THEGLASS
' ~ R A N S I T ION I N
AMORPHOUS WATER
possible that iodine molecule will move partly with solvated ethanol moZecule. If we can assume that the stability of Iz -t. EtOH complex in the present mixed media, say, in an equimolar mixture, is larger than that in pure ethanol, such a two diffusing unit mechanism can also account for f i e fact that D I I ' V ~ ~is/ Textremely small in comparison with that for pure solvents in the present mixed sdvent systems. We can thus oonciude that solute-solvent interactions have a large effect on the variation of Duo with the composition in i he present mixed solvent systems. One may question possible inconsistency between the present interpretatlon and that given in P-I, as we had emphasized 1 he importance of structural anomalies in interpreting the behavior of D1L0q23/T us. x2 relation in alcohol-water 5oiutions. Perhaps we can ascribe this
967 difference t o the facts that the hydrogen-bonded structures which prevail in aqueous alcohol solutions are three dimensional, while, in alcohol nonpolar liquid solutions, the associations are less pronounced, the associated complex is of linear type, and there should be no structure-making effect due to the second solvents. It should also be pointed out that the modes and strength of interactions with Izin water and alcohol are similar to each other, This implies that alcohol water systems are of EtOH toluene type rather than of EtOH n-hexane type, though this classification should not be taken as definitive. At the present stage, it should be indispensable for establishing a general relation to accumulate the diffusivity data for ternary systems where the interactions among three components involved are of different strength and specificity.
+
+
+
+
The Glass Transition in Amorphous Water.1 Application of the Measurements io Problems Arising in Cryobiology
on H. Rasmussen and Alan P. MacKenzie* Cryobiology Research Institute, Madison, Viseonsin
68704
(Received A p ~ i 16, l 1070)
Publication costs assisted by the U. S. Public Health Service
Values for the glass transition temperature, Tg,of aqueous solutions of glycerol, ethylene glycol, and methanol, measured by differential thermal analysis, were extrapolated to obtain values for T , in amorphous water. l h r a heating rate of 5 deg min-l, T , for amorphous water is observed at - 137 1'. The results were correlated with Jenckel's expression for the concentration dependence of T , in binary solutions. A kinetic analysis after ?vlchiiilan led to "kinetic" parameters for the glass transition that were linear functions of weight fraction and yielded a time-temperature dependence for T , in amorphous water correlating well with previously published values.
*
Introduction
temperature for water by extrapolation t o zero solute concentration. Yannas5 used a buoyancy method that Thermal techniques have been applied t o the study determined the temperature a t which the specific gravof the glass transition in pure water on several occaity of a solution of glycerol in water changed abruptly sions. Pryde and J onesza reported an approximate upon warming. Yannas' warming rates ranged from determination of the glass transition temperature. 0.5 t o 2 deg min-' and resulted in a glass transition temMcJIillan and L,os2bdetermined, by differential thermal perature for water, by extrapolation, of -146 f 4". analysis, that, at a warming rate of 20 deg min-l, the glass transition occurred at - 134". Sugisaki, Suga, and Seki,3aftis a calorimetric determination, reported a (1) Supported by Grant No. GM-15143 from tho U. S . Public value of -138" (though their published data suggest Health Service. a maximum in dC~/clTaround -140"). G h ~ r m l e y , ~ (2) (a) J. A. Pryde and G. 0. Jones, Nature, 1'90,685 (1952); (b) J. A. McMillan and S.C. Los, ibid., 206, 806 (1965). using thermal analysis of warming curves, could not (3) M. Sugisaki, W. Suga, and S.Seki, J . Chem. SOC. Jag., 41, 2591 detect a glass transition. (1968). Several workers have examined glass transitions in (4) J. A. Ghormley, J . Chem. Phys., 48, 503 (1968). aqueous solu5ons and estimated the glass transition ( 5 ) I. V. Yannas, Science, 160, 298 (1968). The Journal of Physical Chemistry, S o l . 76,No, 7 , 1971
DONH. RASMUSSEN AND ALANP. MACKENZIE
968
