUPPI.), 1012-1026.

122- 127.

454 G. DIAZ. W. WOLF, A.E. KOSTAROPOULOS and W.E.L. SPIESS

properties of some natural polysaccharide sulfates (1). J. Polymer Sci. Part

RICE, P. and SELMAN, J.D. 1984. Apparent diffusivities of ascorbic acid in peas during water blanching. J. Food Technol. 19, 121-124.

ROTSTEIN, E. 1987. The prediction of diffusivities and diffusion-related transport properties in the drying of cellular foods. In Physical Properties of Foods - 2, (R. Jowitt, F. Escher, M. Kent, B. McKenna and M. Roques, eds.) pp. 131-145, Elsevier Applied Science, London.

SCHNEEBERGER, R., STAHL, R. and LONCIN, M. 1975. Experimentelle Bestimmung der Diffusionskoeffizienten einiger wasserloslicher Vitamine in Wasser. Lebensm. Wiss. Technol. 8, 222-224.

SCHNEIDER, F., EMMERICH, A., FINKE, D. and PANITZ, N. 1976. Untersuchungen uber die Diffusion in reinen und technischen Saccharoselo- sungen. Zucker 29, 222-229.

SELMAN, J.D., RICE, P. and ABDUL-REZZAK. 1983. A study of the apparent diffusion coefficients for solute losses from carrot tissue during blanching in water. J. Food Technol. 18, 427-440.

SLADE, L., CREMERS, A. and THOMAS, H. 1966. The obstruction effect in the self-diffusion coefficients of sodium and cesium in agar gels. J. Phys. Chem. 70, 2840-2844.

STAHL, R. and LONCIN, M. 1979. Prediction of diffusion in solid foodstuffs. J. Food Processing Preservation, 3, 213-223.

TANFORD, 0. 1961. Physical Chemistry of Macromolecules, John Wiley 8t Sons, New York.

THOMSON, D.R. 1982. The challenge in predicting nutrient changes during food processing. Food Technol. 36, 97-1 15.

TYRREL, H.J.V. and WATKISS, P.J. 1979. Diffusion in viscous solvents. Part 3. J. Chem. SOC. Faraday Trans. 1. 75, 1417-1432.

VOILLEY, A. and LE MESTE, M. 1985. Aroma diffusion: The influence of water activity and of molecular weight of the other solutes. In Properties of Wafer in Foods, (D. Simatos and J.L. Multon, eds.) pp. 357-373, NATO AS1 Ser., Ser E, 90. Martinus Nijhoff Publishers, Dordrecht.

WANG, J.H. 1954. Theory of the self-diffusion of water in protein solutions. A new method for studying the hydration and shape of protein molecules. J. Am. Chem. SOC. 76, 4755-4763.

WILKE, C. R. and CHANG, P.C. 1955. Correlation of diffusion coefficients in dilute solutions. AIChE J. 1, 264-270.

WOESSNER, D.E. and SNOWDEN, B.S. 1970. Pulsed NMR study of water in agar gels. J. Col. Int. Sci. 34, 290-299.

C. 28, 261-276.

DIFFUSION of LOW-MOLECULAR COMPOUNDS IN FOOD MODEL SYSTEMS

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