Predictions of Glass Transition Temperature for Hydrogen Bonding

Article pubs.acs.org/JPCB

Predictions of Glass Transition Temperature for Hydrogen Bonding Biomaterials R. G. M. van der Sman* Agrotechnology and Food Sciences Group, Wageningen University and Research Centre, 6708 PB Wageningen, The Netherlands ABSTRACT: We show that the glass transition of a multitude of mixtures containing hydrogen bonding materials correlates strongly with the effective number of hydroxyl groups per molecule, which are available for intermolecular hydrogen bonding. This correlation is in compliance with the topological constraint theory, wherein the intermolecular hydrogen bonds constrain the mobility of the hydrogen bonded network. The finding that the glass transition relates to hydrogen bonding rather than free volume agrees with our recent finding that there is little difference in free volume among carbohydrates and polysaccharides. For binary and ternary mixtures of sugars, polyols, or biopolymers with water, our correlation states that the glass transition temperature is linear with the inverse of the number of effective hydroxyl groups per molecule. Only for dry biopolymer/sugar or sugar/polyol mixtures do we find deviations due to nonideal mixing, imposed by microheterogeneity.

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often discussed only in a qualitative way.11,12 For example, the high glass transition temperature of food materials has been attributed to the strong hydrogen bonding between the molecules of the food materials (polysaccharides and proteins), and with their plasticizers (water, polyols and sugars).9,13 Differences in Tg are ascribed to differences in molecular packing, which are associated with differences in intermolecular hydrogen bonding.11,12,14 A more dense molecular packing implies a lower free volume. More quantitative relations between T g and hydrogen bonding are given in the literature,6−8,15 which build upon the work of Nakanishi and Nozaki. The latter study relates in a way to the topological constraint theory, which was originally developed for covalentbonded glasses.16 Recently, the theory has been shown to apply to simple saccharides.17 Before presenting our results, we give a critical review of theories for predicting Tg of mixtureseither based on free volume or on topological constraints. Subsequently, we analyze the Tg of binary and ternary mixtures of compounds from the following group of hydrogen-bonding materials: water, sugars, polyols, and biopolymers (polysaccharides and structural proteins). Several of these systems have a tendency to phaseseparate.18−21 However, it is shown that the phase separation evolves very slowly, and it is easily arrested via either gelation or a glass transition.19,22 Consequently, these arrested systems are practically stable, and find many applications in food and pharma. In these fields, the arrest of phase separation is often obtained via freeze-drying or spray-drying, which are the systems that we have investigated.

INTRODUCTION Many biomaterials, like food and pharma, interact predominantly with water via hydrogen bonding. Contrary to the common view in food science, there appears to be some degree of universality among food materials that interact via hydrogen bonding.1 This is also evident from some of our recent papers, where we have shown universal behavior concerning moisture sorption and moisture diffusivity in polysaccharides and sugars.2−5 The applied theories for both physical properties include effects of free volume, and they have the glass transition temperature as a parameter. We have argued that small water molecules probe only their local environment, which act an as a hydrogen bonded network. Hence, the chain length of solutes is quite irrelevant for moisture sorption and moisture diffusivity, which explains the universality. In this paper we present yet another universal scaling, which holds for the glass transition temperature Tg of mixtures of these biomaterials. Their Tg is shown to be related to the (averaged) number of hydroxyl groups per molecule. Our scaling is based on recent works of Nakanishi and Nozaki,6,7 with corrections for intramolecular hydrogen-bonding, as proposed by Pawlus and co-workers.8 Traditionally, the glass transition of mixtures is computed using free volume theories, such as the Gordon−Taylor, Debye−Bueche, or Couchman−Karasz relations.4,9 The glass transition of food materials like carbohydrates does depend on the chain length of the solute molecule, as expressed by the Fox−Flory relation.2 There, we showed that the glass transition of many food ingredients follows the Couchman−Karasz relation, with a universal value for ΔCp,s = 0.42 kJ/kg·K, which is the jump in the specific heat of the pure solute at its glass transition.2,4,10 Various earlier papers indicate a relation between the glass transition and the hydrogen bonding. However, this relation is © 2013 American Chemical Society

Received: August 15, 2013 Revised: October 21, 2013 Published: December 5, 2013 16303

dx.doi.org/10.1021/jp408184u | J. Phys. Chem. B 2013, 117, 16303−16313

The Journal of Physical Chemistry B

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GLASS TRANSITION OF MIXTURES Free Volume Theory. The free volume theory states the transition from the rubbery to the glassy state is accompanied by an abrupt change in the thermal expansion of the material. In glassy aqueous mixtures, the solutes become immobilized, and the thermal expansion becomes quite small. Above the glass transition temperature, the fluctuating free volume (the interstices between molecules) allows for the mobility of the solutes. The Williams−Laundel−Ferry (WLF) theory assumes that the free volume fraction has a universal value at the glass transition temperature: f(Tg) = fg ≈ 0.025, which is also called the iso-free volume hypothesis. Furthermore, it is assumed that the free volume fraction grows linearly with temperature. Above the glass transition, it holds:

