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Shear modulus-temperature meltdown profiles of gelatin and pectin gels. A cascade theory description
Shear modulus-temperature meltdown profiles of gelatin and pectin gels. A cascade theory description A. H. Clark*, K. T. Evans and D. B. Farrer Unilever Research Laboratory, Co~worth House, Shambrook, Bedford, MK44 1LQ, UK Received 4 November 1993; revised 12 January 1994
In this paper, a cascade theory approach to biopolymer gelation is developed to describe variation of the shear modulus with temperature for thermoreversible gels. The broadness of this 'melting transition' is seen to depend crucially on the enthalpy of cross-linking, while the critical gel melting temperature is determined by additional factors such as the entropy of cross-linking, the polymer concentration, the molecular weight, and the number of cross-linking sites. On using the model to fit experimental data for specific gelatin and pectin systems, it was found that the behaviour of gelatin (sharp melting transition and low melting temperature) is a consequence of the large negative enthalpy and entropy of cross-linking, while, in contrast, the broad melting transition and high melting point of the pectin system are consistent with much smaller negative values for these parameters.
Keywords: biopolymer gelation; cascade theory; shear modulus
Thermoreversibility is a well-known property of biopolymer gels. The gelatin-water system probably provides the most familiar example of a 'melting' gel, but there are several other systems showing this property, particularly polysaccharide gels such as those based on agar, carrageenan, certain pectins, modified starches, etc. Probably the first theoretical description of the melting event is that by Eldridge and Ferry x who gave mathematical relationships relating gel melting temperature to polymer concentration and molecular weight. They focused particularly on the gelatin system, and explained melting in terms of a shift in a reversible cross-linking equilibrium to a point where the degree of cross-linking falls below the critical threshold necessary for gelation (the Flory-Stockmayer condition2'a). Thus, certainly for homogeneous biopolymer gels, melting is seen not as a first-order phase transition but rather as the passage of the system through a condition of critical branching (the critical gel point). This is consistent with the fact that gel melting can be quite a gradual transition in terms of the behaviour of the gel modulus in relation to temperature. As Eldridge and Ferry made clear, the dependence of the melting temperature on concentration depends crucially on the size of the enthalpy of cross-linking, i.e. the greater this value, the lower the concentration dependence and the sharper the melting transition. However, the absolute value of the critical gel melting temperature will depend on factors additional to cross-linking enthalpy, such as the cross-linking entropy, the polymer concentration, the *To whomcorrespondenceshould be addressed 0141-8130/94/030125-06 © 1994Butterworth-HeinemannLimited
molecular weight, and the number of cross-linking sites per molecule (see below). The present article seeks to redevelop the EldridgeFerry approach to biopolymer gel thermoreversibility using a more recently developed model of gelation based on cascade theory 4-6. This approach will be used not only to re-emphasize the importance of the mathematical plots suggested by Eldridge and Ferry to relate melting temperature to polymer concentration and molecular weight, but it will also allow mathematical description of the shear modulus versus temperature meltdown profile itself in terms of the various molecular factors cited above. The model will be applied to meltdown profiles measured for two specific biopolymer systems, onebased on gelatin and the other based on pectin. Theory A previous treatment of biopolymer network formation 6 using cascade theory 4 has led to development of the following relationship between the gel shear modulus (G) and factors such as biopolymer concentration, molecular weight, degree of cross-linking, temperature etc.: G = {Nf0t(l - o)2(1 - fl)/2}aRT
(1)
where N is the number of moles of polymer per unit volume present initially (equal to C/M, where C is the concentration in mass terms and M is the molecular weight);fis the number of sites (or functionalities) along each molecule's length which are potentially available for cross-linking to other sites on other chains; • is the fraction of all such sites which have reacted at any stage
Int. J. Biol. Macromol. Volume 16 Number 3 1994 125
Modulus-temperature profiles of gelatin and pectin gels: A. t4. Clark et al. during the gel's history; and the quantities v and fl, which are functions o f f and :t, and which are key elements of the cascade approach to network formation, are given by the equations: v=(1 "JC-O~"l-O~V)f - 1
(2)
and fl = ( f - l)~tv/(1 - at + ~v)
(3)
Together, these quantities compose a function (shown in parentheses {} in equation (1)) which is a measure of the number of moles of elastically active chains (EANCs) per unit volume, where an EANC is a segment of the network which is deformed as the network is deformed, and which contributes to the overall increase in free energy which accompanies deformation 4. The factor a R T which also appears in equation (1) is a measure of the average contribution per mole of EANCs to the free energy increase per unit strain, and hence to the modulus. As has been suggested from ideal rubber treatments of polymer networks 7, a is expected to be close to unity. However, for many biopolymer gels, particularly at higher concentrations, there is evidence that a can be substantially greater than this 5'6. Under these conditions, flexible Gaussian chains characteristic of rubber networks are replaced by much less flexible EANC elements, and there is probably an important enthalpic contribution to the modulus. From equation (1), it is evident that, for fixed values of N and f and temperature (T), the gel modulus is crucially determined by the value of ~t, the fraction of reacted sites. For thermoreversible gels cured for long periods, this can be calculated on the assumption of a cross-linking equilibrium, determined by an appropriate equilibrium constant (K). In this approach, the crosslinking sites are treated as independent reacting species in solution, entering into a binary association. From the usual equilibrium relationship appropriate to this situation, ~ can be expressed 6 as a function of the product N f K (or CfK/M) and can be calculated easily. In order to evaluate equation (1) using ~, v and fl must be calculated using expressions (2) and (3). Equation (2) is a recurrence relation defining the so-called extinction probability (v) which measures the probability that a reacted functionality on an arbitrary molecule of the system becomes 'extinct', i.e. does not connect to infinity, or, more realistically, does not connect to the boundaries of a macroscopic gel via connections through other molecules. In equation (3), the quantity fl, which is a function of v, ~t and f, has been formulated 4 so that the full expression in parentheses {} in equation (1), when divided by N, measures the average number of EANCs per polymer molecule. The question now is how can equation (1) be used to describe modulus-temperature data for thermoreversible gels, and, in addition, how can it be used to provide some expression for the gel melting temperature? First, it is clear from equation (1) that there is an explicit dependence of G on temperature through the factor R T, but of course there is an important implicit dependence, in that • will be temperature-dependent through the variation of K with temperature. Also, it is not impossible that a will also show temperature dependence, if there is a significant enthalpic contribution to the free energy increment per EANC. In the present work, such variation of a is neglected, but the consequences of variation in a could
