Synchrotron x-ray scattering has been used to investigate three liquid polyalcohols of different sizes (glycerol, xylitol, and D-sorbitol) from above the glass transition temperatures Tg to below. We focus on two structural orders: the association of the polar OH groups by hydrogen bonds (HBs) and the packing of the non-polar hydrocarbon groups. We find that the two structural orders evolve very differently, reflecting the different natures of bonding. Upon cooling from 400 K, the O⋯O correlation at 2.8 Å increases significantly in all three systems, indicating more HBs, until kinetic arrests at Tg; the increase is well described by an equilibrium between bonded and non-bonded OH with ΔH = 9.1 kJ/mol and ΔS = 13.4 J/mol/K. When heated above Tg, glycerol loses the fewest HBs per OH for a given temperature rise scaled by Tg, followed by xylitol and by D-sorbitol, in the same order the number of OH groups per molecule increases (3, 5, and 6). The pair correlation functions of all three liquids show exponentially damped density modulations of wavelength 4.5 Å, which are associated with the main scattering peak and with the intermolecular C⋯C correlation. In this respect, glycerol is the most ordered with the most persistent density ripples, followed by D-sorbitol and by xylitol. Heating above Tg causes faster damping of the density ripples with the rate of change being the slowest in xylitol, followed by glycerol and by D-sorbitol. Given the different dynamic fragility of the three liquids (glycerol being the strongest and D-sorbitol being the most fragile), we relate our results to the current theories of the structural origin for the difference. We find that the fragility difference is better understood on the basis of the thermal stability of HB clusters than that of the structure associated with the main scattering peak.

X-ray scattering has long been used to study the structures of liquids and glasses,1,2 yielding structure factors and pair distribution functions (PDFs) that are essential for developing theories of liquids3–5 and understanding their properties.6 There has been recent interest in the structural origin for the strong and fragile dynamics of glass-forming liquids.7–11 The availability of high-energy high-brilliance synchrotron x-ray sources has enabled measurements up to high momentum transfer necessary to obtain PDFs at high spatial resolution.12 

The structures of liquid alcohols have attracted long-standing interest because the same molecule contains both polar (OH) and non-polar (CH2 and CH3) groups. The OH groups tend to associate through strong and directional hydrogen bonds (HBs), while the CHn groups organize themselves into closely packed structures through weaker van der Waals forces. This leads to a mosaic structure in the liquid. Nearly a century ago, Stewart observed that the addition of a polar OH group to a hydrocarbon molecule has no strong effect on the main scattering peak of the liquid but introduces a “prepeak” at low angles, suggesting a new supramolecular order.1 Fourier transform of the structure factors of liquid alcohols reveals a peak in the PDF near 2.8 Å, which corresponds to the O⋯O correlation in an OH⋯O hydrogen bond and whose area directly yields the number of HBs per OH (nHB ≈ 2).13–15 This ability to directly obtain the number of HBs in the liquid state proved to be a major advantage of the scattering method. Despite its early beginning, the investigation of liquid alcohols remains active today stimulated by the mosaic structure from the preferential association of polar and non-polar groups16 and its dynamic ramifications.17 For example, the dynamics at the prepeak of 1-propanol is slower than at the main peak, suggesting a link between structure and dynamics.18 

In this work, we investigate three polyalcohols of different sizes: glycerol, xylitol, and D-sorbitol (Scheme 1). Polyalcohols are important ingredients in food and drug formulations; glycerol is a widely used cryoprotectant. Polyalcohols represent a higher level of complexity relative to monoalcohols for which scattering studies have been performed with success.13–15 In a polyalcohol molecule, the OH and CHn groups are closely integrated, likely creating a different mosaic structure than that of monoalcohols. We used synchrotron x-ray scattering to study the two structural orders introduced above, namely, the O⋯O correlation associated with HBs and the C⋯C correlation associated with hydrophobic interactions. These two aspects of the structure are interesting in their own right and are also the ingredients in the recent work on the structural origin of dynamic fragility.7–11 In Angell’s classification of liquid dynamics, the three polyalcohols have significantly different fragility, with glycerol being the strongest and D-sorbitol being the most fragile, and this difference has been linked to the thermal stability of HB clusters7,19 or to that of the structure associated with the main scattering peak.8–11 The result of this work helps evaluate these ideas.

SCHEME 1.

Molecular structures of glycerol, xylitol, and D-sorbitol.

SCHEME 1.

Molecular structures of glycerol, xylitol, and D-sorbitol.

Close modal

Several scattering studies have been reported on glycerol,20–22 and to our knowledge, none have been conducted on xylitol and D-sorbitol. For glycerol, the interpretation of scattering data proved challenging because of its conformational flexibility23 and overlapping intra- and intermolecular distances;24 there have been different reports on its conformation in the liquid state20–22,25,26 and on the state of hydrogen bonding.22,25,27 In this work, we employ a subtraction method to isolate the HB peaks in the PDF without complications from the imperfectly known liquid-state conformations. We show that upon cooling, all three polyalcohols exhibit increased O⋯O correlation due to the formation of HBs. Cooling from 400 K, the gain of HBs per OH is nearly the same in each system until the glass transition halts the progress. The overall trend of the increase is well described by a two-state equilibrium between bonded and non-bonded OH with ΔH = 9.1 kJ/mol and ΔS = 13.4 J/mol/K. When compared at Tg, the HB network in glycerol is the most robust against thermal degradation, losing the fewest HBs per OH for a given temperature rise scaled by Tg, followed by xylitol and by D-sorbitol. We find that all three systems show far-reaching density modulations in their PDFs, with the persistence length being the longest in glycerol, followed by D-sorbitol and by xylitol. These density modulations are associated with the main scattering peak and with the C⋯C correlation. Our results suggest that the thermal stability of the HB network provides a better foundation for understanding glycerol’s low fragility than the temperature effect on the structure associated with the main scattering peak.

