CRYOBIOLOGY

27, 378-400 (1990)

Experimental

Dissection of Devitrification of 1,3-Butanediol’

in Aqueous Solutions

PATRICK M. MEHL American

Red Cross Transplantation Laboratory, Jerome Holland Laboratory for the Biomedical 15601 Crabbs Branch Way, Rockville, Maryland 20855

Sciences,

Devitrification is a major problem which must be overcome for successful organ cryopreservation. Devitrification can be initiated on fracture planes and on bubbles, but the focus of attention here is on devitrification by ordinary heterogeneous and homogeneous mechanisms, which are the most relevant for organ preservation by vitrification. The purpose of the present studies was to define the devitrification process: to determine nucleation rates, ice-crystal growth rates, and the distribution of icecrystal size and to evaluate the applicability of existing quantitative models of these processes which have successfully approximated the behavior of other aqueous systems. The present work was done using differential scanning calorimetry and cryomicroscopy. The amount of ice formed has been estimated for highly concentrated solutions. Kinetic parameters are presented here for isothermal conditions and continuous heating rate experiments. The classical theory based on the JohnsonAvrami equation has been evaluated and the results are compared with the theory of Boutron. The agreement is good for the continuous heating rate conditions, but results differ for the isothermal conditions. 0 1990 Academic Press, Inc.

Cryopreservation of organs by vitrification has been our goal for a number of years (18-21). Although a number of small systems can now be successfully vitrified, vitrification of organs has not yet been achieved. Small systems but not organs can be cooled and warmed quickly by conduction, assuring both vitrification on cooling and the prevention or near prevention of devitrification on warming. An important goal for organ cryopreservation by vitrification is to understand devitrification: how does it occur, how can it be prevented, and in what circumstances might it be innocuous? An understanding of devitrification will help to set engineering goals for electromagnetic heating of organs which may help to close the gap in success presently distinguishing organs from smaller systems. Present models of devitrification are seReceived June 9, 1989; accepted January 30, 1990. ’ Presented at the Symposium on Vitrification at the 26th Annual Meeting of the Society for Cryobiology, Charleston, South Carolina, June 1989. 378 Ool l -2240/90 $3 .OO Cow’ight All rights

0 1990 by Academic Press, Inc. of reproduction in any form reserved.

verely limited by deficiencies in knowledge of the underlying elementary processes of nucleation rate and ice-crystal growth. Until these elementary processes are well defined, it will be difl?cult to interpret the relationship between devitrification and bioIogical injury and to predict accurately the critical warming rate for suppression of devitrification. What is the cause of the damage occurring inside organs when devitrification occurs? Takahashi has calculated that cubic ice is thermodynamically unstable for grain sizes in excess of 200 8, (46), and this has been confirmed by observations of Vigier, Vassoille, and co-workers for glycerol and 1,2-propanediol aqueous solutions (48, 49). Boutron and co-workers have shown that it is sufficient to avoid this transition from cubic into hexagonal ice during warming in order to avoid lethal damage in biological systems (3-6, 12, 13, 40, 41). Thus the size of the crystals might determine the amount of damage produced by devitrification. Takahashi et al. have observed a correlation

DEVITRIFICATION

IN

1,3-BUTANEDIOL

between the intracellular ice-crystal size and the death of cryopreserved monocytes (47). The size limit for innocuousness seems to be around 300 A, which is larger than the limiting size of around 200 8. for cubic ice crystals. Clearly, a more detailed understanding of ice crystallization would facilitate cryopreservation of organs by vitrification. This study has been done on aqueous solutions of 1,3-butanediol, which has been shown to be a good glass former and also a good cryoprotectant for red blood cells used as a biological model (7, 13, 14, 3840). The objects were to define devitrification using a combination of cryomicroscopy and differential scanning calorimetry and to evaluate existing models of devitrification. Studies done three decades ago showed the usefulness of direct observation by cryomicroscopy of ice-crystal patterns and growth (28, 29, 35, 36, 43, 44) but did not provide mathematical descriptions of these phenomena.

SOLUTIONS

379

tions of devitrification are described, a general orientation may be helpful. The following qualitative description of phenomena observed will serve as a general background for the quantitative data which follow. Examples of one kind of result obtained are compared in Figs. 1 and 2, which show events during warming for samples of 44% 1,3-butanediol in deionized water. On the thermogram of Fig. 1A are superimposed the results of cryomicroscopy. Crack formation occurs during cooling or after cooling during the relaxation of the medium (Fig. 2a). During the subsequent warming (Fig. 2b) the cracks begin to disappear slowly or suddenly around (1) on Fig. lA, at temperatures which are close to the beginning of the glass transition. This disappearance ends at higher temperatures (2) after the glass transition is achieved, Later at (3) in Fig. lA, all the shapes corresponding to the previous cracks reappear (Fig. 2~). As the temperature increases, the crystallization for 44% 1,3-butanediol aqueous soluMETHODS tion at (4) in Fig. 1A begins as homogeneous nucleation visible in Fig. 2c. The Calorimetric data were obtained using a temperature corresponding to (4) in Fig. 1A Perkin-Elmer DSC-4. Cryomicroscopy was seems to coincide with the beginning of the done with a Zeiss Jena Pereval “interdevitrification peak on the thermogram, phaco” Interference Microscope with a The crystallization seems to stop at (5). Linkam thermal control from MicroDeAfter the crystallization visually seems vices. Calorimeter sample weights were becomplete, the medium becomes homogetween 5 and 15 mg, and cryomicroscopy neous around (6), when the medium begins sample total volumes were around 20 ~1. to turn darker and the fracture ghosts begin Cryomicroscope samples were generally to disappear (Fig. 2d). This darkening effect quenched directly in liquid nitrogen and appears to correspond to a weak exotherplaced inside the cryomiscroscope prema1peak measured by calorimetry between cooled at - 140°C. Calorimetry samples (6) and (7) in Fig. lA, which may represent were cooled at a rate between 80 and 3OOW a secondary crystallization. Figure 2e cormin to - 140°C as usual (3-14, 3Wl). The responds to the darkening event, (The piccryomicroscopic observations were reture brightness is due to overexposure corded on videotape. Other details will be caused by the automatic setup for the time described below. of exposure.) Bubbles are very weakly observable in Fig. 2e due to the overexposure. QUALITATIVE OBSERVATIONS At higher temperatures after the darkenBefore the detailed quantitative observa- ing event is completed, the crystals begin to

380

PATRICK

M.