block. Higher sample cooling rates were necessary t o AngeP1, Sares and. BresseP estimated the temperature permit the retention of the metastable amorphous state of the glass ixansirion in water, from a study of the in solutions in which the water was present in higher glass transition in concentrated salt solutions by difconcentrations. ferential thermal analysis, t o be - 133 5". For the purposes of this analysis of the glass transiI n the course of the determination of various nontion behavior of aqueous solutions, we used Baker equilibrium 1 rtlnsit ions observed in several aqueous Analyzed reagent grade ethylene glycol, glycerol, and systems subjected i o rapid cooling, Luyet and the presmethanol. These materials contained, according t o ent authors yeportled glass transition temperatures of the supplier's specifications, 0.20%, 4.5%, and 0.04% aqueous solutions of ethylene glycol, glycerol, glucose, water, w/w, respectively. All the solutions were presucrose, dimethyl sulfoxide,* and polyvinylpyrrolipared gravimetrically with deionized, distilled water. done.g From these data the present aut)hors deterAfter the sample and the reference tubes were filled mined the glass transition temperature for pure water (with 0.01-ml volumes delivered from micropipets), to be -140 =I= 3". Noting the wide divergence in values (from --133 A 5 t o - 146 f 4") we undertook a thermocouples were inserted. The reference (deionized study, by differential thermal analysis, of the concendistilled water) was frozen (slowly by comparison with tration dependence of the glass transition in aqueous the sample) in cold nitrogen vapor; the sample tube was solutions OF several solutes at fixed heating rates and an immersed in liquid nitrogen (cooling rates approxianalysis of the effect of different heating rates on the mated 75 deg seei1 between 0 and - 150") or in a bath glass transition in solutions of ethylene glycol. It is of liquid nitrogen containing frozen slushy nitrogen, at our purpose here .to demonstrate agreement between a temperature of -210" (cooling rates approximated the seveTal extrapolations of the glass transition tem200 deg sec-l between 0 and - 150"). The sample and perature curves t o pure water for the different solutes reference were then transferred to the precooled block and to derivc parameters relating to the "kinetics" of and the warming rate adjusted t o the desired value the glass tramition in amorphous water from those rebetween 1and 21 degmin-'. Heating rates were found lating to the process observed in aqueous solutions of to be reproducible to f 5 % of the values desired. ethylene gly~ol. From the nature of the concordance Results of our results and the temperatures for the glass transition of amorphous water previously r e p ~ r t e d ~ ~ ~The ~ ~curves obtained by differential thermal analysis 50% w/w ethylene glycol solutions that were cooled of the reasons for the discrepancies among the latter by immersion in boiling nitrogen and warmed a t values will become ,zpparent. various rates are displayed by Figure 1. These and Apparatus, Materinlis, and Methods the other glass transitions observed in this study necessarily relate to quenched glasses, the cooling rates being The apparatus for differential thermal analysis contoo high to allow a so-called "quasi-equilibrium glass"1° sisted of: (1) a brass block that could be precooled t o to form. Without these extreme cooling rates, how-190" and heated electrically at a constant rate beever, crystallization of water was invaiiably initiated, tween l and 2 i deg min-'; ( 2 ) sample tubes of l mm bore even in solutions of 50% w/w solute, changing the conand 0.23 rnm wall thickness made of Pyrex glass; (3) centration of the remaining glass. A more detailed a thermocouple system made from 3-mil copper and discussion of the crystallization of water during the consz.antan wires which measured (a) absolute referwarming of quenched solutions has been given by ence tempereture, and (b) the temperature differential Luyet and Rasmussen.' We believe that data reported between snniple arid reference; and (4) an amplifierhere correspond to measurements obtained with totally recorder system f o r plotting, on an X-Y basis, the difof behavior of the glass vitrified solutions. Continuity ferential temperature and the absolute reference temtransition as a function of concentration as well as perature. The khermocouple-recorder system was calicontinuity in other transitions' lea us to accept the brated at thib boiling point of liquid nitrogen (- 196") values determined from solutions above about 35% and the freering point of ethyl acetate (-83.6"). The w/w solute. depending upon the solute. setting of th? zero point (at -196") was checked beThe glass transition endotherm occupies an interval fore each run. Temperatures were measured always t o of 5.3-5.8 deg and exhibits a nonsymmetrlcal sigA0.5" and, relalive t o one another, to f0.2". The moid shape (see again Figure 1). The relationship sensitivity of the recorder to changes in the differential temperature was (enhanced by preamplification via a Leeds and Northrup dc amplifier; the resulting sensi(6) C. A . Angell, E. J. Sares, and R. D. Bressel, J . Phys. Chem., 71, 2759 (1967). tivity could be varied up to 0.5 pV/in. of display (7) B. Luyet and D. Rasmussen, Biodynamica, 10, 167 (1968). with minimal noise. The thermocouples were ar(8) D. H. Rasmussen and A. P. MaoKenzie, Nature, 220, 1315 ranged t o intjert into the sample and reference tubes in (1965). such a wa,y that the sample assembly, minus the block, (9) B. Luyet and D. Rasmussen, Biodynamica, 10, 137 (1967). could be coded at a much higher rate than could the (10) G. Adam and J. H. Gibbs, J . Chem. Phys., 43, 139 (1965).