Glycerol was purchased from Alfa Aesar as a liquid and dried under vacuum at 400 K before use. Xylitol and D-sorbitol were purchased from Sigma-Aldrich as crystalline powders and used as received. Synchrotron x-ray scattering was performed in the transmission geometry at beamline 6-ID-D at the Advanced Photon Source (APS), Argonne National Laboratory (IL, USA). The x-ray wavelength was 0.125 43 Å. Each sample was placed in a capillary tube (2 mm O.D. quartz tube with a wall thickness of 0.01 mm, Charles Supper Company) and flame-sealed. The tube lies horizontally on a holder 35 cm from the detector (the distance was known precisely from the diffraction angles of a CeO2 powder). The detector was a Perkin–Elmer amorphous silicon 2D detector with an active area of 409.6 × 409.6 mm2 containing 2048 × 2048 pixels each 0.2 × 0.2 mm2 in size. The instrumental resolution was 0.081 Å−1, determined from the FWHM of each resolved diffraction peak of CeO2 fitted with a Gaussian function. The instrumental width is small but not negligible relative to the width of the main scattering peak (0.3–0.5 Å−1), and its effect was removed using the method of Ref. 28 as performed for Ar.3 For this purpose, the instrument function used was s(q) = (2/π)1/2w−1 exp[−(2/w2) (qq0)2], where w = 0.069 Å−1, which corresponds to a FWHM of 0.081 Å−1 and α = 420 Å2 in Ref. 28. Sample temperature was controlled using an Oxford Cryosystem 700 in the range of 100–400 to ±0.1 K.

Each 2D scattering pattern was integrated azimuthally using the software FIT2D29,30 to yield a 1D plot of intensity vs momentum transfer q=4πλsinθ, with 2θ being the scattering angle and λ being the x-ray wavelength. Corrections for the container, flat panel, sample absorption, multiple scattering, oblique incidence, and Compton scattering were performed using the program PDFgetX2.31 The corrected intensity of coherent scattering intensity I(q) was normalized to obtain the atom-averaged x-ray structure factor,

Sq=1+Iqf2(q)f(q)2,
(1)

where ⟨f2(q)⟩ is the atom-averaged self-scattering power and ⟨f(q)⟩2 is the atom-averaged scattering power of one molecule. The differential PDF, D(r), was obtained by a Fourier sine transform of the function Fq=q(Sq1),12 

Dr=2π0qmaxF(q)sinqrdq,
(2)

where qmax is the upper bound for usable S(q) data. For our datasets, qmax = 19.4 Å−1 [chosen to be a zero of F(q)32]. The S(q) values below the lower bound of measurement (qmin = 0.45 Å−1) were obtained by linearly extrapolating F(q) to q = 0. Other than truncating the data at qmax [equivalent to multiplying F(q) by a box function of unit height], no other modification function was applied.

The following equation is used to calculate the total PDF T(r),12 

Tr=4πnar+D(r),
(3)

where na is the atomic density (atoms/Å3) and is calculated from the bulk density ρ (g/cm3) using na = n0 ρNA/M, where n0 is the number of atoms per molecule, NA is Avogadro’s number, and M is the molecular weight. Table I gives the density values used for this calculation. As detailed in the supplementary material, we have obtained these densities from the literature, our own measurement (for xylitol, Table S1), and estimates based on the linear relation between the density and the position of the main scattering peak (see below and Fig. S2). A peak in T(r) corresponds to an atom–atom correlation and can be fitted33,34 to yield the corresponding bond length, coordination number, and Debye–Waller factor.

TABLE I.

Density ρ, molecular number density n, wavelength L1 and correlation length ξ of density modulation, and L1n1/3 of glycerol, xylitol, and D-sorbitol as functions of temperature.a

SystembT (K)ρ (g/cm3)n (10−3 molecules/Å3)L1 (Å)ξ (Å)L1n1/3
Glycerol Tg = 190 K, m = 50 100 1.336 8.738 4.340 7.84 0.89 
125 1.333 8.718 4.342 7.86 0.89 
150 1.330 8.699 4.345 7.87 0.89 
175 1.327 8.679 4.348 7.89 0.89 
200 1.316 8.607 4.362 7.83 0.89 
225 1.301 8.508 4.381 7.66 0.89 
250 1.286 8.408 4.398 7.50 0.89 
275 1.271 8.308 4.416 7.36 0.89 
300 1.255 8.209 4.435 7.23 0.89 
325 1.240 8.109 4.451 7.12 0.89 
350 1.225 8.010 4.469 6.99 0.89 
375 1.207 7.896 4.489 6.84 0.89 
400 1.189 7.778 4.504 6.72 0.89 
       
Xylitol Tg = 250 K, m = 78 150 1.445 5.721 4.538 4.62 0.81 
175 1.444 5.714 4.541 4.62 0.81 
200 1.442 5.708 4.544 4.65 0.81 
225 1.440 5.701 4.548 4.66 0.81 
250 1.435 5.680 4.558 4.68 0.81 
275 1.418 5.610 4.593 4.64 0.82 
300 1.402 5.546 4.627 4.60 0.82 
325 1.386 5.483 4.661 4.55 0.82 
350 1.370 5.419 4.698 4.51 0.83 
375 1.354 5.356 4.735 4.50 0.83 
400 1.338 5.292 4.766 4.48 0.83 
       