MEHL

/ Td -120

-100

-a0 -64 TEHPERATURE

-40 i ‘c i

1 -20

L. 0

FIG. I, Comparison of events occurring during warming at lO”C/min as determined by cryomicroscopy and DSC-4 thermograms. Samples consist of aqueous solutions of 1,3-butanediol. For cryomicroscopy, the samples in A-C have been quenched directly in liquid nitrogen, placed in the cryomicroscope at ~ l4O”C, and warmed back. For the thermograms the samples were quenched at the maximum rate of the DSC4 (around 100”Cimin) and warmed back. For D, the sample in the cryomicroscope was first cooled at 4OWmin and warmed back or the sample was quenched directly in liquid nitrogen as for A-C. Observations: (1) beginning of the disappearance of cracks by relaxation, (2) apparent end of the previous disappearance of cracks, (3) reappearance of crack shapes with eventual bubble formation, (4) beginning of the crystallization, (5) apparent end of the crystallization, (6) beginning of the darkening effect, (7) apparent end of the darkening.

melt with the appearance of a large number of small crystal granules (Fig. 20. As the temperature increases, the crystals melt and begin to move convectively (Fig. 2g; convection is not demonstrable in this singie photograph). For concentrations higher than 44% 1,3butanedio1, fundamentally different behavior is seen. The most important difference is that nucleation density becomes Iow enough to permit analysis of individual crystals (44% is a critical concentration at which observations of different samples show either homogeneous nucleation or a “low-density” nucleation). For 46% 1,3butanediol, there is still agreement between the calorimetric and the microscopic results, but for higher concentrations, the be-

havior changes: the cryomicroscopically visualized phase transitions occur at lower temperatures than are seen by calorimetry (Figs. 1B and C). However, this discrepency is due to fracturing of the cryomicroscopic samples, which were cooled to lower temperatures than the DSC samples. Less discrepancy is visible when the DSC samples are also fractured for 48% I,3-butanediol as shown in Fig. ID. Figures 3a to e show a nonfractured sample of 48% 1,3-butanediol cooled at 40Wmin and warmed at lO”C/min. Figures 3f to k show the same sample quenched in liquid nitrogen then warmed back at the same rate. The latter observations are reported in Fig. 1D with good agreement with the corresponding thermo-

DEVITRiFlCATION

IN 1,3-BUTANEDIOL

SOLUTIONS

381

b

Q

d FIG. 2. Forty-four percent 1,3-butanediol quenched in liquid nitrogen, placed in the cryomicroscope at - 14O”C,and warmed at lO”C/min. (a) - 128”C, (b) -96”C, (c) - 84”C, (d) -4YC, (e) -4O”C, (f) -3S”C, (g) -21°C. The scale bar represents 50 pm after corrections and is the same for all photographs. the field in d is actually dark but was printed overexposed.

grams on warming. It is evident that ice nucleation in quenched samples of 48% 1,3butanediol is mainly “heterogeneous” and occurs at fracture defects inside the sam-

ple. For the lower concentrations fracturing is less important in determining Td because the temperature range for crystal growth is closer to that for nucleation and because

382

PATRICK

M. MEHt

FLG. 3. Forty-eight percent 1,3-butanediol cooled at WC/min to - 135°Cin the cryomicroscope and warmed at IOWmin: (a) -6l”C, (b) -57”C, (c) -53”C, (d) -4YC, (e) -27°C; or quenched in liquid nitrogen and placed in the cryomicroscope at - 1WC, warmed at IOWmin: (Q - 12O“C,(9) -97T, (h) -79”C, (i) -6l”C, (j) -5l”C, (k) -35°C. Same scale as for Fig. 2.

the nucleation density is high enough to mask heterogeneous nucleation on fractures. The quantitative measurements which

follow all pertain to devitrification which is not mediated by visible fractures. DSC, as above, was performed on nonfractured samples. Cryomicroscopy was, as above,

DEVITRIFICATION

IN 1,3-BUTANEDIOL

SOLUTIONS

383

FIG. 34 Tonhued

performed on fractured samples, but all observations were confined to nonfractured regions of these samples. QUANTITATIVE

MEASUREMENTS

Direct observations by cryomicroscopy

of the volume density of nuclei and of raRes of crystal growth have been done on tlhinlayer samples (thickness around 0.1 nnm, sample volume of 20 t.~l)for concentrati ons of 46% (w/w) and above. The general w pearance of quantitatively analyzed s:amples is illustrated in Figs. 4a to i for the c:ase

384

PATRICK M. MEHL

FIG. 4. Forty-six percent 1,3-butanediol quench[ed in liquid nitrogen, placed in the cryomicroscope at - 14O”C,and then either warmed at 10”Cimin and photographed at a series of temperatures [(a) -139°C (b) -9l”C, (c) -5o”C, (d) -WC, (e) -28”C] or warmed at SO”C/min to -78.5”C and photographed at a series of times, I (isothermal CIrystallization [(f) I min, (g) 2 min, (h) 5 min, (i) IO mm]. (I does not take into account the induction Itime for nucleation.) Same scale as for Fig. 2.

DEVITRIFICATION

IN 1,3-BUTANEDIOL

SOLUTIONS

h

FIG. 4-Conrinued

of 46% 1,3-butanediol. The ice crystals were found to have the same size and to grow at the same rate. This was true for both isothermal and continuous heating rate (CHR) experiments. The striking fact

that all growing crystals have the same diameter supports the hypothesis that nucleation occurs at lower temperatures than crystal growth. In fact, the number of visible crystals does not change during heating,

