Single-phase and heterophase solid-state chemical kinetics of thermally induced methyl transfer in tetraglycine methyl ester

Article abstract of DOI:10.1002/(SICI)1097-4601(1997)29:5<339::AID-KIN4>3.0.CO;2-V

To better understand the general interrelationships between chemical transformations and physical transformations in solid-state reactions, we have studied the kinetics of methyl transfer in polycrystalline samples of tetraglycine methyl ester (TGME) over the temperature range of 83 °C - 115 °C. Changes in the concentrations of the reactant and various intermediates (sarcosyltriglycine methyl ester METGME, and tetraglycine, TG) and products (sarcosyltriglycine, METG, and N,N-dimethyl glycyl triglycine, Me₂TG) were measured over the entire time course of the reaction using HPLC. Corresponding measurements of physical transformations occurring during the course of the reaction were made using X-ray powder diffractometry and differential scanning calorimetry. Kinetic curves for the loss of TGME in the range of 83 °C - 115 °C have a sigmoidal shape and collapse into one curve when plotted in terms of reduced time, t/t_0.5, as do plots of intermediate and product concentration plotted in the same manner. The first 25% of the reaction proceeds homogeneously through what is believed to be the formation of a crystalline solid solution of the intermediates and products in the reactant The acceleratory character of the kinetic curves in the single-phase portion of the reaction has been described by a kinetic scheme that contains a concentration-dependent rate constant. The appearance of a new crystalline phase beyond 35% of the reaction changes the reaction mechanism from a bulk reaction to an interface-controlled process that causes further acceleration of the methyl transfer. The apparent activation energies for both single-phase and heterophase stages of the reaction are about 100-130 kJ/mole.

Full text of DOI:10.1002/(SICI)1097-4601(1997)29:5<339::AID-KIN4>3.0.CO;2-V

Single-Phase and
Heterophase Solid-State
Chemical Kinetics of
Thermally Induced Methyl
Transfer in Tetraglycine
Methyl Ester
1
2
3
EVGENYI YU. SHALAEV, STEPHEN R. BYRN, GEORGE ZOGRAFI
1
,3
School of Pharmacy, University of Wisconsin-Madison, Wisconsin 53706
Department of Industrial and Physical Pharmacy, School of Pharmacy, Purdue University, West Lafayette, Indiana 47906
2
Received 7 August 1996; accepted 23 September 1996
ABSTRACT: To better understand the general interrelationships between chemical transfor-
mations and physical transformations in solid-state reactions, we have studied the kinetics of
methyl transfer in polycrystalline samples of tetraglycine methyl ester (TGME) over the tem-
perature range of 83ЊC–115ЊC. Changes in the concentrations of the reactant and various in-
termediates (sarcosyltriglycine methyl ester, METGME, and tetraglycine, TG) and products
(
sarcosyltriglycine, METG, and N,N-dimethyl glycyl triglycine, Me TG) were measured over the
2
entire time course of the reaction using HPLC. Corresponding measurements of physical
transformations occurring during the course of the reaction were made using X-ray powder
diffractometry and differential scanning calorimetry. Kinetic curves for the loss of TGME in
the range of 83ЊC–115ЊC have a sigmoidal shape and collapse into one curve when plotted in
terms of reduced time, t/t0.5 , as do plots of intermediate and product concentration plotted
in the same manner. The first 25% of the reaction proceeds homogeneously through what is
believed to be the formation of a crystalline solid solution of the intermediates and products
in the reactant. The acceleratory character of the kinetic curves in the single-phase portion of
the reaction has been described by a kinetic scheme that contains a concentration-depen-
dent rate constant. The appearance of a new crystalline phase beyond 35% of the reaction
changes the reaction mechanism from a bulk reaction to an interface-controlled process that
causes further acceleration of the methyl transfer. The apparent activation energies for both
single-phase and heterophase stages of the reaction are about 100–130 kJ/mole. © 1997 John
Wiley & Sons, Inc. Int J Chem Kinet 29: 339–348, 1997.
INTRODUCTION
Correspondence to: G. Zografi
Contract grant Sponsor: Joint Purdue/Wisconsin Industrial Pro-
gram on Molecular Mobility in Solids
An understanding of organic solid-state chemical re-
activity is of significant importance to those inter-
©
1997 John Wiley & Sons, Inc.
CCC 0538-8066/97/050339-10