D-sorbitol Tg = 269 K, m = 102 150 1.470 4.860 4.461 5.72 0.76 
175 1.467 4.850 4.466 5.74 0.76 
200 1.466 4.845 4.468 5.78 0.76 
225 1.465 4.843 4.469 5.69 0.76 
250 1.463 4.837 4.471 5.71 0.76 
275 1.459 4.822 4.480 5.68 0.76 
300 1.443 4.771 4.500 5.53 0.76 
325 1.427 4.719 4.521 5.36 0.76 
350 1.411 4.665 4.549 5.18 0.76 
375 1.394 4.609 4.579 5.02 0.76 
400 1.377 4.551 4.603 4.91 0.76 
SystembT (K)ρ (g/cm3)n (10−3 molecules/Å3)L1 (Å)ξ (Å)L1n1/3
Glycerol Tg = 190 K, m = 50 100 1.336 8.738 4.340 7.84 0.89 
125 1.333 8.718 4.342 7.86 0.89 
150 1.330 8.699 4.345 7.87 0.89 
175 1.327 8.679 4.348 7.89 0.89 
200 1.316 8.607 4.362 7.83 0.89 
225 1.301 8.508 4.381 7.66 0.89 
250 1.286 8.408 4.398 7.50 0.89 
275 1.271 8.308 4.416 7.36 0.89 
300 1.255 8.209 4.435 7.23 0.89 
325 1.240 8.109 4.451 7.12 0.89 
350 1.225 8.010 4.469 6.99 0.89 
375 1.207 7.896 4.489 6.84 0.89 
400 1.189 7.778 4.504 6.72 0.89 
       
Xylitol Tg = 250 K, m = 78 150 1.445 5.721 4.538 4.62 0.81 
175 1.444 5.714 4.541 4.62 0.81 
200 1.442 5.708 4.544 4.65 0.81 
225 1.440 5.701 4.548 4.66 0.81 
250 1.435 5.680 4.558 4.68 0.81 
275 1.418 5.610 4.593 4.64 0.82 
300 1.402 5.546 4.627 4.60 0.82 
325 1.386 5.483 4.661 4.55 0.82 
350 1.370 5.419 4.698 4.51 0.83 
375 1.354 5.356 4.735 4.50 0.83 
400 1.338 5.292 4.766 4.48 0.83 
       
D-sorbitol Tg = 269 K, m = 102 150 1.470 4.860 4.461 5.72 0.76 
175 1.467 4.850 4.466 5.74 0.76 
200 1.466 4.845 4.468 5.78 0.76 
225 1.465 4.843 4.469 5.69 0.76 
250 1.463 4.837 4.471 5.71 0.76 
275 1.459 4.822 4.480 5.68 0.76 
300 1.443 4.771 4.500 5.53 0.76 
325 1.427 4.719 4.521 5.36 0.76 
350 1.411 4.665 4.549 5.18 0.76 
375 1.394 4.609 4.579 5.02 0.76 
400 1.377 4.551 4.603 4.91 0.76 
a

The error is ∼0.007 Å in L1 and 0.06 Å in ξ.

b

Fragility index m is from Ref. 7.

Figure 1 shows the representative data from this work using glycerol as an example. Similar data were collected for xylitol and D-sorbitol. The structure factor S(q) of glycerol was measured between 100 and 400 K [Fig. 1(a)], which covers its Tg (190 K). The first and main peak, labeled q1, appears near 1.5 Å−1. This peak shows significant temperature dependence [see the enlarged view in Fig. 1(b)], shifting to higher q upon cooling, as seen for many liquids.35,36 For glycerol, the shift is strongly correlated with density such that the peak position is nearly constant when plotted against qn−1/3 [Fig. 1(c)], where n is the molecular number density. For glycerol, the scaled peak position is qn−1/3 ≈ 7, approximately the value for densely packed spheres.37,38 Later, we show that the q1 peak is associated with the intermolecular C⋯C correlation. In contrast to the q1 peak, the structure factor at q > 10 Å−1 is insensitive to temperature. In this region, scattering is dominated by intramolecular correlations.13 The contrasting temperature dependence described above reflects the fact that intramolecular distances are maintained by strong covalent bonds and insensitive to temperature, while intermolecular distances are controlled by weaker forces and more sensitive to temperature.

FIG. 1.

(a) Structure factors S(q) of glycerol between 100 and 400 K. The darker line indicates the lower temperature. The main scattering peak is labeled q1. It is enlarged in (b) and plotted against density-scaled q in (c). (d) The distribution functions D(r) of glycerol obtained from S(q) by Fourier transform [Eq. (2)]. L1 is the wavelength of density modulation beyond molecular size (∼5 Å).

FIG. 1.

(a) Structure factors S(q) of glycerol between 100 and 400 K. The darker line indicates the lower temperature. The main scattering peak is labeled q1. It is enlarged in (b) and plotted against density-scaled q in (c). (d) The distribution functions D(r) of glycerol obtained from S(q) by Fourier transform [Eq. (2)]. L1 is the wavelength of density modulation beyond molecular size (∼5 Å).

Close modal

Figure 1(d) shows the distribution function D(r) obtained by a Fourier transform of the structure factor [Eq. (2)]. D(r) characterizes the deviation of local atomic density from its bulk value. The sharp peaks at short distances are mainly intramolecular correlations, while the broad features at longer distances are intermolecular correlations, with some overlap between the two as indicated. Ripples are observed in D(r) well beyond the molecular size (∼5 Å) at a wavelength of L1 = 4.4 Å for glycerol, indicating penetration of local order into the bulk. As we will discuss later, these ripples are associated with the main scattering peak and with the C⋯C correlation.

Figure 2 shows the total PDF T(r) for the three liquids. T(r) was calculated from D(r) [Eq. (3)] and the densities given in Table I. The peak at 1.45 Å corresponds to the (unresolved) CC and CO covalent bonds, in good agreement with the bond lengths in crystals (1.51 Å for CC and 1.43 Å for CO on average).39–41 The 1.45 Å peak is insensitive to temperature, as expected for covalent bonds. The peak at 2.4 Å is also intramolecular, corresponding to the “1–3 distances” (for atoms across two consecutive covalent bonds). This peak is also insensitive to temperature, reflecting the stability of covalent bond lengths and angles.

FIG. 2.

Total PDF T(r) of glycerol, xylitol, and D-sorbitol at different temperatures. The darker line indicates the lower temperature. The dense grouping of lines corresponds to temperatures at or below Tg. The vertical lines indicate the ROO distances in crystals. Assignment of intermolecular correlations is shown at the bottom. ROO is the nearest-neighbor O⋯O distance in a HB.