386

PATRICK

proving that all the observable devitrification is due to growth of preexisting nuclei. It also strongly suggests that all nuclei formed are statistically free to grow on warming and, therefore, that the number of nuclei observed is equal to the number of critical nuclei formed during the initial cooling and the subsequent warming. The results with respect to nucleation density are reported in Fig. 5, where the density is expressed in nuclei per 0.02 mm3 versus the warming rate v on a logarithmic scale. Figure 5 shows that the dependence of nucleation density on v seems not to be a simple power law for all the samples. MacFarlane referred to a law giving a proportionality relation between the density of nuclei N and the warming rate Y, when Y is raised to the power 11and a equals - 1, 0, or 1 (34). For 46% 1,3-butanediol in water, as shown in Fig. 5, this exponent varies continuously from 0.5 to I.7 for the highest warming rates used. For the lowest concentrations the density of nuclei was too high to be calculated or estimated (Fig. 2~). The count could be done for concentrations higher than 44% in the case of isothermal conditions and higher than 46% for CHR conditions. For concentrations higher than 48%, the number of nuclei per unit volume decreases quickly. Figures 4a-e and 4f-i show respectively the ice-crystal growth under isothermal and CHR conditions for samples of 46% 1,3butanediol. The radii of the crystals were measured to determine the growth rate and the Avrami exponent. Figure 6 shows the growth of the crystals in 44% 1,3-butanediol versus time. The assumption that the crystals are full spherulites (i.e., are spherical) is supported by the observations of Figs. 4f to i on crystal shape, although it must be noted that some of the crystals look discoidal. Many additional quantitative observations have been made by both DSC and cryomicrosctopy. The results are best pre-

M.

MEHL

sented, however, in the light of theories permitting the data to be analyzed mathematically. THEORIES

OF ICE

GROWTH

The following theories have been used as bases for the analysis of the present experimental results as shown in the next section. Both the classical Johnson-Avrami theory (16) and Boutron’s theory have been employed in the analysis of both isothermaI and CHR data. The former theory was adapted to CHR conditions as described below. Isothermal crystallization: Johnson-Avrami Equation

Christian has reviewed the development of the classical Johnson-Avrami (JA) equation relating the crystallization fraction X and the time of crystallization t for an isothermal experiment at temperature T (16)) X = 1 - exp[ - (Kr)“],

[II

where n is a constant called the Avrami exponent and K is a kinetic parameter which depends on temperature, K = K,exp[ - E*/RB],

PI

with 9 = T for Arrhenius behavior or 0 = T - TO for Vogel-Fulcher kinetics, depending on the material or the glass (1). E* is a crystallization activation energy and K, is a kinetic constant depending on the composition of the glass former and also on the nature of the nucleation process. TO is a temperature, also called the “ideal glass transition temperature,” below the observed glass transition temperature Tg which corresponds to the temperature at which the viscosity of the medium is infinite. For what follows, the Arrhenius form has been chosen as a simplification. From the JA equation [l], as T = constant for the isothermal experiments, by differentiating the fraction X two times versus f and considering that the maximum crystallization rate (as measured by the

DEVITRIFKATION

IN

1,3-BUTANEDIOL

I

‘ioo

50

ICI 5 WARMING RATE (“C/MIN )

SOLUTIONS

387

'IO0 1

FIG. 5. Nucleation density for aqueous solutions containing 46, 47, 48, or 50% 1,3-butanediol quenched directly in liquid nitrogen and placed at ~ 140°Cin the cryomicroscope, versus the warming rate. The count of the nuclei has been done in areas without apparent cracks. A mean value is obtained by counting the crystals in five or more different places within the same sample recorded by video. The volume of the field is determined by the magnification of the optics, the total volume of the sample, and the surface of the coverglass slides. For the highest concentrations, the count is limited by the size of the sample and the presence of the cracks and also by the drastic decrease in the number of nuclei.

peak of the devitrification thermogram on DSC) is characterized by d2X/dtZ = 0, this particular condition gives (mm,,)” = (n - 1)/n

r31

or log KO - EVRT + log t,, = log( 1 - l/n)/n.

[4]

A plot of log t,,, versus l/T must then give a straight line with slope E*/R. The determination of the Avrami exponent n

is obtained from [l] with a plot log( - log( 1 - X)) versus log t: log(-log(l

of

- x)) = n[log K + log t]. [5]

JA Kinetics: Crystallization at Continuous Heating Rates The last decade has seen many studies on the determination of crystallization kinetics on cooling or on warming from a glassy state by trying to generalize the JA equation. Different authors (25, 26, 50) have reviewed the calculation methods for CHR

388

PATRICK M. MEHL

44%

1,3-8UTANEDIOL

t

ISOTHERM:

2

T,-84*C

-13O”C-

T

AT 50”C/MIN

P ,O’ ,B/ IQ ,p/p $,O.@ .P ,P’ $ p,/g’ ./I ,0’ f T=-62°C

Tr97”C

FIG. 6. Isothermal crystallization in 44% 1,3-butanediol by cryomicroscopy at different temperatures. The radius R in micrometers of the observed spherulite crystals is plotted versus the time Ai of exposure at different temperatures. Both R and At are reported on a logarithmic scale. The induction time for nucleation is taken into account in At. Reproduced, with permission from the publisher, from Ref. (38).

conditions given different assumptions. MacFarlane and co-workers have reviewed the nonisothermal kinetics for aqueous solutions (30-32, 34) on cooling and during the warming of a glass as suggested by Christian (16) and have made an important contribution by providing a basis for understanding ice crystallization in aqueous vitrification solutions (30-32) and a method for the calculation of critical warming rates for organ cryopreservation by vitrification (3&32). The Ozawa method (25) allows the Avrami exponent to be determined under CHR conditions from a plot of log( - log( 1 - X)) versus log((T - T,)lv) where v is the warming rate and T, the temperature at the beginning of the crystallization. The activation energy E* can be determined in several ways using in common the isoconversional method (23). One of these methods, called the Ozawa-Chen method, is presented here. It is close to the method

used by Flynn (23). The Ozawa-Chen method considers a set of experiments for which X can be set constant for a set of paired values of T and V. The result is (24, 50) log(&R/E*)

+ log(T*/v) - E*IRT = constant.

[61

A plot of log(T%) versus l/T for a set of experiments at a fixed crystallization fraction X gives a straight line with a slope of E*/R. All these plots have already been reported in (38). Relating Direct Optical Measurement of Crystal Growth to the Avrami Exponent Calculation of the Avrami exponent can be done for the particular case of full spherulites. Three assumptions have been made for the present analysis: (1) the nuclei form inside the bulk solution; (2) the growth is isotropic; (3) the number of growing nuclei is constant. These assumptions were

DEVITRIFICATION

IN

1,3-BUTANEDIOL

checked and appeared to be valid. The following calculations are a simplification of those of MacFarlane et al. (33). Let d be the diameter of the crystals after a time At which corresponds to the time after which the crystal becomes visible; the time for the induction of nucleation is then obviously neglected. A linear relationship between the logarithm of the diameter and the logarithm of At gives d = d,(At)”

[71

where d, is a parameter with a dimension of length divided by time to the power m. Then the volume V of the spherical crystal versus time will be V(At) = ,rrd03(At)3”/6.