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be significant for certain systems and will be discussed later. Therefore, the central issue regarding the use of equation (1) to calculate G(T), i.e. the variation of the shear modulus of equilibrated gels with absolute temperature, is the temperature dependence of the equilibrium constant K. This will be expected to follow the customary expression: K = exp(AS°/R) exp( - AH°/R T)
(.4)
where AH ° and AS° are the standard enthalpies and entropies of cross-linking, respectively, and are assumed in the present work to be independent of temperature over the gel melting temperature range. From equations (1) to (4), it follows that G can he expressed as a function of temperature for any fixed values of C, f and M, through the parameters a, AH ° and AS°. Since f is not usually known in advance for a given polymer, this quantity should, in principle, be included as an adjustable parameter, but, as has been found in previous analogous treatments of modulus-concentration data 5"6, f is too highly correlated with the other variables to be satisfactorily included in least-squares optimizations, i.e. convergence cannot be obtained. Instead, in current work, as in previous G versus C calculations, modelling exercises will be performed for fixing (increasing)f values, and the resulting fits and corresponding parameter sets will be compared. Finally, in relation to theory, the issue remains of the critical gel melting temperature itself. Since, at this temperature (Tin), the polymer concentration C is necessarily the critical gelling concentration, then, from previous discussion of the mathematical formulation of the critical concentration 5'6, it follows that C = M ( f - 1)/2K(Tm)f( f - 2)2
(5)
where K(Tm) is the value of the equilibrium constant at temperature Tm. Assuming that equation (4) holds, equation (5) can be rearranged to give: Tm=AH°/{AS°+R l n ( 2 C f ( f - 2 ) 2 / M ( f - 1))}
(6)
and this provides a relationship between Tm and AH °, AS°,f, M and C, consistent with the earlier Eldridge and Ferry approach 1, which advocated plotting In C versus 1/Tm to obtain a straight line whose slope established AH °. It is also evident from equation (6) that the gel melting temperature is expressible as a quotient of enthalpy and entropy terms, the latter containing contributions both from the intrinsic entropy of crosslinking and from an entropy change associated with the network-building process itself. The way in which the gel melting temperature is determined by molecular characteristics will receive more discussion at a later stage. However, in the next two sections, the applicability of the theory developed aoove to G versus temperature (t, °C) data for real systems is explored in relation to two fundamentally different biopolymer gel systems, gelatin and pectin.
Experimental Gelatin meltdown curve
A food grade gelatin sample (pigskin, bloom strength =250, isoelectric point=4.8) was used without further purification. The sample concentration was
Modulus-temperature profiles of gelatin and pectin gels: A. H. Clark et al. 50 g 1-1, Measurements of gel formation and subsequent meltdown were performed using a strain-controlled Weissenberg Rheogoniometer in oscillatory mode (1% strain; frequency 1 Hz). Gelation was induced by rapid quenching (40 to 20°C in less than 30 s) and the dynamic modulus was recorded for 180 min at 20°C. The resulting gel was then reheated from 20 to 40°C at 2°C per min, and a G' versus t (°C) curve was recorded. These data were reproducible to better than 5%, and were not significantly altered by changing to slower scan rates (e.g. 0.5°C per min). However, extremely slow scan rates were not examined. Pectin meltdown curve
A commercial pectin sample (Pomosin type Y-1902: degree of esterification--35.3%) was subjected to further purification involving complexation with copper and regeneration using ethanolic EDTA. The sample concentration was 28 g1-1. The pH was adjusted to 7.4, and the pectin was studied in the presence of 0.2 Mbackground sodium chloride electrolyte and a stoichiometric ratio of added calcium ions (R)=0.4. Measurement of gel formation and subsequent meltdown was performed using a Bohlin rheometer in oscillatory mode (2% strain; frequency 0.5 Hz). Starting solutions were made by rapidly mixing hot sodium pectinate and calcium chloride solutions (80°C) and then loading onto the rheometer. The mixture was rapidly cooled to 20°C, and the cure curve was recorded for 120 min. The G' versus t meltdown curve was then recorded by reheating the sample to 80°C at a rate of 0.5°C per min. In some additional experiments, gels were cured at temperatures up to 50°C and long-time moduli were recorded. These results were compared with the meltdown data obtained for the 20°C cured system. Reproducibility of pectin data was not as good as for gelatin (10% on average) and scan-rate effects were still significant at 0.5°C per min (see later).
circumstances. Although the parameter values derived from least-squares optimization are to some extent correlated with M, and hence influenced by the exact value of M employed, this source of error will be regarded as of secondary importance for the purposes of the present study, and is unlikely to invalidate the overall conclusions. Even when using constrained values for f, M and C in least-squares calculations, problems were still experienced when fitting meltdown curves to the remaining variables. In this situation, considerable correlation remains between the enthalpy and entropy of cross-linking, and, in the pectin case, for reasons that will be discussed below, the front factor a also correlated critically with these parameters. However, for the gelatin data, final best values for a, AH ° and AS°, and their uncertainties, could be obtained for f values lying in the range 3 to 100. Fits for the f = 3 and f = 100 models are indicated in Figure 1, and are of comparable quality, the f = 1 0 0 model being slightly better. Corresponding numerical results for parameters a, AH °, AS°, the melting temperature and the overall goodness of fit index (E) appear in Table 1. As Table 1 makes clear, it is impossible to make a convincing distinction between f models on the grounds of fit quality, but definite conclusions can be reached despite this uncertainty. First, the enthalpy of crosslinking is always large and negative, being comparable
G'Pn
2000
~ -
~ ,
..........