FIG. 2.

Total PDF T(r) of glycerol, xylitol, and D-sorbitol at different temperatures. The darker line indicates the lower temperature. The dense grouping of lines corresponds to temperatures at or below Tg. The vertical lines indicate the ROO distances in crystals. Assignment of intermolecular correlations is shown at the bottom. ROO is the nearest-neighbor O⋯O distance in a HB.

Close modal

In contrast to the intramolecular peaks, we observe significant temperature effect on the peaks in T(r) at longer distances. With cooling, we observe an increase in intensity at 2.8 Å, a decrease at 3.1 Å, and an increase at 3.7 Å. These regions of changes are separated by two quasi-isosbestic points at 2.9 and 3.3 Å where intensity is nearly constant. These features are attributed to the temperature effect on intermolecular HBs, as observed for water42 and methanol.14 As shown at the bottom of Fig. 2, the 2.8 Å distance is the O⋯O distance, ROO, in a HB. This distance matches the values in crystals shown as the vertical dotted lines: ROO = 2.74 Å on average for glycerol,39 2.77 Å for xylitol,39 and 2.84 Å for D-sorbitol.41 Our assignment is also supported by MD simulations (see the supplementary material for details). Our simulation results are shown in Fig. 3; they are fully consistent with the previous work22,27,43 but extended to longer distance to investigate the long-range density ripples (see below). The first intermolecular correlation between heavy atoms (which dominates x-ray scattering) is the O⋯O peak at 2.8 Å where the C⋯C or C⋯O contribution is negligible. Based on this assignment, the rise of intensity at 2.8 Å with cooling indicates the formation of more HBs, and the decrease at 3.1 Å is due to the loss of poorly formed HBs or non HBs (R2 in the schematic in Fig. 2). The increase at 3.7 Å corresponds to the increase of C⋯O correlations as more HBs are formed (R3). Since every O atom of a polyalcohol is linked to a C atom, each HB formed introduces an O⋯O contact at 2.8 Å and an O⋯C contact near 3.7 Å. In Fig. 3, we see the O⋯C peak at 3.7 Å in the simulated distribution function. A further argument for associating the temperature effect on the PDF in Fig. 2 is the similar response of the three polyalcohols to temperature. In principle, a temperature change could alter the molecular conformation and thus the PDF, but the effect is likely molecule specific. The fact that the three liquids exhibit a similar pattern argues for a common origin, namely, the temperature effect on HBs.

FIG. 3.

Intermolecular C⋯C, O⋯O, and C⋯O partial PDFs of glycerol at 300 K from MD simulations (see the supplementary material for details). The O⋯O correlation shows a sharp peak at 2.8 Å where no signal comes from the other pairs. The long-range oscillation is dominated by the C⋯C correlation. The red curve is a fit of the C⋯C correlation to Eq. (5).

FIG. 3.

Intermolecular C⋯C, O⋯O, and C⋯O partial PDFs of glycerol at 300 K from MD simulations (see the supplementary material for details). The O⋯O correlation shows a sharp peak at 2.8 Å where no signal comes from the other pairs. The long-range oscillation is dominated by the C⋯C correlation. The red curve is a fit of the C⋯C correlation to Eq. (5).

Close modal

To quantify the temperature effect on HBs, we calculate the differential function ΔT(r) using 400 K as a common reference. The results are shown in Fig. 4. In this format, the growth of the 2.8 Å peak and the loss of intensity at 3.1 Å are more clearly seen. Because covalent bond lengths and bond angles are weekly dependent on temperature, this subtraction removes these intramolecular contributions and isolates the intermolecular effect. Given that the polyalcohols’ molecular conformations are still imperfectly known in the liquid state, this subtraction method circumvents the difficulty, as long as the conformation is weakly dependent on temperature. The positive peak at 2.8 Å and the negative peak at 3.1 Å in each ΔT(r) trace are the result of subtracting two intermolecular O⋯O correlation functions, one at a low temperature and the other at a high temperature, as illustrated at the bottom of Fig. 4. The low-temperature peak in blue has shorter ROO and is sharper, while the high-temperature peak in red has longer ROO and is broader. Their difference gives rise to the positive–negative modulation in ΔT(r) in green. Thus, we can fit each ΔT(r) to obtain the change in the number of HBs per OH group, ΔnHB, as a result of temperature change. The NXFit program was used for this fitting33 with atomic form factors from International Tables for Crystallography,44 and the results are discussed below.

FIG. 4.

The ΔT(r) functions of glycerol, xylitol, and D-sorbitol relative to 400 K. The darker line indicates the lower temperature. The dense grouping of lines corresponds to temperatures at or below Tg. At the bottom, the decomposition of a ΔT(r) function into its low- and high-temperature components is shown using glycerol as an example (in this case, 150 and 400 K are the two temperatures).

FIG. 4.

The ΔT(r) functions of glycerol, xylitol, and D-sorbitol relative to 400 K. The darker line indicates the lower temperature. The dense grouping of lines corresponds to temperatures at or below Tg. At the bottom, the decomposition of a ΔT(r) function into its low- and high-temperature components is shown using glycerol as an example (in this case, 150 and 400 K are the two temperatures).