Wl

We now can make use of the fact that all crystals reach visible size at the same time, as indicated by the fact that ali crystals are observed to have the same shape and diameter. Then for a number N of nuclei growing as crystals, the crystallization fraction is as defined by Christian (I 6) as

389

SOLUTIONS

with 3m when growth is in three dimension. Similarly, for an apparent growth in two dimensions (disks), the exponent will be n = 2m, and for growth of a needle, n = m. This assignment is supported by the fact that a limited first-order expansion of [IO] gives Eq. [ll]. Model of Boutron:

Isothermal

Conditions

Boutron’s initial equation is (3) dX/dt = k, (1 - X)X2’3(Tm - T)

exp( - Q/R2J. For isothermal crystallization, of [I21 gives

[121

integration

- log( 1 - rr) + (log(1 + u + U2))/2 + 31’*arctg((3’W)/(2 + U)) - tk,(T - T,)exp(-QIRZ”‘), (131

where U = X”3. A plot of the left side of [I31 versus time should give a straight line with a slope of k,(T - T,Jexp( - Q/N) if T is constant. A secondary plot of the logarithm of the slope minus the logarithm of (T X(Ar) = 1 - exp[ -(NV(At)lV,l, 191 - T,,,) versus l/T should also give a straight line with a slope of - Q/R. where V, is the crystallizable volume of the The model of Boutron for isothermal exsolution. By replacing V(At) by its expres- periments assumes that at the maximum sion IS], one has crystallization rate (d%/dt2 = 0), the crystallization fraction X, is constant. DifferenX(At) = 1 -- exp[ -(N,ird,,3/6V0)(At)3m], tiating Eq. [12] gives [lOI q

Equation [lo] is similar to the JA equation with an exponent of 3m. Equation [lo] also takes into account overlapping of the growing crystals. For our practical measurements, only the beginning of the crystallization was analyzed, when observable crystals are not in contact with each other. For this case, the crystallization fraction is only determined by the ratio of the volume of ice crystallized to V,,:

d2X/dt2 = {k,(T - T,)exp( - Q/R7’)}2 (1 - X)X2’3 d{(l - X)X2’3}/dX

[I41 from which it follows that d{(l - X)X2’3}/dA’ = 0,

1151

with X = X,. This leads to X, = 2/5 for the ice crystallization fraction at the maximum crystallization rate. Experimental data are generally near if not identical to this value X(Al) = (N~rd,~/6V,) (At)3m. IllI (12, 39). The approximation that X, is constant will be shown to be necessary for exWe will assume that, as has been demonperimental determination of the activation strated in the case of Eq. [lo], the Avrami energy Q as shown in the next section. exponent n can be identified in Eq. [I 11

390 Model of Boutron:

PATRICK

CHR Condihvts

CHR conditions for Eq. [12] have also been considered by Boutron and coworkers (3-5, 7-14) and have been shown to be modeled well (3, 9) using a limited expansion of the exponential function in [12] to allow the integration of [12] with T. This method also permits determination of the activation energy Q under CHR conditions. A recent improvement of Boutron’s theory has been to eliminate the use of a finite expression (12, 39). The application of the results to analysis of CHR experiments give

M.

MEHL

Knowing the approximate value of Q obviously allows the calculation of k, directly from [16]. COMPARISON OF THEORY AND EXPERIMENT

Classical Approach:

Isothermal

Data

The plots of log( - log( 1 - X)) versus log t and of log t,, versus l/T for determination of the Avrami exponent n and the activation energy E* under isothermal conditions are respectively reported in Fig. 7 for 44% 1,3-butanediol and Fig. 8 for all concentrations studied. The values of n and E* v(QT,/R - QTJR - Td2)= for all the concentrations were calculated Wc~(Trn - Td)12exp( - Q/R&) from these plots by least-squares linear re(5X, - 2)/3xd1’3 Ml gression with a correlation coefficient where Td is the temperature at the peak higher than 0.995 and are reported in Tacrystallization rate, k, is a constant, Q is ble 1. The Avrami exponent n as determined analogous to an activation energy, and the other parameters are as previously defined. calorimetrically is equal to 3 for 42 and 44% Equation [16] was solved for kr and Q by solutions, which is in agreement with the the following process. Initially, k, was as- observations by cryomicroscopy of spherisumed to be constant for a given concen- cal crystals and with the value of 2.75 caltration Then if [16] for one experiment culated from these observations for 44%. (one set of values for V, Td, and X,) is di- The Avrami exponent can be analyzed usvided by [16] for a second experiment (a ing a formula (17) derived from a review by second set of V, Td, and X,), k, is elimi- Christian of metallic glasses, n = a + bp, nated and the equation can be solved for Q+ where a varies from 0 for a constant numUnfortunately, when all possible pairs ({V, ber of crystals to 1 for a constant nucleation T,, Xd},, {V, Td, X& were considered, in- rate, b is the dimension of the growth, and consistent values of Q were obtained, and p is equal to 1 for interface-controlled sometimes no value of Q existed which growth or to 0.5 for diffusion-controlled growth. According to this formula, our data could satisfy this equation. To evaluate the theory, it was therefore giving IZ = 2.75 to 3 corresponds to a connecessary to assume that the problem was stant ice-crystal number during threedue to imprecision of the X, data, i.e., that, dimensional interface-controlled crystal as assumed explicitly during the formula- growth (16). tion of the theory, X, is actually a constant. For concentrations lower than 42% the To determine the value of X, to use, exper- calorimetrically defined Avrami exponent n imental X, values were averaged to obtain a is around 2.5. This might be due to a higher mean value which will be referred to as X,,. nucleation density and an overlapping beWhen this was done for the present data tween nucleation and crystal growth. These set, the mean value of Q had a relative vari- results are similar to those of MacFarlane ance of less than 3%, showing that an inter- and Forsyth on 40% 1,2-propanediol (31) nally consistent value for Q can be ob- with an Avrami exponent of 2.6, intertained , preted as diffusion-controlled growth with a