. ~ o oo~' • ....
" ~
.................
Results and discussion
The meltdown profiles for the gelatin and pectin samples are shown in Figures 1 and 2. Figure 2 also shows modulus values obtained for the pectin system by curing at different temperatures as described in the previous section. The gelatin sample clearly has a much lower critical melting temperature and melts much more sharply than the pectin. In fact, the pectin melting is so protracted in temperature terms that the initial plateau behaviour prior to melting has not been accessed. To analyse the data in Figures 1 and 2, a computer program was written to fit equation (1) to meltdown data by a non-linear least-squares method. The variable parameters were the front factor a, and the standard enthalpies and entropies of cross-linking, AH° and AS°, as described previously. As mentioned earlier, severe correlation of these parameters with f meant that f was treated as a constant during fitting, and fits were made for a series ofincreasingfvalues ( f = 3, 5, 10, 20 and 100). Another constant of the treatment was the molecular weight M. For pectin, a number-average value of 36 800 from GPC light scattering was available for the present sample and was assumed, although the index M w / M n = l . 6 which was also available warned of significant polydispersity. For the gelatin sample, no comparable molecular weight data were available, and a typical value of 100000 was assumed in these
1500
!
I
10
20
30
t'C Figure 1 Experimental meltdown data for the gelatin system (O). The best fitting f = 3 (--) and f = 100 (...) models are shown. See text for the experimental conditions and fitting procedure
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127
Modulus-temperature profiles of gelatin and pectin gels: A. H. Clark et al. to the value expected for a covalent bond. This explains the sharpness of the gelatin melting transition. Second, the entropy of cross-linking is also negative and quite large, indicating the entropy penalty imposed as the gelatin triple helix cross-link forms. The size of this entropy parameter explains the low melting point of gelatin gels, since this temperature is determined mainly by the quotient AH°/AS °, with the additional network contribution to the entropy (see the denominator of
G'P|
8000
6000
\ 4000
3,f = 100
\
2000
,5(3
1OO
1,50
t°C
Figure 2 Experimental meltdown data for the pectin system (O), and experimental data for pectin systems cured at temperatures in the range of 20 to 50°C (©). The best fitting f = 3 and f = 100 models are shown as a broken line and as small points, respectively. See text for the experimental conditions and fitting procedure
equation (6)) being much smaller. Third, although its exact value varies with f, the calculated front factor a is of the same order as the ideal rubber theory expectation (except for f = 100, where it seems unrealistically small). This is expected given the nature of the gelatin network, as it is believed to contain a high proportion of randomly coiled peptide chain segments not involved in junction zone formation. Lastly, it is interesting that there is a slight rise in the gelatin modulus prior to melting which is also described by the theory. This is a consequence of the factor aRT in equation (1) dominating the temperature dependence prior to the sudden drop in K. Turning to the pectin results in Figure 2 and Table 2, a rather different picture emerges. In the pectin case, considerable difficulties were experienced in obtaining converged fits when all three variable parameters were adjusted. In practice, even when fits were carried out for a series of fixed cross-linking enthalpy values, no clearly defined optimum fit to the data was obtained. It was found that, over the range of 0 to - 0 . 6 k c a l m o l -~, the fit to data achieved was insignificantly different, although it did become much poorer as the cross-linking enthalpy became more negative than -0.6. In addition, severe correlation between the front factor and the other parameters led to enormous uncertainty in this quantity (Table 2), making it essentially undetermined. To construct Table 2, final fits were carried out assuming a fixed value of AH ° = - 0.2 kcal mol- 1 (mid-point of flat E variation), and uncertainties were estimated by allowing all three variables to change in a last least-squares cycle. From the results in Table 2, the following conclusions can be drawn. First, as in the gelatin situation, no clear distinction can be made betweenfmodels on the grounds of goodness of fit. Indeed, the f = 3 and f = 100 models give almost identical representations of the data, fitting the higher temperature measurements rather better than the data points close to 20°C. Comparison of these last data points with modulus values obtained by curing gels in the temperature range 20°C to 50°C (also shown in Figure 2 and treated as zero scan-rate estimates) suggests that there is a small response lag in operation at the temperature scan rate imposed, where the meltdown procedure is involved. This will be ignored in the following discussion, but it should be recognized that this kinetic feature of the current pectin data will have some influence on the values of the parameters extracted in data fitting and shown in Table 2. From the results in Table 2 it is clear that, for all models, the suggested best estimate for the enthalpy of
Table 1 Results of least-squares analyses of gelatin meltdown data (see text for details). Z is the sum of squared deviations based on log G values. Estimated standard deviations are given in parentheses f values 3 a AH ° (kcal m o l - 1) AS° (kcal tool- i K - 1)
5
10
20
100
1.09
0.67
0.33
0.161
0.031
(0.02)
(0.01)
(0.010)
(0.002)
(0.001)
- 180 (2) - 0.580 (0.015)
- 150 (2) - 0.485 (0.012)
- 180 (4) - 0.588 (0.027)
- 240 (3) - 0.789 (0.021)
- 400 (10) - 1.323 (0.069)
Tm (°C)
30.3
30.3
30.3
30.2
30.3
Z
0.0041
0.0037
0.0076
0.0038
0.0034
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Modulus-temperature profiles of gelatin and pectin gels: A. H. Clark et al. Table 2 Results of least-squares analyses of pectin meltdown data (see text for details). ~ is the sum of squared deviations based on log G values. Estimated standard deviations are given in parentheses
f values 3 a AH° (kcal mol- ~) AS* (kcal tool- 1K- t) Tm(°C)
5150 (30000) -0.2 (0.2) 0.012 (0.001) 130.1 0.0051
5 2372 (10000) -0.2 (0.2) 0.0076 (0.0006) 130.7 0.0051