Close modal

Figure 5 shows the gain of HBs per OH group with cooling for glycerol, xylitol, and D-sorbitol. Upon cooling from 400 to 325 K, each system gains the same number of HBs per OH; at 325 K, ΔnHB = 0.2 for all three systems. For a given system, the gain of HBs is halted by the glass transition at Tg (down arrow). With its lowest Tg, glycerol gains the most HBs on cooling from 400 K before kinetic arrest. Our results on the three polyalcohols are consistent with the view45 that each OH group has a higher probability to form a HB as temperature decreases, and the probability is relatively insensitive to molecular details. Assuming a simple equilibrium for OH groups between bonded and non-bonded states, the equilibrium constant is given by

K=[bondedOH]/[nonbondedOH]=pHB/(1pHB)=exp[ΔH/(RT)+ΔS/R],

where pHB is the probability for OH to be hydrogen-bonded and ΔH and ΔS are the enthalpy change and the entropy change characterizing the equilibrium. Rearranging the equation above, we obtain pHB = 1/[1 + exp(−ΔH/(RT) + ΔS/R)]. Since each OH can serve as both the donor and the acceptor of HBs, the number of HBs per OH is nHB = 2 pHB or

nHB=2/[1+exp(ΔH/(RT)+ΔS/R)].
(4)

Here, nHB corresponds to the coordination number for the O⋯O correlation at 2.8 Å. nHB is 2 in the crystals of the polyalcohols.39–41 The curve in Fig. 5 is a fit of the data points to Eq. (4). Given the similar trends of the three polyalcohols at T > Tg, we fitted their results together. This fit describes our data well, yielding ΔH = 9.1 ± 0.5 kJ/mol and ΔS = 13.4 ± 0.9 J/mol/K. The goodness of this fit supports the two-state description of HBs in these polyalcohols. This two-state model immediately explains the isosbestic points in Fig. 2: these points occur at the distances at which the T(r) functions for the bonded and non-bonded structures have common values. This fit also yields the absolute number of HBs per OH, nHB, which is shown in Fig. 5 using the right y axis. The results indicate that with cooling, nHB approaches the maximal value of 2 per OH in glycerol on arrival at the glassy state, whereas nHB is frozen at lower values (∼1.8) in xylitol and D-sorbitol owing to their higher Tg. From nHB, we obtain the number of HBs per molecule by multiplying nHB with nOH, the number of OH groups per molecule. For glycerol, nOH = 3 and we obtain 5.3 HBs per glycerol molecule at 298 K, in good agreement with 5.7 from the work of Towey et al. based on simulations constrained by neutron scattering data.22 The ΔH value from this work is comparable to the values from vibrational spectroscopy that characterize the equilibrium between two hydrogen-bonded states of glycerol (5.7 kJ/mol)18 and water (10.9 kJ/mol).46 

FIG. 5.

Gain of HBs per OH upon cooling from 400 K in glycerol, xylitol, and D-sorbitol. Each down arrow indicates Tg (190 K for glycerol, 250 K for xylitol, and 269 K for D-sorbitol). The solid curve is a joint fit of all data for T > Tg to Eq. (4). The resulting absolute number of HBs per OH is shown using the right y axis. The dashed curve is extrapolation of the fitting curve.

FIG. 5.

Gain of HBs per OH upon cooling from 400 K in glycerol, xylitol, and D-sorbitol. Each down arrow indicates Tg (190 K for glycerol, 250 K for xylitol, and 269 K for D-sorbitol). The solid curve is a joint fit of all data for T > Tg to Eq. (4). The resulting absolute number of HBs per OH is shown using the right y axis. The dashed curve is extrapolation of the fitting curve.

Close modal

In Fig. 6, we replot the data in Fig. 5 using the glassy state as the reference to investigate the effect of heating on the number of HBs per OH, nHB, in each system. nHB is stable in each system in the glassy state, as expected, and decreases with heating above Tg. Glycerol suffers the smallest loss of HBs with heating above Tg, followed by xylitol and by D-sorbitol. By this measure, glycerol’s HB network is the most robust against thermal degradation. Note that the comparison above is made on a per OH basis. On a per molecule basis, the difference between the three systems is even greater since the number of OH groups per molecule increases in the same order: glycerol (3), xylitol (5), and D-sorbitol (6). The temperature effect on this and other aspects of structure provides the “structural fragility”7–11 that will be compared with the dynamic fragility.

FIG. 6.

Stability of the number of HBs per OH nHB against heating above Tg. Glycerol shows the smallest loss of nHB for the same temperature change relative to Tg.

FIG. 6.

Stability of the number of HBs per OH nHB against heating above Tg. Glycerol shows the smallest loss of nHB for the same temperature change relative to Tg.

Close modal

Having investigated the temperature effect on one aspect of structure (HBs) in the polyalcohol liquids, we now turn to the second aspect associated with the main scattering peak near q1 = 1.4 Å−1 in Fig. 1(b). Given that this peak occurs at the lowest q, its real-space manifestation in D(r) is the farthest-reaching ripples [Fig. 1(d)]. Below, we describe our results on this structural order and then discuss its origin in the C⋯C correlation. In Fig. 7(a), we compare the main scattering peaks q1 of the three systems near Tg. The glycerol peak is the sharpest, followed by D-sorbitol and by xylitol. In Fig. 7(b), we show the real-space structure using the “excess coordination” integral: ΔZ(r) = 0rxDxdx, which yields the total number of atoms within a sphere centered on a tagged atom relative to the bulk density. Integrating in this way removes artificial oscillations in D(r) introduced by Fourier transform, revealing true density modulation at long distances. Glycerol’s density modulation propagates the farthest, followed by D-sorbitol and by xylitol, as expected from the sharpness of their main scattering peaks [Fig. 7(a)].

FIG. 7.

(a) The main scattering peaks q1 of glycerol, xylitol, and D-sorbitol. The systems are compared near Tg (at 175 K for glycerol, 250 K for xylitol, and 275 K for D-sorbitol). (b) Persistence of pair correlation displayed using the function ΔZ(r) [integral of rD(r)]. For clarity, the curves are offset vertically by 4. L1 is the wavelength of density modulation; ξ characterizes its persistence length. The colored curves in (b) are calculated using the parameters obtained by fitting the q1 peak to Eq. (6), matching the experimental curves in black starting from the second peak. ΔZ(r) is expected to approach [S(0) − 1] as r → ∞, and this is observed with our data.

FIG. 7.