*
The Journal of Physical Chemistry, Vol. 76, N o . 7, 1971
THEGLASS
969
TH,ANSITIQN IN aM0RPHOUS WATER
4,
deg/niin
1.0
0
.B .6 .7 .8 .ii .1 .2 .3 .4 SOLUTE CGNCEHTRATION REICKT TRACTION, Q , nnd IlGI.1: I R A C T I O N , tD
i.0
Figure 2. Glass transition temperatures for aqueous solutions of glycerol, ethylene glycol, and methanol.
-is0
-140
-130
-120
T E W E l I A T U R E OF R t F E H L N C f : S A I P L E , ' C .
Figure 1 . Thermograms for 50% ethylene glycol.
of the glass trmsition endotherm as observed by differential thermal andysis to the actual change in the specific heat of the sample has been discussed by McMil1ane1l According to his analysis, the rate of change of specific lieat i s a maximum at the point of inflexion in the thermogram. However, since the curves are asymmetric and the inflexion points difficult to determine, we adopted a tlechnique for measuring T , shown in Figure 1. While this method does not take direct account of the ssymrnetry of the transition it must be noted that, for a change in the heating rate from 1 t o 21 deg min-I, the temperature range which the transition spans (from point A t o B in Curve 6, Figure 1) is about constant :it 5.5 rt 0.3". The glass trcansition temperatures of aqueous solutions of ethylene glycol, glycerol, and methanol have been measured by this method at a heating rate of 5 deg min-l in the comenuration range from 35 to 100% w/w solute (05$7,in the case of glycerol). These data, presented in Figure 2, delimit tlie range of temperatures within which one would place, by extrapolation, the glass transition for amorphous water a t a corresponding heatiiig rate. We made several attempts to add other curvm t o Figure 2 by choosing solutes, isopropyl alcohol and 2-methoxyethanol, for example, having glass transition temperatures (- 153 and - 140", respectively) different from each of the three substances first examined. Unfortunately, we were unable, with the means of rttpid cooling at our disposal, to vitrify these solutions over w r y wide ranges of concentration.
Various semiempirical expressions, developed to describe the dependence of the glass transition upon composition in polymer-diluent systems, have been reviewed by Shen and Tobolsky.l2 The formula most acceptable t o them, and the one used by Yannas6 to obtain a value for the glass transition temperature of water, is that of JenckelX3
Tg
=
Tgiwl
+ TgZW2 + K ~ I W Z
(1)
where T , is tlie glass transition temperature in deg K, w is the weight fraction of each constituent, subscripts 1 and 2 refer to water and solute, respectively, and K denotes a proportionality constant dependent on the particular compounds involved. A plot of ( T , T,,)/wI us. W I yields a slope of -K and an intercept of [T,, - T,, K ] . From data that include T,, and the glass transition temperature, T,, at various concentrations, the temperature T,, can be derived. The results of such an analysis on our solutions (Figure 2 ) at a heating rate of 5 deg min-1 are shown in Figure 3. The derivation yields K values of -26 & I for waterglycerol, -- 13 f 1 for water-ethylene glycol, and 15 i 1 for water-methanol. The resultant values for the glass transition temperatures "TTgl)' are in good agreement with each other and with the visual extrapolation of the curves in Figure 2 . That is, a value of - 137 l oat a heating rate o f 5 deg min--' is found both by visual and empirical extrapolation. Though Jencliel's expression was originally intended to describe the reduction in the glass transition temperature of a polymer matrix with increased concentration of monomer or plasticizer, it appears to hold also for the concentration dependence of the glass transition in binary aqueous solutions.
+
+
*
(11)
J. A. ,MoMillan, J . Chem. Phys.,
42,
3497 (1965).
(12) M. C. Shen and A. V. Tobolsky, Advan. Chem. Ser., Eo. 48, 27
(1965).