cross-linking is very small in absolute terms, although it is of course still negative (melting with temperature increase could not occur for a positive enthalpy of cross-linking). The broadness of the pectin melting transition is clearly consistent with this very small heat of cross-linking. On the other hand, the entropy of cross-linking turns out to be either positive or negative, depending on which f model is examined, but again it is always very small in absolute value. For the pectin system, the high melting point is clearly partly a consequence of the small entropy of cross-linking, but, in this case, in contrast to the gelatin situation, the network contribution to the overall entropy change is of comparable size, and hence is also important. For pectin-calcium systems, a small entropy of cross-linking is to be expected, as previous structural studies 8 of pectin aggregation have suggested that little conformational change takes place as chains align to form junction zones. However, an enthalpy of cross-linking as close to zero as - 0 . 2 kcal mol-1 is much less easy to explain, particularly in the light of a much larger value reported in recent literature 8. However, for a number of reasons, the value of - 0 . 2 kcal mol-z may well have a substantially greater error than is indicated by the uncertainty quoted in Table 2. A lack of data at lower temperatures is certainly one problem (and the implications of this may not be fully reflected in the estimated error just referred to), but failure of one underlying assumption of the present treatment may be more important. This is the assumption that a is temperature-independent. While such temperature invariance is predicted for ideal rubber networks, a would be expected to fall with increasing temperature if there were a significant enthalpic contribution to the gel modulus (which is not unlikely for polysaccharide gels). In addition, there is some evidence 9 that, for biopolymer gels, a is an increasing function of the ratio of polymer concentration (C) to the critical concentration (Co). Since Co increases up to the value C as temperature increases, this means that a could also fall off with increasing temperature because it depends on C/C o. Although such temperature variation of the front factor is unlikely to explain the entire melting profile of the pectin system (Figure 2), it may make a significant contribution, tending to broaden out the underlying effect arising from cross-link dissociation. This could (as some preliminary calculations suggest) lead to an erroneously small negative enthalpy being calculated during fitting of the constant a model to the data.
10 1564 (6000) -0.2 (0.2) 0.0039 (0.0006) 130.8 0.0051
20 1304 (6000) -0.2 (0.2) 0.0008 (0.0008) 130.9 0.0050
100 1140 (6000) -0.2 (0.2) - 0.0059 (0.0008) 130.9 0.0051
Finally, in view of the enormous uncertainties appropriate to estimates of the front factor a in Table 2, no conclusion can be reached about the value for this quantity appropriate to pectin gels. These uncertainties appear to relate principally to the broadness of the pectin melting transition, and to the fact that, as a consequence of this, the experimental melting transition data are limited in extent, i.e. do not and cannot extend down to a plateau value characteristic of the gel prior to melting.
Conclusion In principle, the gel melting profiles for thermoreversible gelling systems can be described by the cascade approach to biopolymer gelation. However, there are difficulties with parameter correlation which are particularly severe in cases where only part of the melting transition is accessible (as for the pectin data above). This tends to happen when the cross-linking enthalpy is low, making the melting transition extremely broad. However, such parameters as have been determined convincingly in the analyses of the last section are in accordance with current molecular descriptions of the gelatin and pectin networks and cross-linking mechanisms. The absolute value of the gel melting temperature in any circumstances is expressed (equation (6)) as a quotient of enthalpy and entropy contributions. For large values of the enthalpy of cross-linking (e.g. approaching that characteristic of covalent bonds), and for melting temperatures in the range of 0 to 200°C, the entropy Of cross-linking must also be sizeable and negative. It is then much more significant in determining the melting temperature than the additional entropy contribution arising from network assembly of the interacting molecules. Under these conditions, the gel melting temperature is not strongly concentration-dependent, and melting occurs over a narrow temperature range (i.e. there is a tendency towards what is conventionally understood as melting, i.e. a phase transition). On the other hand, where the enthalpy is much smaller (with an absolute value of a few kcal mol-1), and for the same melting point as the high enthalpy system, the entropy of cross-linking is much smaller. In this case, it will become comparable to the network contribution (which is a function of f, C and M) and this latter contribution then partly determines the melting temperature of the gel. In these circumstances, melting becomes more
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Modulus-temperature profiles of gelatin and pectin gels: A. H. Clark et al. gradual, and much more concentration- and molecularweight-dependent. Finally, as discussed at some length in the previous section, variation of the front factor a with temperature has been assumed to be absent in the present treatment. This could be an incorrect assumption in some circumstances, such variation contributing significantly to the gel melting profile and placing in doubt the reliability of physical quantities extracted by the fitting procedures. Further work is required to assess the importance of this alternative 'gel melting mechanism' for biopolymer gel systems.
Acknowledgements A. H. C. is grateful to Professor S. B. Ross-Murphy of King's College, London, for much useful discussion of
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the gel melting problem. The authors are also grateful to Mrs L. Linger for technical assistance in the preparation of the manuscript.
References 1 2 3 4
Eldridge,J.E. and Ferry, J.D.J. Phys, Chem. 1954, 58, 1054 Flory,P.J.J. Am. Chem. Soc. 1941, 63, 3083 Stockmayer,W.H.J. Chem. Phys. 1943, 11, 45 Gordon, M. and Ross-Murphy,S.B. Pure Appl. Chem. 1975,43, l .