Couchman−Karasz theory, we have been able to predict the sorption isotherms of the ternary solutions using the Flory− Huggins Free Volume theory, which requires Tg as a function of moisture content for the prediction. Hence, for complex (ternary) mixtures, the key assumption of ideal mixing (of free volume) is not totally true. Moreover, several studies performed with PALS (positron annihilation lifetime spectroscopy) put doubts on the other key assumption: the iso-free volume state at the glass transition.25−27 Such doubt concerning the validity of the free volume theory also arises with our recent findings concerning moisture diffusivity in hydrogen bonding food material.5 For relatively dry food materials, we find universal behavior in the water selfdiffusion, which is independent of the molar weight. The selfdiffusivity can be predicted using the free-volume theory of Vrentas and Duda,28 with a slight adaptation. The original theory of Vrentas and Duda for solvent self-diffusion takes the glass transition of the material, which we have shown to depend on the molar weight, as described by the Fox−Flory theory.29 Hence, according to the original theory, the diffusivity should depend on the molar weight, but that is in conflict with experimental observations5 and numerical simulations.30 We have resolved this issue, via assuming that the free volume of all hydrogen bonding food material behaves very similarly as a function of temperature and volume fraction of water. This behavior is also apparent in the similar specific volume of aqueous mixtures of carbohydrates or polysaccharides, in both the rubbery and the glassy state.5 This is in contradiction with the Fox−Flory theory, which assumes that chain-ends of solutes/polymers add relatively more to the free volume, which thus should depend on the molar weight, i.e., chain length. In the predictive theory for water self-diffusion, we take the values obtained for sucrose,31 and assume they hold for any other hydrogen-bonding biomaterial (with molar weights equal or larger than sucrose). Topological Constraint Theory. From the above brief review it follows that it is doubtfull that the glass transition of hydrogen-bonding materials can be explained via the free volume concept, despite the fact that good correlations with the theories have been obtained. An alternative explanation for the glass transition of these materials is apparent from the recent works of Nakanishi and Nozaki on pure polyols.6,7 Here, the glass transition is found to be related to the number of hydroxyl groups per molecule, NOH, which provide the hydrogen bonding in polyols. We view that the hypothesis of Nakanishi and Nozaki bears large similarity with the topological constraint theory for glasses.16 The topological constraint theory of Phillips states that Tg can be related to the (average) coordination number per node (molecule) in the bonded network. For mixtures, the averaging is performed using the mole fraction as the weight factor. This theory is mainly applied to covalent glasses. Phillips claims that the constraint theory also holds for simple saccharides, taking into account constraints by intramolecular covalent bonds and intermolecular hydrogen bonds.17 In contrast to covalent glasses, the coordination number of polyhydroxy compounds is highly temperature dependent, due to the relatively low energy level of the hydrogen bond, as compared to covalent bonds.32 Earlier studies have also made the link between hydrogen bonding and the glass transition. Via Fourier transform infrared spectroscopy (FTIR), Wolkers and Imamura have shown that the glass transition in sugars is manifested by a sharp decrease of the rotational mobility of the hydrogen bonds.11,12 They

f = fg + (αliq − αglass)(T − Tg) = fg + Δα(T − Tg) (1)

α is the thermal expansion coefficient, which is different in the liquid and glassy state. In an ideal mixing binary system, the free volume in the rubbery state is thought to be additive: f = ys fs + yw fw

(2)

with yw is the mass fraction of water, and ys that of the solute. Their free volume is indicated by f w and fs respectively, which also scales linearly with temperature: fs = fg + Δαs(T − Tg,s) fw = fg + Δαw(T − Tg,w )

(3)

Furthermore, in ideal mixing systems, the thermal expansion coefficient is also additive: Δα = ys Δαs + yw Δαw

(4)

From the above assumptions follows the Kelley−Bueche theory:23 Tg,mix =

ys ΔαsTg,s + yw ΔαwTg,w ys Δαs + yw Δαw

(5)

This equation is equivalent to the Couchman−Karasz relation,24 if we assume that the jump in the thermal expansion scales linear with the jump in the specific heat at the glass transition: ΔCp,i ∼ Δαi: Tg,mix =

ys ΔCp,sTg,s + yw ΔCp,wTg,w ys ΔCp,s + yw ΔCp,w

(6)