DEVITRIFICATION

IN I ,3-BUTANEDIOL

196 K

391

SOLUTIONS

192 K

188 K ./ .’

194 K -4

19.0 K, / - 1- ,..,L 0.5

1,3-EWT’ANEDIOL 1 5

1

10

TIME (MIN) FIG. 7. Johnson-Avrami plot of the log( -log( I - I’)) versus time in minutes reported in a logarithmic scale for 44% 1,3-butanediol under isothermal conditions at different temperatures. X is the crystallization fraction, which is assumed to be equal to the ratio of the heat evolved up to the time t divided by the total heat evolved when crystallization has been completed. It has been assumed that the temperature dependence of the crystallization heat in the thermal range used is negligible as shown by Mikhalev et al. (42). An Arrhenius dependence on temperature is also assumed for the crystallization constant defined for the Johnson-Avrami theory. Although the viscosity of aqueous solutions of polyalcohols has been shown to follow a Vogel-Fulcher law for temperatures higher than that of the glass transition (I), the Arrhenius approximation is made to lower the number of kinetic parameters and to simplify the calculations. This appears justified at the temperatures of interest here, which are considerably above Tg.

constant nucleation rate, The same process can be considered for the case of 38 and 40% 1,3-butanediol for which the early stage of crystallization might be a threedimensional diffusion-controlled process. The microscopic observation of planar growth for 48% 1,3-butanediol is in good agreement with the calorimetrically derived value of the Avrami exponent calculated for the isothermal experiments at temperatures between -70 and - 50°C and reported in Table 1, i.e., IZ = 2.09. This exponent is very close to that of the theory for interface-controlled crystallization with a growth dimension of 2 (16). As will be shown below, optical measurements con-

firm that crystal growth is slower at the beginning before reaching the rate defining the Avrami exponent. The Avrami exponent was also calculated from the ice-crystal growth rates observed microscopically and reported earlier in Fig. 6. The exponent n was calculated assuming three-dimensional growth, These direct observations are in good agreement with the previous analysis of the calorimetric measurements, especially for isothermal conditions with 44 and 46% 1,3-butanediol and also for 48% 1,3-butanediol with twodimensional growth. Log K,, where K,, is in units of set - I, has been determined for 38, 40,42, 44, 46, and

392

PATRICK M. MEHL

that the crystals are actually growing in two dimensions and that the Avrami exponent is therefore really two-thirds of its tabulated value of 3.82. For all concentrations, the calculations from calorimetry for the CHR conditions give lower values for the Avrami exponent than were found for isothermal experiments, Surprisingly, the calorimetrically derived Avrami exponent increases with concentration. This behavior of the Avrami exponent has also been reported elsewhere (38). For 48% 1,3-butanediol, planar crystals were observed, In Fig. 3b, two crystals are shown growing superimposed on each other and then fusing, As noted also for 46% 1,3-butanediol in Fig. 4d, above a critical radius the spherulites show a dendritic growth which can be called an evanescent form. In Table 1 the optical measure of the Avrami exponent is not reported for 48% TIME OF MAXIMUM RATE 1,3-butanediol. This is because crystalliza(MINI tion of this solution seems to occur in two FIG. 8. Plot of 1000/T versus time at the maximum crystallization rate for different concentrations of 1,3- steps at the highest temperatures under the butanediol under isothermal conditions. The points isothermal conditions and at the highest correspond to experimental data. warming rates under the CHR conditions. The crystals seem to grow as disks until, at 48% 1,3-butanediol. The values are, respec- a certain limiting size, the growth pattern tively, 21, 19.4, 18,3, 17, 11.6, and 8.6. The switches to a spicular morphology. In the accelerating change in K, with increasing discoidal form, which is similar to the geconcentration is similar to the effect of con- ometry of macroscopic hexagonal ice cryscentration on activation energy and is ob- tals, the stability of the interface is reduced viously related to the drastic effect of con- as the concentration at the interface increases, giving rise to rapid growth in the centration on the nucleation process. form of spicules. The angle between two adjacent needles stays constant as deterClassical Approach: CHR Data mined by the inherent symmetry of the ice Now let us consider the results for CHR crystal. this behavior is reflected in Fig. 9: conditions. In Fig. 4a to e, the crystals the slopes of the curves have very high valseem to be between spheres and disks for ues, showing a more complex process for 46% 1,3-butanediol. In fact most of the crystallization than constant growth in one, crystals look discoidal when they overlap. two, or three dimensions, For the CHR conditions, cryomicroscopy For concentrations higher than 48%, nugives a very high value for IZ when three- cleation is lower and the crystals grow didimensional growth is assumed, suggesting rectly in the dendritic shape (data not

DEVITRIFICATION

IN 1,3-BUTANEDIOL TABLE

393

SOLUTIONS

1

Parameters of Ice Crystallization Kinetics for Aqueous Solutions of 1,3-Butanediol in the Classical Model or in the Model of Boutron Classical method

Boutron method t

CHR

Isothermal(I)

Microscopy CHR

I

CHR

% wlw

E*

It

Ef

?t

Q

k,

Q

h

n

n

38

16.9 16.3 16.0 15.6 -

2.59 (0.26) 2.66 (0.24) 3.04 (0.35) 3.02 (0.W 2.63 (0.16) 2.09 (0.10)

15.4 VW 14.5

1.77 (0.22) 1.79

13.4

3.1 El5

16.7

2.4 EL0

N.O.”

N.O.

18.1

5.4 no

15.6

1.2 El8

N.O.

N.O.

03.2)

(0.07)

12.7

2.08 (0.15) 2.14 (0.07) 2.42 (0.14)

20.3

2.7 E22

13.8

1.1 El5

N.O.

N.O.

17.6

3.5 El8

12.5

9.2 El2

N.O.