(13) E. Jenckel and It. Heusch, Kolloid-Z., 130, 89 (1958).
The Journal of Physical Chemistry, Vol. 76, N o . 7 , 1971
DONH. RASMUSSEN AND ALANP. MACKENZIE
970 -loo
r
I
festation of To. I n the region of the glass transition, the average relaxation time follows the Vogel equation In
T
=
A
+ B / ( T - To)
(2)
where r is the relaxation time, A and B are constants, T is the temperature of measurement of +, and TOis the "zero mobility" temperature. Provided sufficient information regarding the value of To is knowri, or sufMETHANOL ficiently extensive experimental data relating + to T,, I 49 K = 15 the values of constants A and B and To can be evaluated. Our results, obtained by varying the heating I = -31 -20 K -13 rate from one scan by dta to another, indicate a timeT, (%O) = -137 k 1 ' C . temperature dependence for the glass transition, but the variation in T,, from the highest Lo the lowest heating 0 1 .2 .3 .4 .5 .6 .7 .8 .9 10 WElGI3T F R A C T l O l BATER, rate, was only 5 deg. These somewhat limited experimental data, indicating as they do that each individual Figure 3. Analysis of the glass transition as a function of transition spans 5.5 deg, preclude evaluation of the weight fraction aF solute. Tg = Tglwi + T,~wzf Kwiwz. three-parameter equation and of the graphical tests of the usefulness of the Toconcept. It was, consequently, In an attempt t o interpret the observed variation in not possible to test such derivations as that of Adam the glass transition temperature with heating rate, the and Gibbs, who proposed the relationship T g / T 0 = times taken by the samples to undergo the changes in 1.29; we could not select a value for Tomore consistent specific heat detected by dta were calculated from the than another with the primary data. temperatuyes spanned by the respective transitions ; for If values for TOwere derived for the ethylene glycolexample for 50% EIG,the transition spanned 5.5 f 0.3". water system from some other type of measurement, For a change in heating rate from 1 to 21 deg min-l, (e.g., viscosity) they would suffice t o permit the evaluathe time therefore varied from 5.5 min at -131.6 k tion of the "kinetics" of the glass transition using 0.2 t o 0.26 min a t -127.4 A 0.2". These times may Vogel's equation. T o the authors' knowledge, howbe considered representative of the time scales of parever, this information is not available, ticular experiments and the temperatures at which I n the absence of the necessary information regarding T , is observed as the temperatures of the relaxation in TO,the derivation due to McMillan, based on classical the respectivi: experimental time scales. Under these kinetic theory, was adopted in the belief that its applicaconditions, the time required for the observation of a tion might provide some insight into the concentration T, could be compared with relaxation times for prodependence of the "kinetics" of the relaxation and allow cesses such RS aliflusion, viscous flow, and dielectric the extrapolation of the parameters obtained to pure relaxation; that is the formulas describing these prowater. Such an analysis corresponds t o the evaluation cesses might be used t o describe the relaxation at T,. of an equation of the form of eq 2, where Tois set equal That there must be a lower temperature limit, TO, to O"K, though, as previously pointed out, TOmust in above absolute zero, to the glass transition measured reality exceed absolute zero. during an e!xperiment of infinite time scale (zero heating McMillan l 1 derived kinetic expressions relating rate) was pointed out by Kauzmann.'d Below this TO, thermal behwior in the neighborhood of the glass if liquid state properties were t o persist, negative entransition temperature with measurements obtained by tropies, volumes, etc,, with respect to the crystal, would differential thermal analysis. To determine the apresult. The use of such a lower temperature limit t o plicability of his procedure t o our data, we repeated his liquid state properties is inherent in the Williamsanalysis of the effect of heating rate on the glass transiLandel-Ferry l5 equation for relaxation in viscoelastic tion in glycerol. Though the samples we