We would like to emphasize two key assumptions of the Kelley−Bueche/Couchman-Karasz theories: (1) the iso-free volume hypothesis, and (2) the ideal mixing assumption. In recent papers, we have shown that the Couchman−Karasz theory is in agreement with the experimental data of binary aqueous mixtures with various food ingredients, like sugars, polysaccharides and proteins. Moreover, it appears that these compounds have a universal value of ΔCp,s. For polysaccharides it follows that Tg,s is dependent on the molecular weight and follows the Fox−Flory relation.2−4 However, for dry binary mixtures of biopolymer and disaccharides and for ternary solutions containing biopolymers, disaccharides, and water, we have had to modify the Couchman−Karasz theory to account for nonideal mixing between the sugars and biopolymers.4 Using the modified 16304



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clearly link the glass transition to the configurational mobility of the hydrogen-bonded network of sugars. Greater configurational freedom allows for more efficient molecular packing, which reduces free volume and increases the glass transition temperature. Via FTIR spectra they have derived the length of the hydrogen bond, which is shown to scale linearly with the Tg for a select number of sugars and polysaccharides. Furthermore, they have found that the increase of the hydrogen bond length with temperature is very similar for all investigated sugars and polysaccharides, which confirms our finding that there is large similarity in specific volume and free volume among these compounds. Also, they indicate the importance of the glycosidic bond in di- and polysaccharides, which contributes strongly to constraining the configurational mobility. The mobility can be constrained further by intramolecular hydrogen bonds.33 Miller and dePablo indicate that there is also a difference in conformational flexibility between pentose and hexose rings (which is largest).34 Hence, the actual glass transition is an interplay between intermolecular and intramolecular bonds, as is also indicated by the paper by Philips on sugars glasses,17 and other recent studies.35,36 Nakanishi and Nozaki themselves have also indicated the relation of their findings with the constraint theory. They have verified that the glass temperature is indeed dependent on the number of hydroxyl groups, NOH, rather than the number of carbon atoms, NC, via comparing sugar alcohols with trihydric alcohols with varying chain length (NC).6 A correlation of Tg with NC would have been in compliance with free volume theory, while the correlation with NOH is more in compliance with the topological constraint theory. A modification of the theory is proposed by Pawlus and co-workers,8 who have shown that some of the polyhydric alcohols (isomers of pentiols) have intramolecular hydrogen bonds, which must be excluded from NOH. They state that the effective NOH has to be determined via dielectric spectroscopy measuring the relaxation dynamics of the molecule. Inspired by the work of Nozaki and co-workers, another relation between glass transition temperature and the (effective) NOH has been established by Djabourov and coworkers for ternary mixtures of gelatin, glycerol, and water.15 They show that the two plasticizers can be viewed as a single effective plasticizer via plotting experimental data on the glass transition as a function of the number of hydroxyl groups per monomer of gelatin (R). They have obtained a master curve for all Tg data if it is plotted against R. Moreover, they have also obtained a similar master curve for the melting temperature of gelatin. The master curve of Tg is showing a discontinuity, which has also been observed by Kawai and Hagura37 for mixtures of maltodextrins and sugars. In an earlier paper,38 Nozaki and his co-workers also treated mixtures of plasticizers as an effective liquid. However, in that earlier paper, they still regarded NC as the governing parameter instead of NOH (which is understandable, as for sugar alcohols it holds that NC ∼ NOH). They have shown that the glass eff temperature can be scaled with Neff C ∼ NOH, which is a weighted average based on molar fractions. In Table 1 we have listed the properties of the investigated polyols, in terms of NC and NOH, 8 and with Neff OH as postulated by Pawlus and co-workers.

Table 1. Properties Polyols and Sugars polyol

NC

NOH

water ethylglycol glycerol (ery)threitol xylitol adonitol D-arabitol L-arabitol sorbitol mannitol erytrose xylose glucose sucrose maltose maltotriose

0 2 3 4 5 5 5 5 6 6 4 5 6 12 12 18

2 2 3 4 5 5 5 5 6 6 3 4 5 8 8 11

NOH,eff 0.8 1.8 2.8 3.8 3.5 4.6 5.0 4.8 4.8

Mw(g/mol) 18 62 92 122 152 152 152 152 162 162 120 150 180 342 342 504

volume theory. We have analyzed whether the glass transition temperature Tg as a function of mass fraction of water yw can be fitted by the Couchman−Karasz theory. In the analysis, we have used experimental data from the literature, which is listed in Table 2. Results are shown in Figure 1. For the fitting we have Table 2. Data Sets for Tg of Binary Mixtures Carbohydrates and Water polyol

reference

symbol

polyols glycerol/water sorbitol/water xylitol/water Saccharides glucans glucans/water fructans/water

6 39−42 43,44 44,45

k○ r g b

12 29 46,47

k× m c

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RESULTS Binary Aqueous Mixtures. As a follow up of our previous work on Tg of carbohydrate solutions, we have been investigating the glass transition of polyol solutions using free

