Biotechnol. Prog. 1994,
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70,652-654
Mathematical Characterization and Graphical Presentation of the Stiffness-Temperature-Moisture Relationship of Gliadin Micha Peleg Department of Food Science, University of Massachusetts, Amherst, Massachusetts 01003
Published data for storage modulus ( G ) vs temperature (27 relationships of gliadin at various moisture contents [deGraaf, E. M.; Madeka, H.; Cocero, A. M.; Kokini, J. Biotechnol Prog. 1993,9,210-2131 were transformed into relative stiffness, R(T,M), vs temperature and moisture relationships, with the former being defined as R(T,M) = G ( T ) / G ( - 5 0 "C). These relationships could be described by the model R(T,M) = 1/{1 exp[(T - T,(M)/u(M)l},where both T,(M) and d M ) , as has been experimentally confirmed, are in the form of a single-exponential decay term. This model, with the constants of the T,(M) and u(M) terms incorporated, was also used to create threedimensional plots of the relative stiffness-temperature-moisture relationship of gliadin at and around its glass tranition, in which the combined effects of moisture and temperature are revealed visually. Although the model applicability is only demonstrated with gliadin data, there is evidence t o suggest that it would also be appropriate for other biosolids.
+
Introduction Gliadin, as gluten, is a cereal protein whose mechanical properties are of importance in dough-making. Its stiffness, like that of most biosolids, is affected simultaneously by temperature and moisture content. In terms of modern concepts of polymer rheology, the water acts as a plasticizer that lowers the glass transition temperature (Slade and Levine, 1991; Levine and Slade, 1993). Consequently, moist biosolids, including gluten and its components, soften at a lower temperature than when dry (Hosney et al., 1986; Cocero and Kokini, 1991; deGraaf et al., 1993). Recently it has been proposed (Peleg, 1993a) and demonstrated (Peleg, 1994) that the mechanical changes at and around the transition can be described by a model of the following form:
Y(T,M)= YJU
+ exp[(T - T,~M))~u(M)I} (1)
where Y ( T f l )is a stiffness or rigidity parameter (e.g., modulus, E,or storage modulus, G ) ,Y, is its magnitude in the glassy state, which is assumed to be practically constant, T is the temperature, T,(M) is a critical temperature, and a(M)is a measure of the relationship's slope, the units of which are that of T and T,. At a constant moisture content ( M = const), the critical temperature, T,, specifies the location of the inflection point of Y(r) and need not coincide with Tg. According t o the mathematical properties of eq 1, at a constant moisture content about 90% of the drop in Y(r)occurs at T , f 3a and if a-0, the shape of Y(T)approaches that of a step function. There is evidence in the published food literature that the relationships between the sticky point, collapse temperature or Tg,and moisture can be described by a single-term exponential model (Flink, 1983;Peleg, 1993a). Thus, if T, has a similar type of moisture dependency, as previously had been assumed (Peleg, 1993b1, then T,(M) can be written in the form
T,(M) = T,,exp(-klM)
(2)
where the subscript 0 refers to the dry state, i.e., where M=0.
It has also previously been conjectured (Peleg, 1993b) that a(M) is governed by a relationship of the same kind:
which implies that as moisture increases, the drop in the magnitude of the stiffness parameter becomes steeper. However, most published data on stiffness changes as a result of glass transition were obtained under constant moisture or isothermal conditions, and therefore the constants of eq 2, and especially those of eq 3, could not always be determined. If T,does coincide with Tg,then at least in principle one can use the Gordon-Taylor , that the dry material's model to produce T g ( W provided T g is known or has been determined experimentally (deGraaf et al., 1993). The Gordon-Taylor model can also be used to fit experimental T,(M) relationships (Kalichevski et al., 1993a,b; deGraaf et al., 1993). It can be shown that up to about 30%moisture, the GordonTaylor model and eq 2 can be used interchangeably in many instances. Without knowledge of the exact nature of T,(M) and a(M), only a simulated R(T,MI relationship can be created using assumed values of T,o, ao, and the k's (Peleg, 199313). However most recently, deGraaf et al. (1993) reported a storage modulus ( G I vs T relationship for gliadin, determined at a fairly wide range of moisture contents, that was sufficient to establish the validity and determine the constants of eqs 2 and 3 directly. The objective of this report is to demonstrate for the first time, using their published data (deGraaf et al., 1993) with permission, how the experimental stiffness-temperaturemoisture relationship can be described by a single mathematical model with a format similar to that of eq 1.