(a) The main scattering peaks q1 of glycerol, xylitol, and D-sorbitol. The systems are compared near Tg (at 175 K for glycerol, 250 K for xylitol, and 275 K for D-sorbitol). (b) Persistence of pair correlation displayed using the function ΔZ(r) [integral of rD(r)]. For clarity, the curves are offset vertically by 4. L1 is the wavelength of density modulation; ξ characterizes its persistence length. The colored curves in (b) are calculated using the parameters obtained by fitting the q1 peak to Eq. (6), matching the experimental curves in black starting from the second peak. ΔZ(r) is expected to approach [S(0) − 1] as r → ∞, and this is observed with our data.

Close modal

For all three systems of this study, the D(r) oscillation at long distance has the form of exponentially damped sine wave,47,48

Dr=aexpqcorrrsin(q1r+θ),
(5)

where a characterizes the amplitude of oscillation, q1 is the spatial frequency, qcorr is the inverse correlation length (qcorr = 1/ξ), and θ is the phase. Equivalently, we find that the q1 peak is well fitted by the Fourier inversion of Eq. (5) [see Eq. (2)],48 

Sq=a2qcorrq1cosθ+qcorr2q12+q2sinθqcorr2+(qq1)2qcorr2+(q+q1)2.
(6)

In Fig. 7(a), the smooth curves in color are the fits of the experimental data (black circles) to Eq. (6). The resulting parameters, when entered in Eq. (5), yield the smooth colored curves in Fig. 7(b); these curves reproduce the experimental curves in black starting from the second peak. This agreement indicates that the far-reaching density modulation in real space corresponds to the q1 peak through Fourier transform. Although the density ripples can be fitted in real space, fitting the q1 peak in the Fourier space is numerically advantageous since a single peak with high fidelity is fitted.48 The parameters from this fitting are collected in Table I.

We now turn to the origin of the far-reaching pair correlation in Fig. 7(b). The observation1,13 that adding an OH group to an alkane does not strongly alter the main scattering peak near 1.4 Å−1 argues that this peak mainly corresponds to the C⋯C correlation, a conclusion supported by MD simulations.49 For glycerol, this conclusion has been confirmed by simulations from the literature22,27 and this work (Fig. 3). From Fig. 3, we see that the O⋯O and C⋯O correlations have sharp first peaks but relatively weak long-range ripples. In contrast, the C⋯C correlation has a broad first peak but robust ripples at long distances. These ripples have the same wavelength (4.4 Å) and functional form [exponentially damped sine wave, Eq. (5)] as observed experimentally. At r > 5 Å, the C⋯C ripples are well fitted using Eq. (5), yielding L1 = 4.42 Å and ξ = 8.38 Å at 300 K, in good agreement with the experimental values (L1 = 4.43 Å and ξ = 7.23 Å; Table I and Fig. 8). The slightly faster damping of experimental D(r) could reflect the inaccuracy of the force field and the small contributions from the faster-damping O⋯C and O⋯O correlations. The different pair correlations described above reflect the different kinds of bonding in the system. As shown by the schematic at the bottom of Fig. 2, the HBs between OH groups create a sharp O⋯O peak at 2.8 Å (ROO) and a relatively sharp C⋯O peak near 3.7 Å because of the covalent C–O linkage (R3). In contrast, the hydrocarbon groups interact by weaker van der Waals forces to form closely packed structures. Density modulations in the form of exponentially damped sine wave are well known for closely packed systems.37,38,50–52 Overall, the results indicate that the q1 peak of glycerol and the corresponding real-space ripples in D(r) correspond to the C⋯C correlation. The same conclusion is assumed to hold for xylitol and D-sorbitol.

FIG. 8.

(a) Wavelength of density modulation L1 and (b) persistence length ξ associated with the main scattering peak q1 plotted against temperature. The down arrows indicate Tgs. The black triangles are the results of MD simulations for the C⋯C correlation in glycerol.

FIG. 8.

(a) Wavelength of density modulation L1 and (b) persistence length ξ associated with the main scattering peak q1 plotted against temperature. The down arrows indicate Tgs. The black triangles are the results of MD simulations for the C⋯C correlation in glycerol.

Close modal

Figure 8(a) shows the temperature dependence of the density-modulation wavelength associated with the main scattering peak q1: L1 = 2π/q1. For each system, L1 is ∼4.5 Å and decreases with cooling. The decrease is halted at Tg (down arrow). For these systems, the temperature dependence of L1 is almost identical with that of the average distance d between molecules: d = n−1/3, where n is the molecular number density. As a result, L1n1/3 is nearly independent of temperature: 0.89 for glycerol, 0.82 for xylitol, and 0.76 for D-sorbitol (see Table I). The decrease of L1n1/3 in the order of glycerol, xylitol, and D-sorbitol results from the fact that L1 is roughly constant, but n−1/3 increases in this order.

Figure 8(b) shows the persistence length ξ for the C⋯C correlation in glycerol and its analogs. ξ is the distance by which the amplitude of density modulation drops to 1/e (34%) of the initial value [see Eq. (5)] and indicates how far local order penetrates into the bulk. At the same temperature, glycerol has the longest ξ, followed by D-sorbitol and by xylitol. Thus, by this measure (C⋯C correlation), the glycerol liquid is the most ordered. For each system, ξ increases with cooling, indicating more structure with respect to the C⋯C correlation, until the glass transition halts the progress (down arrow at Tg). Using Tg as the reference, we find that upon heating, D-sorbitol’s ξ decreases the fastest, followed by glycerol and by xylitol. Thus, xylitol’s C⋯C correlation, although the weakest to begin with, is the most resistant to thermal degradation. It is noteworthy that the temperature effect on the C⋯C correlation is significantly different from that on the O⋯O correlation described above, as expected from a fundamental difference between the two structural orders.