Figure 1. Glass transition of mixture of polyol/water: glycerol, xylitol, sorbitol, fitted with Couchman−Karasz. 16305



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⎛1 1 Tg − Tg,w n ⎞ = − ⎜ ⎟ 2 Tg∞ − Tg,w nOH ⎠ ⎝2

assumed that ΔCp,s is constant, but that it can be different from the value we found previously for carbohydrates. From linear regression analysis, it follows that ΔCp,s = 0.85 kJ/kg/mol, which is about twice the value used previously for carbohydrates. After becoming aware of the recent works of Nakanishi and Nozaki6 and of Djabourov and co-workers,15 we have reanalyzed our Tg data again, with the hypothesis that Tg is a function of the (averaged) number of OH-groups per molecule, nOH/n, which is defined as w s NOH yw /M w + NOH ys /Ms nOH = n yw /M w + ys /Ms

(8)

∞ with T∞ g = 450 K for glucose-oligomers, Tg = 425 K for = 339 K for polyols. Tg,w = 139 K is fructose-oligomers, and T∞ g the glass transition temperature of pure water. T∞ g is thought to be the Tg of (infinitely) long chain molecules from a particular class of material. Note, that T∞ g = 450 K is only slightly lower than Tg,s = 475 K, whihch we found earlier for glucose-based polysaccharides like dextran, pullulan, and starch. This deviation might be due to effects of branching in dextran and starch, or increased rigidity of the backbone as in the case of pullulan. The difference between the three classes of materials, as displayed in Figure 2, are explained by the difference in flexibility/topology of the backbone of carbon atoms.34 Polyols have a linear chain as the backbone. In glucose-oligosaccharides (hexose), rings are part of the linear backbone. In fructoseoligosaccharides the (pentose) rings are side groups of a linear backbone of carboxyl groups. (Inulins have one glucose-unit as an end-group.) The correlation can be made even stronger if one accounts for the intramolecular hydrogen bonds within the polyol or glucose oligomer molecules, which are excluded from forming hydrogen bonds with water, as has been recently shown for polyols.8 The effective NOH per molecule has been determined via structural relaxation measurement via dielectric spectroscopy. We are not aware of detailed dielectric spectra for glucoseoligomers to perform such an analysis. For glucose-oligomers, corrections are needed because it is known that (some isomers of) glucose-oligomers can form intramolecular H-bonds.48,49 Ternary Mixtures of Biopolymer/Polyol/Water. For mixtures of gelatin with glycerol and water, it is shown that Tg is also a function of nOH/n.15 Furthermore, it is posed that there is a discontinuity in Tg versus nOH/n. There is an overlap region with apparently two Tg, which might be a signature of phase separation. However, it is said that in the overlap region, the glass transition is very broad, and the lower Tg corresponds to the onset of the glass transition, while the upper Tg corresponds to the end of the glass transitions.37 We investigate whether the glass transition of ternary mixtures of biopolymer, polyol, and water collapse to a single state diagram, as has been found for the gelatin/glycerol/water system.15 The used data sources are listed in Table 3, and the

(7)

NOH i

is the amount of OH groups per molecule, with i = s,w for solute and water, respectively. Furthermore, we have yi as the mass fraction, and Mi as the molar weight. Relevant properties of the investigated polyols are listed in Table 1. In the spirit of the Fox−Flory relation, we have plotted the results against the inverse of nOH/n. Results are shown in Figure 2. There, we have also included other data sets on (1) pure

Figure 2. Glass transition of various polyols solutions (black line), and glucose-oligomer solutions (purple line), and fructan solutions (blue line) as a function of the inverse of the number hydroxyl groups per molecule (n/nOH). Data from Figure 1 is plotted using the same colors. Black symbols indicate original data of Nakanishi and Nozaki, and the purple and blue symbols are data of glucose-oligomers and fructans, respectively.

Table 3. Data Sets for Tg of Ternary Systems

polyols as given in ref 6 (which are reproduced in Table 1), (2) glucose-oligomer solutions, as collected from our previous paper,29 (3) pure glucose-oligomers from ref 12, and (4) fructose-oligomers (fructans) solutions from.46,47 The used data sources we have also listed in Table 2. In Figure 2 we observe that Tg is linear dependent on n/nOH, for polyols, fructose-oligomers, and glucose-oligomers. However, each class of compounds has a different gradient. This different behavior of glucose-oligomers and polyols we have also observed in our analysis of the Tg data with Couchman− Karasz, where each class of compounds is characterized with its own ΔCp,s. We have fitted the following correlation to the experimental data: 16306

polyol

polymer

reference

symbol

ethyl glycol glycerol glycerol glycerol xylitol xylitol sorbitol sorbitol sorbitol sorbitol

gelatin starch gelatin gelatin/starch starch gelatin starch gelatin gelatin/starch starch/pullulan

50 51,52 15,50,53 54 51 53 51 53 54 55

c○ r□ r○ r◊ g□ g○ b□ b○ b◊ b*

xylose glucose maltose

starch/pullulan dextrins dextrins

55 37 37

m* c○ c□



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results are shown in Figure 3. Here Tg is plotted against R, which is defined as the total number of hydroxyl groups present

Results are shown in Figure 4. We observe that the upper and lower branch of the state diagram result in a linear curve.