5
Clark, A.H.andRoss-Murphy, S.B. Br.PolymerJ. 1985,17,164
6
Clark,A.H. in 'Food Structure and Bchaviour' (Eds P.J. LiUford and J.M.V. Blanshard), Academic Press, London, 1987, p 13 Flory,P.J. 'Principles of Polymer Chemistry', Cornell University Press, Ithaca, New York, 1953, chapter 11 Durand, D., Bertrand, C., Bushel, J.-P., Emery, J., Axelos, M., Thibault, J.F. et al. in 'Physical Networks Polymers and Gels' (Eds W. Burchard and S.B. Ross-Murphy),Elsevier,London, 1990, p 283 Clark,A. H. Carbohydr. Polym. in press
7 8
9
In this paper, a cascade theory approach to biopolymer gelation is developed to describe variation of the shear modulus with temperature for thermoreversible gels. The broadness of this 'melting transition' is seen to depend crucially on the enthalpy of cross-linking, while the critical gel melting temperature is determined by additional factors such as the entropy of cross-linking, the polymer concentration, the molecular weight, and the number of cross-linking sites. On using the model to fit experimental data for specific gelatin and pectin systems, it was found that the behaviour of gelatin (sharp melting transition and low melting temperature) is a consequence of the large negative enthalpy and entropy of cross-linking, while, in contrast, the broad melting transition and high melting point of the pectin system are consistent with much smaller negative values for these parameters.
Keywords: biopolymer gelation; cascade theory; shear modulus
Thermoreversibility is a well-known property of biopolymer gels. The gelatin-water system probably provides the most familiar example of a 'melting' gel, but there are several other systems showing this property, particularly polysaccharide gels such as those based on agar, carrageenan, certain pectins, modified starches, etc. Probably the first theoretical description of the melting event is that by Eldridge and Ferry x who gave mathematical relationships relating gel melting temperature to polymer concentration and molecular weight. They focused particularly on the gelatin system, and explained melting in terms of a shift in a reversible cross-linking equilibrium to a point where the degree of cross-linking falls below the critical threshold necessary for gelation (the Flory-Stockmayer condition2'a). Thus, certainly for homogeneous biopolymer gels, melting is seen not as a first-order phase transition but rather as the passage of the system through a condition of critical branching (the critical gel point). This is consistent with the fact that gel melting can be quite a gradual transition in terms of the behaviour of the gel modulus in relation to temperature. As Eldridge and Ferry made clear, the dependence of the melting temperature on concentration depends crucially on the size of the enthalpy of cross-linking, i.e. the greater this value, the lower the concentration dependence and the sharper the melting transition. However, the absolute value of the critical gel melting temperature will depend on factors additional to cross-linking enthalpy, such as the cross-linking entropy, the polymer concentration, the *To whomcorrespondenceshould be addressed 0141-8130/94/030125-06 © 1994Butterworth-HeinemannLimited
molecular weight, and the number of cross-linking sites per molecule (see below). The present article seeks to redevelop the EldridgeFerry approach to biopolymer gel thermoreversibility using a more recently developed model of gelation based on cascade theory 4-6. This approach will be used not only to re-emphasize the importance of the mathematical plots suggested by Eldridge and Ferry to relate melting temperature to polymer concentration and molecular weight, but it will also allow mathematical description of the shear modulus versus temperature meltdown profile itself in terms of the various molecular factors cited above. The model will be applied to meltdown profiles measured for two specific biopolymer systems, onebased on gelatin and the other based on pectin. Theory A previous treatment of biopolymer network formation 6 using cascade theory 4 has led to development of the following relationship between the gel shear modulus (G) and factors such as biopolymer concentration, molecular weight, degree of cross-linking, temperature etc.: G = {Nf0t(l - o)2(1 - fl)/2}aRT
(1)
where N is the number of moles of polymer per unit volume present initially (equal to C/M, where C is the concentration in mass terms and M is the molecular weight);fis the number of sites (or functionalities) along each molecule's length which are potentially available for cross-linking to other sites on other chains; • is the fraction of all such sites which have reacted at any stage
Int. J. Biol. Macromol. Volume 16 Number 3 1994 125
Modulus-temperature profiles of gelatin and pectin gels: A. t4. Clark et al. during the gel's history; and the quantities v and fl, which are functions o f f and :t, and which are key elements of the cascade approach to network formation, are given by the equations: v=(1 "JC-O~"l-O~V)f - 1
(2)
and fl = ( f - l)~tv/(1 - at + ~v)
(3)
Together, these quantities compose a function (shown in parentheses {} in equation (1)) which is a measure of the number of moles of elastically active chains (EANCs) per unit volume, where an EANC is a segment of the network which is deformed as the network is deformed, and which contributes to the overall increase in free energy which accompanies deformation 4. The factor a R T which also appears in equation (1) is a measure of the average contribution per mole of EANCs to the free energy increase per unit strain, and hence to the modulus. As has been suggested from ideal rubber treatments of polymer networks 7, a is expected to be close to unity. However, for many biopolymer gels, particularly at higher concentrations, there is evidence that a can be substantially greater than this 5'6. Under these conditions, flexible Gaussian chains characteristic of rubber networks are replaced by much less flexible EANC elements, and there is probably an important enthalpic contribution to the modulus. From equation (1), it is evident that, for fixed values of N and f and temperature (T), the gel modulus is crucially determined by the value of ~t, the fraction of reacted sites. For thermoreversible gels cured for long periods, this can be calculated on the assumption of a cross-linking equilibrium, determined by an appropriate equilibrium constant (K). In this approach, the crosslinking sites are treated as independent reacting species in solution, entering into a binary association. From the usual equilibrium relationship appropriate to this situation, ~ can be expressed 6 as a function of the product N f K (or CfK/M) and can be calculated easily. In order to evaluate equation (1) using ~, v and fl must be calculated using expressions (2) and (3). Equation (2) is a recurrence relation defining the so-called extinction probability (v) which measures the probability that a reacted functionality on an arbitrary molecule of the system becomes 'extinct', i.e. does not connect to infinity, or, more realistically, does not connect to the boundaries of a macroscopic gel via connections through other molecules. In equation (3), the quantity fl, which is a function of v, ~t and f, has been formulated 4 so that the full expression in parentheses {} in equation (1), when divided by N, measures the average number of EANCs per polymer molecule. The question now is how can equation (1) be used to describe modulus-temperature data for thermoreversible gels, and, in addition, how can it be used to provide some expression for the gel melting temperature? First, it is clear from equation (1) that there is an explicit dependence of G on temperature through the factor R T, but of course there is an important implicit dependence, in that • will be temperature-dependent through the variation of K with temperature. Also, it is not impossible that a will also show temperature dependence, if there is a significant enthalpic contribution to the free energy increment per EANC. In the present work, such variation of a is neglected, but the consequences of variation in a could