This study has investigated the temperature effect on two aspects of structure in three polyalcohol liquids: the O⋯O correlation associated with the hydrogen bonding of polar OH groups and the C⋯C correlation associated with van der Waals interactions of non-polar CHn groups. We find that the two structural orders evolve differently with temperature. For each system, the number of HBs per OH, nHB, increases with cooling until kinetic arrests at Tg (Fig. 5). Upon cooling from 400 K, the rate at which nHB increases is similar in all three systems and the overall trend is well described by a simple two-state equilibrium between hydrogen-bonded and non-hydrogen-bonded OH groups with ΔH = 9.1 kJ/mol and ΔS = 13.4 J/mol/K. We find that the HB network in glycerol is the most stable against temperature change, losing the fewest HBs per OH on heating above Tg, followed by xylitol and by D-sorbitol (Fig. 6). All three liquids show far-reaching exponentially damped density modulations with a similar wavelength of L1 ≈ 4.5 Å but different persistence lengths ξ (Fig. 8, Table I). This aspect of the liquid structure corresponds to the main scattering peak and with the C⋯C correlation. At a common temperature, glycerol has the longest ξ, followed by D-sorbitol and by xylitol; for heating above Tg, D-sorbitol’s ξ decreases the fastest, followed by glycerol and by xylitol. In what follows, we discuss the two types of structural order and relate them to the current theories of the structural origin of dynamic fragility.

In an alcohol molecule, the polar OH group is “permanently” connected to the non-polar hydrocarbon group. In a liquid, the OH groups tend to associate with each other by HBs, leading to the well-defined O⋯O distance and number of nearest neighbors. The process of forming HBs, in turn, perturbs the manner in which the CHn groups pack together. The regions of HBs exclude the CHn groups and interrupt what is otherwise a contiguous domain of hydrocarbon. This creates a mosaic structure whose details are determined by the molecular structure. In alcohols with relatively large hydrocarbon segments, the mosaic structure is responsible for the prepeaks in x-ray scattering1 and leads to multiple relaxation modes.17 For the polyalcohols studied here, the OH and CHn groups are so tightly interwoven in the same molecule that the mosaic structure must be finer than that of monoalcohols with comparable size. Indeed, the polyalcohol liquids do not show detectable prepeaks [Fig. 7(a)]. Nevertheless, they do show strong C⋯C correlations as the farthest-reaching features in pair correlation, indicating that the long-range regularity of the C⋯C packing is not destroyed by the adjacent OH groups.

To further characterize the C⋯C order in the polyalcohol liquids, we compare their persistence lengths ξ with that of other systems. As a point of reference, we use the hard-sphere liquid for which ξ is well characterized. For this purpose, we calculate the ratio ξ/L1, which indicates the number of ripples within the persistence length ξ. For a hard-sphere liquid, ξ/L1 ≈ 0 at low density and rises exponentially with packing fraction ϕ, reaching a value of 2 at ϕ = 0.52.38,51 (The maximal ϕ is 0.64 for the densest random close packing.53) In comparison, for the polyalcohols of this study near Tg, ξ/L1 = 1.8 for glycerol at 175 K, 1.3 for D-sorbitol at 275 K, and 1.0 for xylitol at 250 K. These values indicate significant correlation between the hydrocarbon groups comparable to hard-sphere liquids at a moderately high density (ϕ = 0.4–0.5).

Our results indicate very different temperature dependence for the O⋯O and C⋯C correlations in the polyalcohol liquids. For example, the three systems show a similar gain of HBs per OH upon cooling (Fig. 5), but their persistence lengths ξ for C⋯C correlation exhibit very different temperature dependence [Fig. 8(b)]. In the case of xylitol, ξ hardly changes with temperature. Given the very different interactions leading to the O⋯O and C⋯C correlations observed (HBs vs van der Waals forces), the different temperature dependence of the two structural orders is not surprising. This duality of the structure is irrelevant for simple liquids such as argon but important for many liquids composed of “amphiphilic” molecules such as alcohols. In the latter case, any structure–property investigation necessarily confronts the question: “Which structure and which property?” We have seen that the C⋯C order correlates well with bulk density, as captured by the relation L1n1/3 ≈ constant (Table I and Fig. S2). Stimulated by recent work,7–11 we will explore below whether the two structural orders investigated here are related to dynamic fragility.

In Angell’s classification of glass-forming liquids,54o-terphenyl and SiO2 represent the fragile and strong behaviors, while the polyalcohols of this study fall in between, in the order glycerol (strongest) > xylitol > D-sorbitol (most fragile). As shown in Fig. 9,55 the viscosity of o-terphenyl increases sharply as it approaches the glass transition temperature Tg by 7 orders of magnitude in the final 10% of the journey (from 1.1 Tg to Tg), while the corresponding increase is much smaller for SiO2 (two decades). Glycerol falls between these two extremes, while xylitol is more fragile than glycerol and D-sorbitol still more fragile. Among glass-forming organic liquids, glycerol is considered the strongest. Whether there is a structural origin for the different dynamic fragility has been the subject of recent studies.7–11 

FIG. 9.

Angell plot for the polyalcohols of this work, o-terphenyl, and SiO2. Of the three polyalcohols, glycerol is the strongest, followed by xylitol and by D-sorbitol. Viscosity data are from the collection in Ref. 55 except for xylitol, whose viscosity is calculated from its structural relaxation time to give a common η at Tg.7 

FIG. 9.