Figure 3. State diagram of ternary systems, mixtures of biopolymer, polyol, and water, showing the glass transition Tg − Tg,w as a function of the number of hydroxyl groups per mole of plasticizer, normalized with the amount of polymer (mU). Black symbols indicate the original results of ref 15, and black lines are fitted to the data in the upper and lower regions.

Figure 4. Glass transition of ternary mixtures of biopolymer, polyol, and water is plotted against the inverse of the number of hydroxylgroups per molecule: 1/2 − n/nOH. The data points break down to two linear lines related to the lower and upper branches of the universal state diagram for ternary mixtures.

in the plasticizers, normalized with the amount of polymer. If the compounds are characterized by their mass fractions yw, ys, and yp for water, solute, and biopolymer respectively, the quantity R is defined as

The gradients of the curves are quite similar to those displayed in Figure 2. The lower branch is similar to the curve of binary water/polyol systems, while the upper branch is similar to binary carbohydrate/water systems. From this behavior we state the hypothesis that in the lower branch the polyol is the percolating phase in the glass, while in the upper branch the biopolymer is the percolating phase of the glass. The percolating phase renders the backbone of the hydrogen bonded network, which immobilizes at the glass transition. Such a picture of a percolating hydrogen bonded network of solutes for a glassy material has also been proposed for glucose, which is based on molecular dynamics (MD).56,57 Dry, Binary Mixtures of Sugars and Polyols. In this section we analyze the relation between glass transition of dry mixtures of two or more sugars, and sugars with polyols. It is known that many sugars with similar molecular weight and similar number of hydroxyl groups have different glass transition temperature. Similar to polyols, we assume that this difference is due to intramolecular hydrogen bonds. Hence, for a particular sugar, the effective number of hydroxyl groups, which can form intermolecular hydrogen bonds, we compute from the Tg of the pure compound as follows:

R=

w s yw NOH /M w + ys NOH /Ms

yp /MU

(9)

with MU the molar mass of the monomeric unit of the biopolymer. As in the original publication, we have taken the value MU = 100 g/mol.15 Despite the scatter in the experimental data, the data points do appear to collapse to a universal state diagram, as is also found in ref 15. Furthermore, there appears to be a discontinuity in the state diagram, as is found by ref 37 for binary sugar/maltodextrin systems. In the study of ref 52, they have measured explicitly the onset and end of the glass transition of starch/glycerol/water system. The glass transition is quite broad. We have included their data in our graph, which shows that the onset and end of the glass transition coincide with the lower and upper branch of the universal state diagram. Further understanding of the state diagram is obtained if we plot the data against n/nOH. For the ternary systems, this is computed as follows: w s yw NOH /M w + ys NOH /Ms + yp nU nOH = n yw /M w + ys /Ms + yp /M p

1 n 1 Tg, i − Tg,w − eff = ∞ − Tg,w 2 2 Tg,glucan nOH, i

(11)

We have taken T∞ g,glucan = 475 K, otherwise the effective number of hydroxyl groups per molecule of trehalose will exceed its actual number. In the analysis of mixtures of sugars and polyols, we cannot directly take the molar average of the number of hydroxyl groups. Because of the difference in flexibility of the backbone, the glass transition of polyols occurs at lower temperatures than for sugars. Hence, we have to transform the neff OH of polyols to correct for that difference. This correction is as follows:

(10)

For biopolymer having long chains of X monomers with molar weight Mp = XMU, we have yp/Mp ≈ 0 and nU = NpOH/Mp = np/ MU, the number of hydroxyl groups per monomer divided by the weight of the monomer MU. For polysaccharides, MU = 162 g/mol and np = 3. We assume that similar numbers hold for gelatin, because of the similar value of Tg for gelatin and starch.2,10 16307



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∞ n 1 1 Tg,p − Tg,w Tg,polyol − Tg,w − eff = ∞ − Tg,w T g∞, glucan − Tg,w 2 2 Tg,polyol nOH,p

(12)

We have taken T∞ g,polyol = 339 K, as we have found above. In Table 4 we list neff OH/n for various sugars and polyols, together with the values of their glass transition, as given in the Table 4. Effective Number of Hydroxyl Groups Per Molecule of Polyols and Sugars compound

Tg,i

(neff OH,i)/n

glycerol sorbitol mannitol fructose glucose sucrose maltose trehalose maltotriose

190 266 286 280 306 325 368 388 398

2.36 3.22 3.56 3.44 4.00 4.78 6.28 7.72 8.72

Figure 5. Glass transition of mixtures of sugars and polyols plotted against the inverse of the average effective number of hydroxyl groups per molecule: 1/2 − n/neff OH.