14.9

1.2 El4

10.9

2.4 El0

b

14.2

7.1 El2

b

b

2.75 (0.34) 2.52 (0.W c

40 42 44 46

11.4

48

12.21 -

(0.2) 11.0

uw 9.7 to.1) b

3.82 (0.07) c

Note. The activation energies E* and Q are expressed in kcaUmol and the constant k, in (K . min)-‘. (El5 = 10”). The values are the means of values calculated by least-squares linear regression; the values of the variances are given in parentheses. The variance does not take into account the error of determining the correct baseline for the devitrification peaks. a Number of growing crystals is too large to measure or the rate of crystal growth is too high to measure. b The values for 48% 1,3-butanediol have not been determined yet because of the overlap between the devitrification peak and the melting peak for warming rates higher than 6”Clrnin. ’ For determination of the value IL, see Fig. 9.

ported in Table I. The apparent activation energy increases with concentration up to 42%, at which concentration it is 20.3 kcal/ mol, then decreases for the higher concenTheory of Boufron trations. The breakdown of monotonic paThe same data have also been analyzed rameter variation at 42% 1,3-butanediol with the theory developed by Boutron. For might be due to a change in the mechanism isothermal conditions, the plots of A(X) of crystallization and is consistent with the (representing the value of the left term of cryomicroscopic observations. The kinetic Eq. [13]) versus time are reported in Fig. IO constant k, determined as independent of for different concentrations of 1,3-bu- the temperature varies in a pattern very tanediol. In this figure, contrary to the the- similar to that of Q, peaking at 42% 1,3ory, the dependence of A(X) on t is not lin- butanediol. k, is in principle highly related ear. The areas of greatest linearity were an- to the nucleation density, based on the aralyzed nevertheless by deriving average gument of MacFarlane (30). Results for CHR conditions are reported slopes of the A(X) curve over crystallizain Table 1 and in Fig. 12. This last figure tion fractions taken between 0.05 and 0.81. From the average slopes P the activa- shows that it is possible to obtain good tion energies Q were determined from the agreement between the experimental points slopes of the curves of log(P/(T, - T)) ver- and the theory when an average value of Q sus lOOO/Tas drawn in Fig. 11 and are re- is used as described above. The resulting shown). The study of these higher concentrations will be presented in a subsequent paper.

394

PATRICK M. MEHL

I

50-

48 % ‘lJ-BUTANEDIOL (11 CHR

5A z Y 2

nti51 S (2 1 I&THERMAL

1

-60.7T

I. .##I 5 10

At(S) 1 I . ..I 50 100

FIG. 9. CHR and isothermal crystallization in 48% 1,3-butanediol by cryomicroscopy at different warming rates and at different temperatures, respectively. As in Fig. 8, the radius of the crystals is plotted versus the time of growth. The slope P of each line is noted next to the line. Other conditions and dimensions as in Fig. 8.

kinetic constant kI is also reported in Table I and again varies in a manner qualitatively similar to that of Q. The results are similar to those published for similar concentrations (13, 14). X,, values used were respectively 0.45, 0.492,0.524,0.538, and 0.576 for 38,40,42, 44, and 46% 1,3-butanediol. These values increase with the concentration of 1,3butanediol and diverge from the theoretically predicted value 2/ under isothermal conditions.

DISCUSSION

Boutron and co-workers originally defined the stabirity of the amorphous state during warming in terms of a theoretical warming rate above which ice does not have time to form. Crystal growth is implicitly assumed to be the limiting step of the crystallization. Forsyth and MacFarlane have shown the role of nucleation on devitrification kinetics, which are increased if the nucleation density is artificially in-

DEVITRIFICATION

- t

I ct

172

if

IN 1.3-BUTANEDIOL

i

t

012345

lay 1.1 a6,484

0123456

rnn

:

178

SOLUTIONS

395

/ I76 /

0123456789 182 ,i

I

196.

216 2’4 717i ,i212

,.

0123456012345 -TIME

( MIN I+

FIG. 10. Isothermal crystallization in different concentrations of 1,3-butanediol. The function A(X) from the model of Boutron of the crystallization fraction X is plotted versus the time of crystallization for different temperatures indicated next to the curves in degrees Kelvin. Points are experimental data; curves were drawn through the experimental points.

creased by incubation above the glass transition temperature (24): the higher the density of nuclei, the lower the devitrifkation temperature and the larger the amount of crystallization (24). Despite the dependence of the devitrification temperature on the density of nuclei, the theory of Boutron (3) gives good approximate values of the critical warming rate vCrwhen the experimental points used for the extrapolation are close to the point corresponding to the critical warming rate and the temperature of melting. This estimation of the critical warming rate v,, has been compared to the calculated critical warming rate derived from the classical theory by MacFarlane, which uses either an Arrhenius or a Vogel-Fulcher approximation for the temperature dependence of intensive physical parameters such as the vis-

cosity and assumes diffusion-controlled crystal growth at the interface (3&32). The difficulty of experimentally testing MacFarlane’s approach arises from the need to determine the real physical parameters that govern the crystallization process such as the self-diffusion coeffkient of water molecules or TO,for example. Boutron’s model until recently gave higher values for the critical warming rate v,, than MacFarlane theory (12, 39), but recent work has brought the predictions of the two theories closer together (12). A simplification of the classical method gives a close estimate of the critical warming rate using a linear extrapolation of l/T, versus log(v) to the hypothetic limit Td = T,, a result which is in agreement with the predictions of both Boutron and Mehl (12) and MacFarlane (30).

396

PATRICK

-2-

\

44%

i

'1 '\ -3 - 'lie '\

M. MEHL

i

'\ '\ -4- \ i j.46% '\ -5 48% -4 FIG. 11. Isothermal crystallization

in different concentrations of 1,3-butanediol. The function log (P/T,- T)) is plotted versus 1000/T for different temperatures T, where P is the slope of the corresponding curves in Fig. 10 and T,,, is the end of melting temperature. Points are experimental data.