used were systems arid in the equation due t o Vogel.16 Gibbs smaller than his by a factor of 750 and our heating rate and DiMarzioIT predicted a thermodynamic secondranged from 1 t o 21 deg min-l, we found, with reagent order trsneitim at To and Adam and GibbslO presented grade glycerol (95.5% glycerol) an activation energy the concept that the configurational entropy of the for the glass transformation of 40.6 i1 kcal/mol, a liquid disappears a$,Toand that during the glass transivalue agreeing favorably with his figure of 42.4 f 1.2 tion the number of states available t o the system is reduced to such an extent that transformation between (14) W. Kauzmann, Chem. Rev., 43, 219 (1948). (15) 1M. L.Williams, R. F. Landel, and J. D. Ferry, J . Amer. Chem. alternative riaates requires prohibitively numerous Soc., 77, 3701 (1955). cooperative rearrangements consistent with the thermal (16) H. Vogel, Phvs. Z., 22, 645 (1921). energy of the system and the time allowed for measure(17) J. H. Gibbs and E. A. DiMarzio, J . @hem. Phys., 28, 373 ment; more simply, T, is supposedly the kinetic mani(1958). GLYCEHOL I = -78 K = -26
-
i
W1
The Journal of Physical Chemistry, Vol. 76,No. 7, 1871
I
THEGLASST ~ a ~ s r v r oINn AMORPHOUS WATER
430 0
97 1
i:
30
32
1 160
150
bD 0 3I
140
2 0
130
4i
100
1
12 0
1
1
.1
.2
I
1
1
1
1
I
.5 .6 .7 .8 WE1G.W FRACTIOS ETHYLEXE GLYCOL .3
.4
I
.9
80
1.0
Figure 5. Activation enthalpy and activation entropy as a function of composition. 6 3
(1
4
e.5
6 6
6.7
6.8
1000
6 9
7.0
7.1
7.2
7.3
/ Tg
Figure 4. Effect of heating rate on the glass transition temperatures of aqueous soluhions of ethylene glycol.
8.8
9’0
1
kcal/mol, especially so when the presence of 4.5% water by weight is considered. An analysiis of ethylene glycol and its aqueous solutions showed that McR!Iillan’s kinetic expressions were applicable siiic~:,when the heating rate was varied, the glass transition Gemperatwe varied as demonstrated in Figure 4. According to McMillanll the slopes of the curves in Figure 4 are related t o the activation enthalpy, AH*, necessary for the relaxation process by the equation
and to the activatiton entropy, AS*, hindering the relaxation, by
AX*
=
R[AEI*,’RT,
+ In ( A H * / R ) In (T,2/d - In (kT,lh)l
1
I
I
0
.1
.2
I
I
I
I
.3 .4 .5 .6 .7 .8 WEIGHT FRACTION ETHYLENE GLYCOL
.9
1.0
Figure 6. Activation free energy, AF*, as a function of composition.
(4)
where R = the gas constant, T , = the measured glass transition temperature, p = the heating rate in deg min-l, k = Boltema,nn’s eonstant, and h = Planck’s constant. These parameters, hH* and AS*, are plotted as functions of conct:ntration in Figure 5 . The linearity of the curves is in each casle strikingly apparent and is, perhaps, related to some additive property, free volume for instance, on 1% weight fraction basis. The extrapolated enthalpy of activation, AH*, for the glass transition in water is 19.2 f 0.5 kcal/mol; the activation entropy, AS*, is 87 f 2 cal/mol O K .
From the entropy and enthalpy of activation, the free energy of activation was calculated from the relation: AF* = AH* - TAX*. Thus a separsbte curve relating AF* to the composition of the system could be obtained at each heating rate. Four such curves, presented in Figure 6, were obtained with values interpolated from the original data exhibifed in Figure 4. The AF* values for water range from 7.55 f 0.1 kcal/ mol at 1 deg min-’ to 7.0 f 0.1 kcal/mol at 21 deg min-I. Table I relates these activation parameters and the glass transition temperatures for water at 1, 5, 10, and 20 deg min-l heating rate. Table PI lists those preThe Journal of Physical Chemistry, ‘Vol. 76, No, 7, 1971
DONH. RASMUSSEN AND ALANP. MACKENZIE
972
100
Table I : Activation Parameters and Glass Transition Temperature for Water as a Function of Heating Rate (AH
4
AN*,
AS*, AF*,
kcal/ mol
oal/ koal/ mol OIh mol
Sugisaki et el.
-
40
“K
TgVi
1/Tg
133.8 136.5 138.4 140.3
17,902 3726 1915 984.2
7.47 X 10-3 7.33 7.23 7.13
20
1 5 10 20
d7
19.2
7.55 7.33 7,16
7.0
-.----
0
AF)/A# = T,,
-
(1
21U-
-x \
I
9 8 -
7G 54 -
Table I1 : Kinetic Pammeters for the Glass Transition of Water as Determined from Results of Other Investigators
Ailgel1
d, Researoh