Angell plot for the polyalcohols of this work, o-terphenyl, and SiO2. Of the three polyalcohols, glycerol is the strongest, followed by xylitol and by D-sorbitol. Viscosity data are from the collection in Ref. 55 except for xylitol, whose viscosity is calculated from its structural relaxation time to give a common η at Tg.7 

Close modal

One line of investigation focuses on the temperature dependence of the structure associated with the main scattering peak q1.8–11 Mauro et al.8 defined a “structural fragility” based on the temperature dependence of the height of the q1 peak. For multi-component glass-forming metallic liquids, they reported a correlation between high structural fragility and high dynamic fragility. Their analysis involves extrapolating the q1 peak’s height of a high-temperature liquid (at 1.7–1.9 Tg) to Tg and compare it with the glass value. This analysis cannot be performed with our data because our temperature range does not cover 1.7–1.9 Tg for all three systems. Voylov et al.10 defined a “structural fragility” using the width of the main scattering peak Δq, calculating its percentage change from 1.3 Tg to 0.9 Tg, δΔqq. For 12 glass-formers of different types, they reported a correlation between large δΔqq and high dynamic fragility. Their correlation plot is reproduced in Fig. 10(a) to which we have added the data points from this study as crosses. In our notation [Eq. (5)], δΔqq = δqcorr/qcorr = 5.2% for glycerol, 2.4% for xylitol, and 10% for D-sorbitol. In Fig. 10(a), we see our xylitol and D-sorbitol points are significantly off the published trend. For glycerol, our value (5.2%) compares with the published value of 10.4%, possibly a result of different fitting functions and fitting range (1.4–1.6 Å−1 in Ref. 10 and 0.5–2 Å−1 in this work). Nevertheless, our glycerol point agrees reasonably with the overall trend.

FIG. 10.

Correlation between the fragility index m and (a) percentage change of the width of the main scattering peak10 and (b) scaled correlation length ξ at Tg.11 The original plots are from Refs. 10 and 11 to which our data points (X) are added.

FIG. 10.

Correlation between the fragility index m and (a) percentage change of the width of the main scattering peak10 and (b) scaled correlation length ξ at Tg.11 The original plots are from Refs. 10 and 11 to which our data points (X) are added.

Close modal

In Ref. 11, Ryu and Egami used the correlation length ξ at Tg as a structural metric and reported a correlation between the fragility index m and ξ/a, where a is the average nearest-neighbor distance [Fig. 10(b)], with larger m correlating with larger ξ/a. To their plot, we have added our data points using L1 ≈ 4.5 Å to represent the average nearest-neighbor distance for the C⋯C correlation. Our data points join the scatter of points in the original plot, lying near the edge. Within the three polyalcohols, glycerol has the largest ξ [see Fig. 8(b)] and is predicted to be the most fragile, which is inconsistent with the experimental result (Fig. 9).

In a second approach, developed for hydrogen-bonded liquids, the stability of the HB network is the structural attribute that is related to dynamic fragility, while the interaction between the hydrocarbon groups is considered secondary.7,19 Nakanishi and Nozaki7 equated a specially defined HB cluster with a Cooperatively Rearranging Region (CRR) of Adam and Gibbs. They treated each OH as having a temperature-dependent probability to be hydrogen-bonded, given by Eq. (4), with the probability being insensitive to molecular details. A polyalcohol with more OH groups is thought to be able to form a larger cluster, but the cluster size is more vulnerable to damage by heating, leading to a fragile behavior. Our results support this model at a qualitative level. We have observed that the probability for an OH group to be hydrogen-bonded is well described by a simple two-state equilibrium with common ΔH and ΔS values for three polyalcohols, regardless of their molecular details (Fig. 5). However, our values of ΔH and ΔS are significantly different from those of Ref. 7. We have also observed (Fig. 6) that when heated above Tg, the robustness of the HB network follows the order glycerol (most robust) > xylitol > D-sorbitol (least robust), matching the ranking of dynamic strength (Fig. 9). Overall, our results suggest that for the polyalcohol liquids, the structure associated with HBs provides a better foundation for understanding the different dynamic fragility than the structure associated with the main scattering peak q1. Of course, this distinction is needed only for the systems that have multiple structural orders as studied here due to preferential association. For simpler liquids without network association, the persistence length of density modulation could be a basis for understanding their dynamic fragility.

We have studied three liquid polyalcohols (glycerol, xylitol, and D-sorbitol) by synchrotron x-ray scattering over a wide temperature range that covers both the liquid and the glassy states. We find that the O⋯O correlation associated with HBs and the C⋯C correlation associated with hydrophobic interactions evolve very differently. Upon cooling, more HBs are formed in all three systems until kinetic arrests at Tg and the effect is well described by an equilibrium between bonded and non-bonded OH with ΔH = 9.1 kJ/mol and ΔS = 13.4 J/mol/K. When heated above Tg, glycerol loses the fewest HBs per OH for a given temperature rise scaled by Tg, followed by xylitol and by D-sorbitol, in the same order the number of OH groups per molecule increases (3, 5, and 6). The pair correlation functions of the three liquids show exponentially damped density modulations of a similar wavelength (4.5 Å) but different persistence lengths. These density modulations are associated with the main scattering peak and the intermolecular C⋯C correlation. In this respect, glycerol is the most ordered with farthest-propagating density ripples, followed by D-sorbitol and by xylitol. Heating above Tg causes faster damping of the density ripples with the rate of change being the slowest in xylitol, followed by glycerol and by D-sorbitol. We find that the fragility difference is better understood on the basis of the thermal stability of HB clusters than that of the structure associated with the main scattering peak. In this regard, further work is still needed for a quantitative agreement between theory and experiment. This work highlights the need to distinguish the different structural orders of liquids in order to understand their properties, especially for liquids of amphiphilic molecules that undergo preferential association leading to mosaic structures. Future progress could benefit from reverse Monte Carlo fitting of the observed structural factors to obtain partial pair correlations, orientational correlation, and detailed topology of the HB network and from elucidating the connection between the static length scales obtained here and other length scales in use in the literature that characterize the glass formation.56–58 

See the supplementary material for S(q) and D(r) of glycerol, xylitol, and D-sorbitol ; xylitol liquid densities measured in this work (Table S1); and details of molecular dynamics simulation.

We thank the NSF for supporting this work through the University of Wisconsin Materials Research Science and Engineering Center (Grant No. DMR-1720415).

The authors have no conflicts to disclose.

The data that support the findings of this study are available within the article and its supplementary material.

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Supplementary Material