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be significant for certain systems and will be discussed later. Therefore, the central issue regarding the use of equation (1) to calculate G(T), i.e. the variation of the shear modulus of equilibrated gels with absolute temperature, is the temperature dependence of the equilibrium constant K. This will be expected to follow the customary expression: K = exp(AS°/R) exp( - AH°/R T)
(.4)
where AH ° and AS° are the standard enthalpies and entropies of cross-linking, respectively, and are assumed in the present work to be independent of temperature over the gel melting temperature range. From equations (1) to (4), it follows that G can he expressed as a function of temperature for any fixed values of C, f and M, through the parameters a, AH ° and AS°. Since f is not usually known in advance for a given polymer, this quantity should, in principle, be included as an adjustable parameter, but, as has been found in previous analogous treatments of modulus-concentration data 5"6, f is too highly correlated with the other variables to be satisfactorily included in least-squares optimizations, i.e. convergence cannot be obtained. Instead, in current work, as in previous G versus C calculations, modelling exercises will be performed for fixing (increasing)f values, and the resulting fits and corresponding parameter sets will be compared. Finally, in relation to theory, the issue remains of the critical gel melting temperature itself. Since, at this temperature (Tin), the polymer concentration C is necessarily the critical gelling concentration, then, from previous discussion of the mathematical formulation of the critical concentration 5'6, it follows that C = M ( f - 1)/2K(Tm)f( f - 2)2
(5)
where K(Tm) is the value of the equilibrium constant at temperature Tm. Assuming that equation (4) holds, equation (5) can be rearranged to give: Tm=AH°/{AS°+R l n ( 2 C f ( f - 2 ) 2 / M ( f - 1))}
(6)
and this provides a relationship between Tm and AH °, AS°,f, M and C, consistent with the earlier Eldridge and Ferry approach 1, which advocated plotting In C versus 1/Tm to obtain a straight line whose slope established AH °. It is also evident from equation (6) that the gel melting temperature is expressible as a quotient of enthalpy and entropy terms, the latter containing contributions both from the intrinsic entropy of crosslinking and from an entropy change associated with the network-building process itself. The way in which the gel melting temperature is determined by molecular characteristics will receive more discussion at a later stage. However, in the next two sections, the applicability of the theory developed aoove to G versus temperature (t, °C) data for real systems is explored in relation to two fundamentally different biopolymer gel systems, gelatin and pectin.
Experimental Gelatin meltdown curve
A food grade gelatin sample (pigskin, bloom strength =250, isoelectric point=4.8) was used without further purification. The sample concentration was
Modulus-temperature profiles of gelatin and pectin gels: A. H. Clark et al. 50 g 1-1, Measurements of gel formation and subsequent meltdown were performed using a strain-controlled Weissenberg Rheogoniometer in oscillatory mode (1% strain; frequency 1 Hz). Gelation was induced by rapid quenching (40 to 20°C in less than 30 s) and the dynamic modulus was recorded for 180 min at 20°C. The resulting gel was then reheated from 20 to 40°C at 2°C per min, and a G' versus t (°C) curve was recorded. These data were reproducible to better than 5%, and were not significantly altered by changing to slower scan rates (e.g. 0.5°C per min). However, extremely slow scan rates were not examined. Pectin meltdown curve
A commercial pectin sample (Pomosin type Y-1902: degree of esterification--35.3%) was subjected to further purification involving complexation with copper and regeneration using ethanolic EDTA. The sample concentration was 28 g1-1. The pH was adjusted to 7.4, and the pectin was studied in the presence of 0.2 Mbackground sodium chloride electrolyte and a stoichiometric ratio of added calcium ions (R)=0.4. Measurement of gel formation and subsequent meltdown was performed using a Bohlin rheometer in oscillatory mode (2% strain; frequency 0.5 Hz). Starting solutions were made by rapidly mixing hot sodium pectinate and calcium chloride solutions (80°C) and then loading onto the rheometer. The mixture was rapidly cooled to 20°C, and the cure curve was recorded for 120 min. The G' versus t meltdown curve was then recorded by reheating the sample to 80°C at a rate of 0.5°C per min. In some additional experiments, gels were cured at temperatures up to 50°C and long-time moduli were recorded. These results were compared with the meltdown data obtained for the 20°C cured system. Reproducibility of pectin data was not as good as for gelatin (10% on average) and scan-rate effects were still significant at 0.5°C per min (see later).
circumstances. Although the parameter values derived from least-squares optimization are to some extent correlated with M, and hence influenced by the exact value of M employed, this source of error will be regarded as of secondary importance for the purposes of the present study, and is unlikely to invalidate the overall conclusions. Even when using constrained values for f, M and C in least-squares calculations, problems were still experienced when fitting meltdown curves to the remaining variables. In this situation, considerable correlation remains between the enthalpy and entropy of cross-linking, and, in the pectin case, for reasons that will be discussed below, the front factor a also correlated critically with these parameters. However, for the gelatin data, final best values for a, AH ° and AS°, and their uncertainties, could be obtained for f values lying in the range 3 to 100. Fits for the f = 3 and f = 100 models are indicated in Figure 1, and are of comparable quality, the f = 1 0 0 model being slightly better. Corresponding numerical results for parameters a, AH °, AS°, the melting temperature and the overall goodness of fit index (E) appear in Table 1. As Table 1 makes clear, it is impossible to make a convincing distinction between f models on the grounds of fit quality, but definite conclusions can be reached despite this uncertainty. First, the enthalpy of crosslinking is always large and negative, being comparable
G'Pn
2000
~ -
~ ,
..........