Other studies have also observed nonideal mixing in sugar/ polyol mixtures, which has been linked to antiplasticization.63 Furthermore, we have previously observed nonideal mixing behavior in binary mixtures of biopolymer and disaccharides and in ternary mixtures of biopolymer, disaccharides, and water.4 Dry, Binary Mixtures of Disaccharides and Biopolymers. We have re-examined our earlier collected data on binary mixtures of disaccharides and biopolymers,4 and have eff plotted that against 1/2 − n/n ̅ OHeff in Figure 6. Again we have used eq 13 for computing n̅/nOH. Again, we note that for biopolymers it holds yi/Mw,ineff OH,i = yinp/Mu, with np = 3.

Table 5. Data Sets for Tg of Mixtures of Sugars and Polyols polyol/sugar

reference

symbol

fructose/glucose/sucrose sorbitol/glucose sorbitol/sucrose glucose/sucrose trehalose/sucrose glucose/maltotriose glycerol/maltose mannitol/trehalose

58 59 59,60 59 59 59 61 62

r○ m△ yv b◊ k* g□ c× y□

data sources given in Table 5. We have learned that the effective number of hydroxyl groups per molecule of disaccharides scales quite well with their hydration number, as discussed in the Appendix. Differences in the hydration number are explained by differences in intramolecular hydrogen bonds among the disaccharides, or rather the number of equatorial hydroxyl groups. Also, for polyols, the difference in glass temperature of isomers is attributed to intramolecular hydrogen bonds.8 Hence, we state Neff OH is the number of hydroxyl groups, which are available for intermolecular hydrogen bonds. For mixtures, we take a molar average of n̅OH/neff: eff nOH ̅ = n

Figure 6. Glass transition of mixtures of disaccharides and polymers plotted against the inverse of the average effective number of hydroxyl groups per molecule: 1/2 − n/neff OH.

eff ∑i nOH, iyi / M w, i

∑i yi /M w, i

(13)

The literature data sources that we have used for our analysis are listed in Table 5. The analyzed data are presented in Figure 5, where we have plotted Tg for sugar/polyol mixtures against the (inverse) average effective number of hydroxyl groups per molecule. Most of the data points follow the straight line, except for asymmetric mixtures, which differ largely in Tg,i or equivalently in neff OH,i/n. The deviation from the straight line we attribute to nonideal mixing. Hence, nonideal mixing is found in mixtures of (1) monosaccharides and trisaccharides, (2) sucrose/sorbitol, and (3) maltose/glycerol. The stronger the asymmetry in Tg,i of the pure ingredients, the larger the deviation from ideal mixing.

Figure 6 shows that the data from these compounds also follow the trend shown by asymmetric mixtures in the previous figure. The strong deviation from the linear trend line indicates nonideal mixing. The data from the last two figures, we have combined in Figure 7, where one can observe that the deviation of disaccharide/biopolymer mixtures from the linear line is of similar magnitude of the most asymmetric mixtures from Figure 5. To describe the nonideal mixing we have fitted the following relation: 16308



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similar hypothesis has been formulated for ternary biopolymer/ sugar/water mixtures.74 Microheterogeneity cannot be equated to phase separation, as the cluster size is very small, and the glass transition is significantly determined by hydrogen bonding across the interface of sugar-rich and polymer-rich domains.37 Hence, only a single glass transition is observed. Mixtures of Biopolymers, Disaccharides and Water. As a final investigation, we have reanalyzed the Tg of the ternary mixtures of biopolymers, disaccharides, and water from our earlier paper.4 There, we have analyzed the following mixtures: trehalose/dextran/water, sucrose/gelatin/starch/water, lactose/ gelatin/starch/water, and maltodextrin/maltose/water, with data obtained from the literature. The results are shown in Figure 8. We have plotted Tg versus n/neff OH, which is computed

Figure 7. Glass transition of various mixtures (from Figures 5 and 6), plotted against the inverse of the average effective number of hydroxyl groups per molecule: 1/2 − n/neff OH. The linear curve indicates the ideal mixing relation, while the curved lines indicate the nonideal mixing relation (cf. eq 14), for maltose/glycerol (cyan), glucose/raffinose (green), and disaccharide/biopolymer mixtures (red, black, and blue for sucrose, maltose, and trehalose). eff fij xNOH, i + (1 − x)NOH, j nOH ̅ = n fij x + (1 − x)