The approach in the present paper is to assumption, i.e., a constant number of nudetermine the exact roles of nucleation and clei growing in a spherical shape at a concrystal growth by direct observation. Both stant rate. Boutron’s data have shown sevclassical theory and Boutron theory give eral cases in which these assumptions are similar results at least for the activation en- valid (3), but the present results show severgies (respectively E* and Q), except for eral cases in which the assumptions break the isothermal experiments. The general down. In fact, the theory of Boutron does behavior is similar despite the different not fit any of the isothermal conditions meaning (31) of the activation energies E* studies here, which is the simplest case and Q. upon which all CHR calculations of BouThe theory of Boutron depends on sev- tron theory are based. eral assumptions (3). One of them, a conE* and Q show in all isothermal experistant nucleation density, is easily checked ments a decrease as the concentration inin the case of the highest concentrations creases. It is not surprising to observe a and appears to be valid as shown here. As slow decrease of the activation energy with pointed out by MacFarlane and Forsyth concentrations up to 42% since this corre(31), the theory of Boutron concerns inter- sponds to increasing temperatures at which face-controlled crystallization, which does the crystallization occurs (increasing Td). seem to occur at the highest concentrations The faster decrease of activation energy at according to the present data. In fact, the still higher concentrations corresponds to a model of Boutron is a particular case of the change in the nucleation process. Interestmodel developed by Sestak and Berggren ingly, both the classical method and that of (45) with two specific exponents as used by Boutron gave similar values for the activaMatusita et al. (37) for nonorganic glasses. tion energies under CHR conditions. The theory of Boutron is limited by its main But determination of the activation en-

DEVITRIFICATION

) WARMING

IN 1,3-BUTANEDIOL

397

SOLUTIONS

RATE

TEMPERATURE

OF

DEVITRIFICATION

Td ( K 1

FIG. 12. CHR crystallization for different concentrations of 1,Sbutanediol at different warming rates. The warming rates are plotted on a logarithmic scale versus the devitrification temperature T,. The points are experimental data measured by calorimetry, and the curves are the calculated curves

adjusted by the determination of activation energies Q so as to minimize the distance between these curves and the experimental points.

ergy is not sufficient for the assessment of the stability of the liquid state under isothermal conditions, The value of the constant KOdepends on the nucleation density, and this constant is therefore also a major parameter determining the stability of the formerly vitreous state. Unfortunately, detailed interpretation of this parameter is for the moment uncertain. It is possible that the techniques and data described here could be extended to address some of the remaining physical problems pertinent to cryopreservation of organs by vitrification, for example, nucleation near Tg. To avoid fracturing, it is necessary to maintain temperatures close to Tg in the immediate environment of the organ as the organ cools (22). However, this means lengthy exposure to temperatures at which nucleation might still be possible (22). A related problem is to know the

location of the glass transition: TB goes down as the cooling rate goes down. Because of this, nucleation might occur during storage below the usually measured Tg (27)) increasing the probability of damage during warming. As Koster suggested (27), under the glass transition the diffusion of water molecules might be sufftciently higher than that of the cryoprotectant to allow density fluctuations to develop inside the glassy matrix. This hypothesis must be evaluated. Nucleation below TB and a resulting enhancement of devitrification have already been observed by Angel1 and MacFarlane (2) in aqueous LiCl solutions and by Chang and Baust in water-glycerol solutions (15). Nucleation, whether homogeneous or heterogeneous, is a statistical phenomenon. This means that as the sample size increases, the total number of stable nuclei increases too, as documented recently by

398

PATRICK M. MEHL

Fahy et al. for aqueous I ,2-propanediol solutions (volumes of about 50 to 500 ml) (22). It is obvious that even the presence of a few nuclei in the sample and especially inside Iiving tissue may cause irreversible damage to the tissue. Therefore, it is important to know both crystal growth rate and nucleation rate for highly concentrated solutions which can be used to predict the effects of relatively rare events. The results described in the present paper represent a contribution to this end. CONCLUDING

REMARKS

The combined use of calorimetry and cryomicroscopy provides a powerful tool for the study of crystallization during warming from the glassy state. Moreover, it can also be a very good approach to the description and quantification of phenomena like crack formation and nucleation. Just as X-ray diffraction is useful for the determination of the nature and structure of phases formed at various transitions, the cryomicroscope is very useful for understanding macroscopic processes occurring during warming, with good correlation with calorimetric measurements. Electron microscopy is an intermediate technique which is complementary to cryomicroscopy and X-ray diffraction. Cryopreservation of organs by vitrification depends on many physical processes in addition to those discussed here, and many of these are currently under investigation. However, devitrification as described here can be regarded as the most obvious problem to be overcome. The present analysis shows that devitrification can be understood in detai1, and that the prediction of conditions for suppressing devitrification in organs should therefore be possible. The cryopreservation of organs by vitrification is a technique with many practical problems which are beginning to be solved one by one. Because of the need to satisfy both biological and physical constraints simultaneously, the formulation of increas-

ingly effective vitrification solutions, including, perhaps, polyol-based solutions, is likely to continue. The success of such mixtures wiI1 depend to a significant extent on their resistance to devitrification. It is hoped that the practical and theoretical observations reported here will help in the evaluation of such solutions and thereby facilitate the successfu1vitrification of organs as well as of simpler systems. ACKNOWLEDGMENTS The author thanks R. J. Williams and A. Hirsh for allowing the use of the necessary equipment, H. T. Meryman for his support, and G. M. Fahy for his support, editorial assistance, and scientitic advice. This work is supported by Grant BSRG 2 S07RRD5737and NIH Grant SROiGM-17959-15. REFERENCES 1. Angell, C. A. Strong and fragile liquids. 2r1“Relaxations in Complex Systems” (K. L. Ngai and G. B. Wright, Eds.), pp. 3-11. National Technical Information Service, U.S. Department of Commerce, Springfield, VA, 1988. 2. Angell, C. A., and MacFarlane, D. R. Conductimetric and calorimetric methods for the study of homogeneous nucleation and crystallization below both T,, and TB. Adv. Ceram. 4, 66-79 (1982). 3. Boutron, P. Comparison with the theory of the kinetics and extent of ice crystallization and of the glass-forming tendency in aqueous cryoprotective solutions. Cryobiobgy 23, 88402 (I986). 4. Boutron, P. Non-equilibrium formation of ice in aqueous solutions: Efficiency of polyalcohol solutions for vitrification. In “The Biophysics of Organ Cryopreservation” (D. E. Pegg and A. M. Karow, Jr., Eds.), pp. 201-228. Plenum, London, 1988. 5. Boutron, P. &vu- and dextro-2,3-butanediol and their racemic mixture: Very efficient solutes for vitrification. Cryobiology 25, 569 (1988). 6. Boutron, P., and Amaud, F. Comparison of the cryoprotection of red blood ceHs by 1,2-propanediol and glycerol. Cryobiology 21, 348-358 (1984). 7. Boutron, P., Delage, D., and Roustit, B. Stability of the amorphous state in the system water1.2-propanediol-1,3-propanediol. J. Chim. Phys. 77, 567-570 (1980).