The G vs T Relationships of Gliadin at and around Its Glass Transition Because the difference in stiffness in many polymeric materials between the glassy and leathery or rubbery states is several orders of magnitude, the G vs T relationships traditionally are reported in the form of log
8756-7938/94/3010-0652$04.50/00 1994 American Chemical Society and American Institute of Chemical Engineers
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Table 1. Characteristics of the Stiffness-Temperature Relationships of Gliadin" storage modulus, relative stiffness) eq 5 parameters eq 1parameters moisture G, Tc a (96) (109dyncm-2) ("C) ("C) h
4
N
E ?!
2
0
,
-50
-25
0
50
25
75
I \
O L -50 -30
-10
10
50
30
11.2 13.0 19.2 24.5 27.2
14.7 16.3 26.8 26.3 10.2
1.0
0
1 0 20
-50 -40 -30 - 2 0 - 1 0
0
Y
I
Figure 1. Storage modulus, G , vs temperature relationship of gliadin with different levels of moisture contents plotted in linear coordinates: 0, experimental data; -, fit of eq 1. The regression parameters are listed in Table 1. Data are from deGraaf et al. (1993).
G vs T plots (Ferry, 1980). [Since the values of G can be about 2 orders of magnitude lower than those of G , the plot of log G"vs T can be presented in the same graph (deGraaf et al., 1993; Anglea et al., 199311. However, since the transition region in biomaterials can be fairly wide in terms of temperature, and the stiffness drop within it no more than 2 orders of magnitude, a plot of G vs T in linar coordinates is perhaps equally appropriate (Peleg, 1993a,b). Examples of such plots-representing the mechanical behavior of gliadin at various moisture contents-are shown in Figure 1. [The units of G ( T )in the figure, as well as in Table 1 and the following discussion, are those reported in the original paper by deGraaf et al. (1990)l. Also shown in the figures is the fit of eq 1. As can be seen in Table 1, it had an r2value on the order of 0.992-0.999 in all cases. It is quite evident from the plots, however, that even at -50 "C the "wet" samples a t least had already been partially plasticized. Thus, the relationship's plateau, that is, the temperature range where G(T) = G,, was barely reached in the experiments. Consequently, and because of an inevitable scatter in the results, the magnitude of G, as calculated by nonlinear regression (Table 1)varied considerably (10.2-26.8 dyn cm-2) and without a clear trend. The magnitudes of Tc and a did show a clear trend, but the scatter was too large to derive an exact mathematical relationship. To eliminate this predicament and to reduce the effect of the experimental data scatter, the relationships between G' and T were converted into relationships beand T. The tween a dimensionless stiffness ratio, R(T), stiffness ratio was defined as the ratio of the storage modulus at a temperature T , G(T),and its corresponding magnitude at -50 "C, G(-50 "C), the lowest experimental temperature:
(4)
Typical plots of R(T)vs T are shown in Figure 2. These relationships too could be fit by a model of the same format as eq 1, that is (Peleg, 1993a,b)
R(T,M)= 1/{1 + exp[(T - T,(M))/a(M)I)
a
("C) 23.9 21.0 16.2 10.3 8.6
r2 0.991 0.997 0.995 0.997 0.998
11.2 19.2 & 27.2%M
I
10
TEMPERATURE (deg C )
R(T) = G'(T)/G(-50 "C)
0.992 0.997 0.998 0.997 0.999
Tc ("C) 13 3 -14 -28 -25
a All of the experimental data are from deGraaf et al. (1993) and are used with the authors' permission. "he relative stiffness is defined as G(T)/G(-50 "C).
\
0 -50 -40-30-20-10
17 21.3 -1 22.7 -24 20.3 -44 16.0 9.8 -26
r2
(5)
,kh'J\
w
I
U
0.0
-50
-25 0 25 50 75 TEMPERATURE (deg C)
Figure 2. Relative stiffness vs temperature relationship of gliadin with different levels of moisture contents plotted in linear coordinates: 0, experimental data; -, fit of eq 5. The regression parameters are listed in Table 1. Data are from deGraaf et al. (1993) [relative stiffness is defined as G(T)/ G(-50 "C)].