(14)

with i indicating the compound with the lowest molecular weight. The above relation is inspired by the earlier one that we used for the Tg of biopolymer/disaccharide mixtures.4 This new relation for nOH/n is inserted in eq 8. In Figure 7 we have plotted the fits of the relations for mixtures of (1) maltose/ glycerol, (2) glucose and raffinose, and (3) disaccharides and biopolymers. Fits with parameter values of f = 3,4,5 respectively, give quite a reasonable fit to the experimental data. A possible physical interpretation of the above relation is that the smaller molecule has an increased chance to hydrogen bonding with molecules of their own kind, as compared to the other larger molecule in the mixture. This is the case if the mixtures display microheterogeneity, as have been observed for various polyol/water and sugar/water mixtures in MD,57,64−68 and in experiments.33,69,70 The MD simulations show that at high concentrations of the larger molecules, the smaller molecules form clusters, with little other molecules present in the cluster. In the clusters, the chance of hydrogen bonding between smaller molecules is indeed higher than in the case of a homogeneous mixture. Such microheterogeneity is also given as an explanation of the discontinuity in ternary systems of biopolymer/sugar/ water.37 It is assumed that macromolecules are confined within the water cluster, and that the sugars are excluded from the surface of the macromolecule.33 This is in agreement with the water entrapment hypothesis,71 which is recently corroborated by MD simulations.72 Shamblin and co-workers have stated that in mixtures of sugars and biopolymers, there is hydrogen bonding between biopolymer and sugar impose steric constraints on adjacent hydroxyl groups, and consequently there is a lower degree of hydrogen bonding compared to the case of ideal mixing.73 A

Figure 8. Glass transition of various ternary mixtures of biopolymer, disaccharide, and water versus the inverse of the average effective number of hydroxylgroups per molecule: 1/2 − n/neff OH. The linear curve indicates the ideal mixing relation. Disaccharides are composed of trehalose (blue), sucrose (red), lactose (black), and maltose (green), and biopolymers are dextran, starch, gelatin, and maltopolymer.

using eq 10. For lactose, we have used the same NOH as for maltose, because their behavior is similar to that found in ref 4. The solid line represents the ideal mixing relation, following eq 11, with T∞ g = 450 K. We find it remarkable that the data more or less follow the ideal mixing line, whereas in our previous paper we had to account for nonideal mixing. Perhaps the scatter in the data is blurring the effect of the nonideal mixing. If present, the nonideal mixing effects are much less pronounced than for dry binary mixtures of biopolymers and sugars or polyols. The presence of water clearly reduces nonideal mixing effects or equivalently the microheterogeneity. In dry binary microheterogeneous mixtures, there are biopolymer-rich and sugarrich regions. Water mixes ideally with both the biopolymer and the sugars. In ternary mixtures, the presence of water in the interfacial region between biopolymer-rich and sugar-rich clusters enhances/mediates the bonding between sugars and biopolymers, making them more compatible. Taking a free volume point of view, water might be filling the holes present in the interfacial region between the clusters. The increased compatibility of the clusters becomes apparent in the facts that 16309



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the glass transition versus n/neff OH follows a more straight line, and that there is only one straight line, in contrast to Figure 4, which holds for ternary mixtures of biopolymers, polyols, and water, and which displays two lines, corresponding to the onset and end of glass transition. Probably, water can fill holes (free volume) normally present in the dry mixtures such that it approaches the ideal-mixing case.