. ~ o oo~' • ....
" ~
.................
Results and discussion
The meltdown profiles for the gelatin and pectin samples are shown in Figures 1 and 2. Figure 2 also shows modulus values obtained for the pectin system by curing at different temperatures as described in the previous section. The gelatin sample clearly has a much lower critical melting temperature and melts much more sharply than the pectin. In fact, the pectin melting is so protracted in temperature terms that the initial plateau behaviour prior to melting has not been accessed. To analyse the data in Figures 1 and 2, a computer program was written to fit equation (1) to meltdown data by a non-linear least-squares method. The variable parameters were the front factor a, and the standard enthalpies and entropies of cross-linking, AH° and AS°, as described previously. As mentioned earlier, severe correlation of these parameters with f meant that f was treated as a constant during fitting, and fits were made for a series ofincreasingfvalues ( f = 3, 5, 10, 20 and 100). Another constant of the treatment was the molecular weight M. For pectin, a number-average value of 36 800 from GPC light scattering was available for the present sample and was assumed, although the index M w / M n = l . 6 which was also available warned of significant polydispersity. For the gelatin sample, no comparable molecular weight data were available, and a typical value of 100000 was assumed in these
1500
!
I
10
20
30
t'C Figure 1 Experimental meltdown data for the gelatin system (O). The best fitting f = 3 (--) and f = 100 (...) models are shown. See text for the experimental conditions and fitting procedure
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Modulus-temperature profiles of gelatin and pectin gels: A. H. Clark et al. to the value expected for a covalent bond. This explains the sharpness of the gelatin melting transition. Second, the entropy of cross-linking is also negative and quite large, indicating the entropy penalty imposed as the gelatin triple helix cross-link forms. The size of this entropy parameter explains the low melting point of gelatin gels, since this temperature is determined mainly by the quotient AH°/AS °, with the additional network contribution to the entropy (see the denominator of
G'P|
8000
6000
\ 4000
3,f = 100
\
2000
,5(3
1OO
1,50
t°C
Figure 2 Experimental meltdown data for the pectin system (O), and experimental data for pectin systems cured at temperatures in the range of 20 to 50°C (©). The best fitting f = 3 and f = 100 models are shown as a broken line and as small points, respectively. See text for the experimental conditions and fitting procedure
equation (6)) being much smaller. Third, although its exact value varies with f, the calculated front factor a is of the same order as the ideal rubber theory expectation (except for f = 100, where it seems unrealistically small). This is expected given the nature of the gelatin network, as it is believed to contain a high proportion of randomly coiled peptide chain segments not involved in junction zone formation. Lastly, it is interesting that there is a slight rise in the gelatin modulus prior to melting which is also described by the theory. This is a consequence of the factor aRT in equation (1) dominating the temperature dependence prior to the sudden drop in K. Turning to the pectin results in Figure 2 and Table 2, a rather different picture emerges. In the pectin case, considerable difficulties were experienced in obtaining converged fits when all three variable parameters were adjusted. In practice, even when fits were carried out for a series of fixed cross-linking enthalpy values, no clearly defined optimum fit to the data was obtained. It was found that, over the range of 0 to - 0 . 6 k c a l m o l -~, the fit to data achieved was insignificantly different, although it did become much poorer as the cross-linking enthalpy became more negative than -0.6. In addition, severe correlation between the front factor and the other parameters led to enormous uncertainty in this quantity (Table 2), making it essentially undetermined. To construct Table 2, final fits were carried out assuming a fixed value of AH ° = - 0.2 kcal mol- 1 (mid-point of flat E variation), and uncertainties were estimated by allowing all three variables to change in a last least-squares cycle. From the results in Table 2, the following conclusions can be drawn. First, as in the gelatin situation, no clear distinction can be made betweenfmodels on the grounds of goodness of fit. Indeed, the f = 3 and f = 100 models give almost identical representations of the data, fitting the higher temperature measurements rather better than the data points close to 20°C. Comparison of these last data points with modulus values obtained by curing gels in the temperature range 20°C to 50°C (also shown in Figure 2 and treated as zero scan-rate estimates) suggests that there is a small response lag in operation at the temperature scan rate imposed, where the meltdown procedure is involved. This will be ignored in the following discussion, but it should be recognized that this kinetic feature of the current pectin data will have some influence on the values of the parameters extracted in data fitting and shown in Table 2. From the results in Table 2 it is clear that, for all models, the suggested best estimate for the enthalpy of
Table 1 Results of least-squares analyses of gelatin meltdown data (see text for details). Z is the sum of squared deviations based on log G values. Estimated standard deviations are given in parentheses f values 3 a AH ° (kcal m o l - 1) AS° (kcal tool- i K - 1)
5
10
20
100
1.09
0.67
0.33
0.161
0.031
(0.02)
(0.01)
(0.010)
(0.002)
(0.001)
- 180 (2) - 0.580 (0.015)
- 150 (2) - 0.485 (0.012)
- 180 (4) - 0.588 (0.027)
- 240 (3) - 0.789 (0.021)
- 400 (10) - 1.323 (0.069)
Tm (°C)
30.3
30.3
30.3
30.2
30.3
Z
0.0041
0.0037
0.0076
0.0038
0.0034
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Modulus-temperature profiles of gelatin and pectin gels: A. H. Clark et al. Table 2 Results of least-squares analyses of pectin meltdown data (see text for details). ~ is the sum of squared deviations based on log G values. Estimated standard deviations are given in parentheses
f values 3 a AH° (kcal mol- ~) AS* (kcal tool- 1K- t) Tm(°C)
5150 (30000) -0.2 (0.2) 0.012 (0.001) 130.1 0.0051
5 2372 (10000) -0.2 (0.2) 0.0076 (0.0006) 130.7 0.0051