The fit of this model is demonstrated in Figure 2, and its regression parameters are summarized in Table 1. As can be seen from the table and Figure 3, where Tc( M ) and a(M) are plotted, the decreases in Tc(M)and a(M) with moisture are more clearly evident when the normalized parameter, R(T,M), is used. Both, in fact, could be described adequately by a single-exponential decay term, that is, by eqs 2 and 3, respectively. The fit of these two equations and the corresponding regression parameters are shown in the figure. (The r2 values for Tc(M)and a(M)were 0.999 and 0.998, respectively.) The reader will notice that, for convenience, T,(M) was expressed in kelvin. This also is how it is expressed in the GordonTaylor model and in subsequent discussions and presentations.
The Relative Stiffness-TemperatureMoisture Relationship Incorporation of the calculated values of Tc(M)and a(M) into the format of eq 5 produces the relationship
+
R(T,M)= 1/{1 exp[(T - 314 exp(-O.OSM))/ (46.7 exp(-O.GM))I) (6) where T is in kelvin and M is in percent. This relationship can now be used to construct three-dimensional displays of R(T,M) of the kind shown in Figure 4. These plots, as well as all other graphics and statistical analyses, were done using the Systat 5.2.1 package with a Macintosh SEI30 computer. The advantage of these three-dimensional plots is that they enable visualization of the combined effects of temperature and moisture on the plasticization of gliadin
Biotechnol. frog., 1994, Vol. 10, No. 6
654 TC=314'EXP(-.009'M)
300
25
a =46.7'EXP(-.06'M)
i
i
" 260 250 240
1 '
0
10 20 % MOISTURE
U 30
,
01 0
10 20 % MOISTURE
30
Figure 3. Relationship between T,, a , and moisture of gliadin: 0, calculated values using eq 5; -, fits of eqs 2 and 3, respectively. EXTENDED RANGE
EXPERIMENTAL RANGE
k+
1.0 0.8 0.6
j
gliadin data confirm a previous conjecture that both T,(M) and a(M) can be described by a single-exponential decay term. Incorporation of these terms into the model's equation (eq 5) produces a single mathematical expression with which the combined effects of temperature and moisture on a material's stiffness at and around the glass transition can be conveniently assessed, numerically or graphically. Although the model has only been fully validated with gliadin data, the pattern in other reported stiffness-temperature-moisture relationships at the transition region suggests that it will also be appropriate for other biosolids.
Acknowledgment This paper is a contribution of the Massachusetts Agricultural Experiment Station at Amherst, MA. The author expresses his sincere thanks to Professor J. Kokini and his collaborators for the permission to use their data.
A
Figure 4. Three-dimensional views of the relative stiffnesstemperature-moisture relationship of gliadin.
at and around its transition. The plot that describes the experimental range (Figure 4,left), which ought to be mentioned, was created without any theoretical assumptions. It merely depicts behavior as recorded experimentally and is based solely on the demonstrated fit of eqs 2,3, and 5. Extrapolation of the relationship beyond the experimental range (Figure 4,right) probably gives only an approximate account of the gliadin behavior, although nevertheless a fairly realistic one. Such a plot, although based on extrapolation, can be used to identify temperature-moisture combinations that are likely to produce high stiffness [R(T,W 31 11 or full plasticization [R(T,W = 01, as well as intermediate stiffness levels. It should also be added that similar plots can be created with a reference temperature other than -50 "C (223 K). Their shapes will be very similar to the one shown in Figure 4,as long as the reference temperature is not too close to T,. If the same reference temperature is selected for different materials, their corresponding stiffness-temperature-moisture relationships plotted on the same scale can reveal, in a glance, differences and similarities that may not be obvious from either a series of individual stiffness-temperature plots at selected moisture levels or stiffness-moisture plots at selected temperatures.
Conclusions The model originally devised for R(T,M)= G(T)/G,(eq 11, where G , is the storage modulus in the glassy state, can also be used with R(T,W = G(T)/G(T,),where T, (T,