molecules that are affected by interactions between solute and solvent.75,76 The precise value or definition of nH clearly depends on the experimental method applied. Only via MD can a more precise definition be given: the average number of water molecules that are H-bonded with the solute molecule. Molecules are said to be H-bonded if the donor/acceptor atoms are within a certain distance and the angle of the O−H− O bond is larger than 120 degrees.75 An alternative way for obtaining the hydration number is via ultrasonic measurement of the solution compressibility. It is stated that the hydration number is obtained from the relation between compressibility and the number of moles of solute, using the assumption that hydration water compressibility is negligible.77,78 Another method for obtaining hydration water is via viscometry, where it is assumed that the intrinsic viscosity is related to the effective size of the solute, which includes the hydration shell. It is argued that the dynamics of water in the hydration shell is slower than in the bulk, and they are dragged with the solute under flow conditions.79 Literature data show that the hydration numbers obtained via MD65 compare reasonably with those obtained via acoustic methods.78 Several studies have stated that there is a linear relation between the (dynamic) hydration number, nH, and the mean eq number of equatorial nOH for oligosaccharides.80,81 The dynamic hydration number is obtained from the concentration dependency of the NMR-dynamics of oxygen.80 Intramolecular hydrogen bonding in carbohydrates depends on the equatorial/ axial orientation of the hydroxyl groups. Equatorial hydroxyl groups have an O−O spacing similar to that of water, which is important for the hydration of solutes (osmolytes).82,83 Due to their position, axial hydroxyl groups form weaker bonds with water, and often form intramolecular bonds.48,49 It is also shown that for the hydration number of cyclodextrins, the number of hydroxyl groups that from intramolecular H-bonds should be excluded for the correlation with the hydration number.84 Quite a large number of studies have shown that phase transitions of biopolymers in mixed plasticizers are governed by the average dynamic hydration number of the plasticizers applied.85−91 This agrees with the recent notion of Djabourov and co-workers15 that gelatin shows an universal supplemented state diagram if the amount of plasticizers is renormalized as the number of hydroxyl groups per biopolymer monomer. Furthermore, for several sugars, a single master curve of the hydration number, nH, as a function of solute concentration is obtained, if plotted against the number of solute OH-groups per water molecule.75 All the above cited papers appear to indicate that neff OH is linear with the (dynamic) hydration number. For sugars it is found that the (dynamic) hydration number is also linear with the (mean) number of equatorial hydroxyl groups per molecule. We have verified these notions via plotting neff OH against nH and neff OH for several polyols and sugars, as shown in Figure 9. The data for nH is obtained from refs 78 and 92 using ultrasonic or acoustic methods, and from ref 65 using MD. The data for neq OH is obtained from refs 87 and 93 .We must note that nH is a function of solute concentration and temperature.75,76 Hence, for our comparison, we take nH in the limit of very dilute solutions at room temperature. Figure 9 shows that indeed neff OH is linear with the hydration number, and for sugars it is also linear with neq OH. Normand and co-workers have found that nH scales linear with nOH for maltooligomers.78 For polyethylene-glycols (PEGs), hydration

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CONCLUSIONS In this paper we have presented a new correlation for the glass transition and hydrogen bonding for a multitude of mixtures of materials, as often used for food, biomaterials, or pharmaceuticals. The correlation relates Tg to the effective number of hydroxyl groups per molecule Neff OH. Our correlation extends the earlier work of Nakanishi and Nozaki, who have given a similar correlation for pure polyol liquids. For mixtures, a molar average has to be taken. Furthermore, our correlation bears large similarity with the topological constraint theory. Our correlation takes the effective number of hydroxyl groups, which are available for intermolecular hydrogen bonding. Differences in Tg among disaccharides can be explained by the different amount of intramolecular hydrogen eff bonds, which are excluded from NOH . Our estimate of intramolecular hydrogen bonds agrees well with the hydration numbers of these sugars, as found by MD. For dry binary mixtures of biopolymers with sugars or polyols, and sugar/polyol mixtures, there are strong nonideal mixing effects that cause deviations from our correlation, which is based on ideal mixing. The stronger the asymmetry in Tg, the stronger the nonideal mixing effects. The nonideal mixing effects are explained by the microheterogeneity often observed for these mixtures. Sugars form small clusters, and this strongly changes the number of intermolecular hydrogen bonds of the sugar molecules as compared to the ideal mixing case. For ternary biopolymer/polyol/water mixtures, we observe that the glass transition occurs over a broad temperature range. Both the onset and the end temperature of the glass transition are a function of the effective number of hydroxyl groups per molecule. In the state diagram it appears that Tg has a discontinuity, if it is plotted against the number of hydroxyl groups of only the plasticizers (water and polyol). We have hypothesized that for each branch of the state diagram holds that either the biopolymer or the polyol forms the percolating network, which forms the backbone of the hydrogen bonded network, which is immobilized at Tg. The effects found with ternary biopolymer/polyol/water mixtures are more or less absent in ternary biopolymer/ disaccharide/water mixtures. The asymmetry between Tg of biopolymer and disaccharide is definitely smaller than for polyols, and the nonideal mixing effect is much smaller or even negligible. We hope that this study stimulates further research into the relevance of hydrogen bonding for the glass transition of food and biomaterials. Especially, MD studies are welcomed for validation of the above hypotheses, including the one concerning the deviations from the ideal (mixing) behavior due to microheterogeneity in complex mixtures.

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APPENDIX: NEFF OH VERSUS HYDRATION NUMBER In the literature the macroscopic properties of H-bonding materials are often correlated to the so-called hydration number, nH, which is commonly defined as the average number of water 16310



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Figure 9. Correlation of neff OH with (a) the hydration number nH for a glycerol and oligosaccharides, and (b) the mean equatorial hydroxyl groups neff OH for oligosaccharides, which contain fructose, glucose, sucrose, maltose, trehalose, and maltotriose.

number scales linear with the degree of polymerization.77 All these findings strengthens our hypothesis that neff OH relates to the number of hydroxyl groups that are available for making intermolecular H-bonds with the hydrogen-bonded network, formed by other solute molecules and water.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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Predictions of Glass Transition Temperature for Hydrogen Bonding

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