8. Boutron, P., Delage, D., Roustit, B., and KSrber, C. Ternary systems with I,Zpropanediol: A

DEVITRIFICATION

IN I $-BUTANEDIOL

new gain in the stability of the amorphous state in the system water-l ,2-propanediol-l-propatrol. Cryobiology 19, 550-564 (1982). 9. Boutron, P., and Kaufmann, A. Stability of the amorphous state in the system water-glycerol-dimethylsulfoxide. Cryobiohgy 15, 93-103 (1978). 10. Boutron, P., and Kaufmann, A. Stability of the amorphous state in the system water-glycerol-ethylene glycol. Cryobiology 16, 83-89 (1979). 11, Boutron, P , and Kaufmarm, A. Stability of the amorphous state in the system water-1,2-propanediol. Cryobiology 16, 557-568 (1979). 12. Boutron, P., and Mehl, P. Theoretical prediction of devitrification tendency: Determination of critical warming rates without using Fnite expansions. Cryobiology 27, (1990). 13. Boutron, P., and Mehl, P. Non-equilibrium ice crystallization in aqueous solutions: Comparison with theory, case of solutions of polyalcohols with four carbons, ability to form glasses, compounds favoring cubic ice. J. Phys. Colloq. Cl, Suppl. 3 48, 44148 (1987). 14. Boutron, P., Mehl, P., Kaufmann, A., and Angibaud, I’. Glass-forming tendency and stability of the amorphous state in aqueous solutions of linear polyalcohols with four carbons. I. Binary systems water-polyalcohol. Cryobiology 23, 453-469 (1986). 15. Chang, Z. H., and Baust, J. G. Physical aspect of glass relaxation: DSC study of vitrified glycerol systems. Cryobiology 26, 573-574 (1989). 16. Christian, J. W. The theory of transformation in metals and alloys. In “Physical Metallurgy” (R. W. Cahn, Ed.), pp. 443-539. NorthHolland, Amsterdam, 1965. 17. Conde, C. F., Miranda, H., Conde, A., and Marquez, I;:. Non-isothermal crystallization and isothermal transformation kinetics of the N&5Cr,4,5P,7 metallic glasses. J. Mat. Sci. 24, 139-142, (1989). 18. Fahy, G. IM. Vitrification. In “Low Temperature Biotechnology: Emerging Applications and Engineering Contributions” (J. 1. MacGrath and K. R. Diller, Eds.), pp. 113-146. ASME, New York, 1988. 19. Fahy, G. ‘M. Biological effects of vitrification and devitrification. In “The Biophysics of Organ Cryopreservation” (D. E. Pegg and A. M. Karow. Jr., Eds.), pp. 265-297. Plenum, London, 1988. 20. Fahy, G. M., Levy, D. I., and Ah, S. Some emerging principles underlying the physical propel.ies, biological actions, and utility of vitrification solutions. Cryobiology 24, 196-213 (1987).

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21. Fahy, G. M., MacFarlane, D. R., Angell, C. A., and Meryman, H. T. Vitrification as an approach to cryopreservation. Cryobiology 21, 407426(1984). 22. Fahy, G. M., Saur, J., and Williams, R. J. Physi-

cal problems with the vitrification of large biological systems. Cryobiology 27, (1990).

23. Flynn, J. H. The isoconversional method for determination of energy of activation at constant heating rates: corrections for the Doyle approximation. In “Proceedings, XIIth NATAS Conference,” paper 125 (1983). 24. Forsyth, M., and MacFarlane, D. R. Recrystallization revisited. Cryo-Letters 7,367-378 (1986). 25. Henderson, D. W. Thermal analysis of nonisothermal crystallization kinetics in glass formSolids 30, 301-315 ing liquids. J. Non-Cryst. (1979). 26. Kemeny, T., and Granasy, L. The evaluation of

27.

28.

29.

30.

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230-244(1986). 31. MacFarlane, D. R., and Forsyth, M. Devitrifica-

tion and recyrstallization of glass forming aqueous solutions. In “The Biophysics of Organ Cryopreservation” (0. E. Pegg and A. M. Karow, Jr., Eds.), pp. 237-257. Plenum, London, 1988. 32. MacFarlane, D. R., Fragoulis, M., Uhlherr, B., and Jay, S. D. Devitrification in aqueous solutions at high heating rates. Cryo-Letters 7, 7380(1986). 33. MacFarlane, D. R., Kadiyala, R. K., and Angell,

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croscope study of the structure of very rapidly frozen gelatin solutions. Biodynamics 9, 47-71 (1962). 36. MacKenzie, A. P., and Luyet, B. J. An electron microscope study of the tine structure of very rapidly frozen blood plasma. Biodynamics 9, 147-163 (1%3). 37. Matusita, K., Miura, K., and Komatsu, T. Kinetics of non-isothermal crystallization of some fluorozirconiate glasses. Thermochim. Acta 88, 283-288 (1985). 38. Mehl, P. Isothermal and non-isothermal crystalli-

zation during warming in aqueous solutions of 1,3-butanediol. Xhermochim. Acta 155, 187-202

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ter in cooled solutions. Chem. Phys. Lett. 21, 547-550 (1985). 43. Rapatz, G., and Luyet, B. Patterns of ice formation ih aqueous solutions of glycerol. Biodynamica 10, 69-80 (1966). 44. Rapatz, G., and Luyet, B. On the instability of

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(1989).

39. Mehl, P. “Correlation between Some Physical Properties of Aqueous Solutions of Polyalcohols with Four Carbons and Their Efficiency in the Cryopreservation of Red Blood Cells.” Ph.D. thesis, Grenoble, 1987. 40. Mehl, P., and Boutron, P. Glass-formation tendency and stability of the amorphous state in aqueous solution of linear polyaicohols. II. Ternary systems with water or 1,2-propanediol or I ,3-butanediol or 2,3-butanediol. Cryobiology 24, 355-367 (1987). 41. Mehl, P., and Boutron, P. Cryoprotection of red

blood cells by 1,3-butanediol and 2,3-butanediol. Cryobiology 25, 4l-54 (1988). 42. Mikhalev, 0. I., Kaplan, A. T., and Trofinov, V. I. On the latent heat of crystallization of wa-

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