cross-linking is very small in absolute terms, although it is of course still negative (melting with temperature increase could not occur for a positive enthalpy of cross-linking). The broadness of the pectin melting transition is clearly consistent with this very small heat of cross-linking. On the other hand, the entropy of cross-linking turns out to be either positive or negative, depending on which f model is examined, but again it is always very small in absolute value. For the pectin system, the high melting point is clearly partly a consequence of the small entropy of cross-linking, but, in this case, in contrast to the gelatin situation, the network contribution to the overall entropy change is of comparable size, and hence is also important. For pectin-calcium systems, a small entropy of cross-linking is to be expected, as previous structural studies 8 of pectin aggregation have suggested that little conformational change takes place as chains align to form junction zones. However, an enthalpy of cross-linking as close to zero as - 0 . 2 kcal mol-1 is much less easy to explain, particularly in the light of a much larger value reported in recent literature 8. However, for a number of reasons, the value of - 0 . 2 kcal mol-z may well have a substantially greater error than is indicated by the uncertainty quoted in Table 2. A lack of data at lower temperatures is certainly one problem (and the implications of this may not be fully reflected in the estimated error just referred to), but failure of one underlying assumption of the present treatment may be more important. This is the assumption that a is temperature-independent. While such temperature invariance is predicted for ideal rubber networks, a would be expected to fall with increasing temperature if there were a significant enthalpic contribution to the gel modulus (which is not unlikely for polysaccharide gels). In addition, there is some evidence 9 that, for biopolymer gels, a is an increasing function of the ratio of polymer concentration (C) to the critical concentration (Co). Since Co increases up to the value C as temperature increases, this means that a could also fall off with increasing temperature because it depends on C/C o. Although such temperature variation of the front factor is unlikely to explain the entire melting profile of the pectin system (Figure 2), it may make a significant contribution, tending to broaden out the underlying effect arising from cross-link dissociation. This could (as some preliminary calculations suggest) lead to an erroneously small negative enthalpy being calculated during fitting of the constant a model to the data.
10 1564 (6000) -0.2 (0.2) 0.0039 (0.0006) 130.8 0.0051
20 1304 (6000) -0.2 (0.2) 0.0008 (0.0008) 130.9 0.0050
100 1140 (6000) -0.2 (0.2) - 0.0059 (0.0008) 130.9 0.0051
Finally, in view of the enormous uncertainties appropriate to estimates of the front factor a in Table 2, no conclusion can be reached about the value for this quantity appropriate to pectin gels. These uncertainties appear to relate principally to the broadness of the pectin melting transition, and to the fact that, as a consequence of this, the experimental melting transition data are limited in extent, i.e. do not and cannot extend down to a plateau value characteristic of the gel prior to melting.
Conclusion In principle, the gel melting profiles for thermoreversible gelling systems can be described by the cascade approach to biopolymer gelation. However, there are difficulties with parameter correlation which are particularly severe in cases where only part of the melting transition is accessible (as for the pectin data above). This tends to happen when the cross-linking enthalpy is low, making the melting transition extremely broad. However, such parameters as have been determined convincingly in the analyses of the last section are in accordance with current molecular descriptions of the gelatin and pectin networks and cross-linking mechanisms. The absolute value of the gel melting temperature in any circumstances is expressed (equation (6)) as a quotient of enthalpy and entropy contributions. For large values of the enthalpy of cross-linking (e.g. approaching that characteristic of covalent bonds), and for melting temperatures in the range of 0 to 200°C, the entropy Of cross-linking must also be sizeable and negative. It is then much more significant in determining the melting temperature than the additional entropy contribution arising from network assembly of the interacting molecules. Under these conditions, the gel melting temperature is not strongly concentration-dependent, and melting occurs over a narrow temperature range (i.e. there is a tendency towards what is conventionally understood as melting, i.e. a phase transition). On the other hand, where the enthalpy is much smaller (with an absolute value of a few kcal mol-1), and for the same melting point as the high enthalpy system, the entropy of cross-linking is much smaller. In this case, it will become comparable to the network contribution (which is a function of f, C and M) and this latter contribution then partly determines the melting temperature of the gel. In these circumstances, melting becomes more
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Modulus-temperature profiles of gelatin and pectin gels: A. H. Clark et al. gradual, and much more concentration- and molecularweight-dependent. Finally, as discussed at some length in the previous section, variation of the front factor a with temperature has been assumed to be absent in the present treatment. This could be an incorrect assumption in some circumstances, such variation contributing significantly to the gel melting profile and placing in doubt the reliability of physical quantities extracted by the fitting procedures. Further work is required to assess the importance of this alternative 'gel melting mechanism' for biopolymer gel systems.
Acknowledgements A. H. C. is grateful to Professor S. B. Ross-Murphy of King's College, London, for much useful discussion of
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the gel melting problem. The authors are also grateful to Mrs L. Linger for technical assistance in the preparation of the manuscript.
References 1 2 3 4
Eldridge,J.E. and Ferry, J.D.J. Phys, Chem. 1954, 58, 1054 Flory,P.J.J. Am. Chem. Soc. 1941, 63, 3083 Stockmayer,W.H.J. Chem. Phys. 1943, 11, 45 Gordon, M. and Ross-Murphy,S.B. Pure Appl. Chem. 1975,43, l .
5
Clark, A.H.andRoss-Murphy, S.B. Br.PolymerJ. 1985,17,164
6
Clark,A.H. in 'Food Structure and Bchaviour' (Eds P.J. LiUford and J.M.V. Blanshard), Academic Press, London, 1987, p 13 Flory,P.J. 'Principles of Polymer Chemistry', Cornell University Press, Ithaca, New York, 1953, chapter 11 Durand, D., Bertrand, C., Bushel, J.-P., Emery, J., Axelos, M., Thibault, J.F. et al. in 'Physical Networks Polymers and Gels' (Eds W. Burchard and S.B. Ross-Murphy),Elsevier,London, 1990, p 283 Clark,A. H. Carbohydr. Polym. in press
7 8
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