A Modified Maxwell and a Nonexponential Model for Characterization of the Stress Relaxation of Agar and Alginate Gels A. NUSSINOVITCH, M. PELEG, and M.D. NORMAND ABSTRACT Compressive stress relaxation curves of agar and alginatc gels of dif- ferent gum concentration (14% and 0.5-2%, respectively) were fitted by a two parameter nonexponential empirical model and a three term modified Maxwell model with two fixed relaxation times (10 and 100s). The asymptotic portion of the residual (unrelaxed) stress cal- culated by the two models for each gel had a similar magnitude, and therefore could serve as an objective measure of the gels dcgrec of solidity. The coefficients of the modified Maxwell model provided a simple but meaningful means to compare, in quantitative terms, the differences in the relaxation time spectra that wcrc associated with the gels’ stiffness and strength. INTRODUCTION THE STRESS-RELAXATION CURVES of gels, as well as many solid foods, have traditionally been described in terms of a discrete linear-Maxwell model (Mitchell, 1976); i.e. E(t) = a,, + E aiexp - i=l where E is the decaying modulus (or sometimes the stress or force), ai coefficients, and 7i relaxation times of the model. It has been repeatedly demonstrated that a model having two to four terms is sufficient to describe experimental curves with a high degree of fit, e.g. Gross et al. (1980), Comby et al. (1986), Costell et al. (1986). Theoretically, the constant a,, in Eq. (1) represents the amount of stressthat remains unrelaxed. If a,, = 0, all the stress relaxes, although at a progressively decreasing rate, and the material is considered liquid. If a,, > 0, that is, there is a residual stress even when t+a, the ma- terial is considered solid and the magnitude of a,, can serve as a measure of solidity. In nonlinear viscoelastic materials, the magnitude of a,, can depend on the deformation history of the specimen and consequently the specimen can exhibit different degrees of solidity at different strains. Since gels are not physically stable and they tend to ex- change moisture with the environment, tests for long term de- termination of their relaxation pattern are difficult to perform. Consequently, the physical meaning of a,, when determined in experiments of short duration, that is on the order of a few minutes, is only relevant to the gel’s short term response. In other words, a gel’s mechanical behavior on a time scale of a few minutes is equivalent to that of a viscoelastic solid with a residual modulus of magnitude a,,. Furthermore, if the con- stants of Eq. (1) are determined by a curve fitting technique their magnitude can depend on the test duration, which makes their significance as true material characteristics highly ques- tionable. In liquid polymers this problem has been bypassed by assigning fixed values to the relaxation times and letting Authors Nussinovitch, Peleg, and Normand are with the Dept. of Food Engineering, Univ. of Massachusetts, Amherst, MA 01003. only the coefficients vary (Chang and Lodge 1972; Wagner and Laun 1978). This resulted in a model of the kind: E(t) = b,exp- (&) + b,exp- ($) + . . . . (2) where the b’s are the coefficients. The advantage of this type of model when used to compare different materials or to assessthe effects of test conditions, such as strain, on the relaxation behavior is obvious. This kind of model, although with a smaller number of terms, was also proposed for the characterization and classification of solid foods (Peleg, 1984). Its capabilities were demonstrated in se- lected foods by Miller et al. (1986) who showed that models with two to three terms are sufficient to capture the main char- acteristics of the stress-strain and stress-relaxation relation- ships. They also showed, however, that the model is not mathematically unique, and that the relaxation time spectrum can be selected in different ways. If, however, the relaxation times were representative of the relaxation behavior then the same basic picture emerged irrespective of the relaxation time selection and the number of terms in the model. The problem with the asymptotic or equilibrium modulus was tackled in a different manner. If the relaxation curve can be representedby (Peleg, 1980): Foci- t ’ F,, k, + k,t or in its linerarized form: (3) F,.t - = F,,- F(t) k, + k,t ’ where F,, is the initial force, F(t) the decaying force, and k, and kz’constants, then a hypothetical asymptotic modulus, E,, can be calculated from EA=% 1-t ( 1 (5) 2 where A is the specimen’s cross-sectional area and E the strain. The physical significance of E, as calculated by Eq. (5) has been demonstrated and discussed elsewhere (Peleg and Pol- lack, 1984; Finkowski and Peleg, 1981; Purkayastha and Pe- leg, 1987). The model expressedby Eq. 3, it should be added, has no direct account of the relaxation time spectrum. The momentary decay rate is expressed by: k, (k, + Uj2 (6) and the initial rate by l/k,. Since the models expressed in Eq. (2) and (3) describe the Volume 54, No. 4, 1989-JOURNAL OF FOOD SCIENCE-1013