3
40
SHALAEV, BYRN, AND ZOGRAFI
ested in new approaches to solid-state synthesis [1]
and in evaluating the storage stability of solids, as
with pharmaceuticals [2]. Many organic solid-state
reactions occur as single-phase reactions with the for-
mation of solid solutions of the product in the reac-
tant, followed by crystallization of the product phase
in the advanced stages of the reaction [3]. Heteroge-
neous solid-state kinetics have been developed in
great detail, especially in application to inorganic de-
composition reactions [4]. Single-phase organic
solid-state kinetics have attracted less attention; there
are very few studies for some specific polymerization
EXPERIMENTAL
Materials
The synthesis of TGME and preparation of crys-
talline samples from methanol solution were de-
scribed earlier [9]. Three samples, separately synthe-
sized and crystallized in this manner, were used
throughout the study. Elemental analysis of these
samples gave the expected results within experimen-
tal error, and the X-ray diffraction pattern of TGME
agreed with that reported earlier [9]. Under 50x and
2
00x magnification in an optical microscope the ini-
[5] and racemization [6] reactions which occur in a
tial crystalline sample consisted of both single crys-
tals and agglomerates of single crystals. The single
crystals were needle-shaped with an average width of
single phase over an entire reaction region. There are
no reported kinetic studies for organic solid-state re-
actions which consist of both single-phase and het-
erophase stages, despite their importance to organic
solid-state stability and synthesis. Recently, we have
pointed out the importance of considering the kinetics
and mechanisms of such solid-state reactions in the
context of chemical transformations (identification of
intermediates and products), coupled with an under-
standing of phase transformations (identification of
amorphous phases, i.e., liquids or glasses, and crys-
talline phases) at different stages of the reaction [7].
In this report we build on previous studies of the
solid-state methyl transfer of crystalline tetraglycine
methyl ester (TGME) [8,9], and use this system as a
model to illustrate the importance of this approach in
the quantitative evaluation of solid-state kinetics. In
earlier studies the loss of TGME was followed at
2
␮
3 ␮m (Ϯ7.6 ␮m) and length of 154 ␮m (Ϯ 63.5
m).
The melting temperature of TGME, determined as
the onset temperature of the melting endothermic
DSC peak at a scanning rate of 10ЊC/min, is equal to
1
93.9 Ϯ 2.0ЊC. In an earlier report [8], the melting
temperature with decomposition was roughly esti-
mated to be about 205ЊC.
Methods
Kinetic Measurements. To follow the kinetics of
methyl transfer 10–30 mg of TGME was placed in 5-
mL ampuls for study at 83ЊC, or in 2-mL ampuls for
all other temperatures. The possible influence of a
sample size change from 3 to 60 mg was checked at
one temperature (100ЊC), and no effect on the kinet-
ics of this reaction was observed. The ampuls, before
sealing, were subjected to vacuum over P O over at
least a 20-h period to remove residual methanol and
any sorbed water. These experiments were carried out
at 83ЊC, 93ЊC, 100ЊC, and 115ЊC using laboratory
ovens. In the experiments conducted at 93ЊC, 100ЊC,
and 115ЊC the ampuls were immersed in an oil to
avoid any thermal inhomogeneity. The temperature of
the oil bath was checked before and after sample
withdrawal using an electronic thermometer with
chromega-alumega thermocouples. Maximal temper-
ature variation during kinetic runs was 2ЊC at 93ЊC
and 100ЊC, 1ЊC at 83ЊC, and 0.5ЊC at 115ЊC.
1
00ЊC in the polycrystalline state as a function of
time, and the overall rate of reactivity was compared
with those of slightly ground samples [8] and freeze-
dried samples [9]. The kinetic curves for polycrys-
talline samples were shown to be sigmoid-shaped
with an apparent induction period. Three main
products and an intermediate have been identified
and it has been shown that this reaction is accom-
panied by crystallization of a product phase [8].
This reaction is believed to involve intermolecular
methyl transfer rather than intramolecular rearrange-
ment.
In the present study we have more closely
examined the kinetics of this reaction in polycrys-
talline samples as a function of temperature, by fol-
lowing with HPLC the loss of reactant and the ap-
pearance of intermediates and products, and any
phase changes through the use of powder X-ray dif-
fraction. We show how knowledge of the chemical
and physical changes that occur during the reaction
can lead to a more physically meaningful kinetic
analysis.
2
5
Ampuls were removed periodically at set time
points and subjected to HPLC analysis, as well as X-
ray or DSC analysis under certain conditions. The
HPLC analysis was carried out in two ways, which
gave identical results within experimental uncertainty.
In one case the entire ampul content was dissolved in

THERMALLY INDUCED METHYL TRANSFER
341
O
C
O
C
H N CH2
N
H
CH2
O
CH3
2
΂
΃
3
Tetra glycine methyl ester, TGME
ϩ
H
O
O
Ϫ
CH3
N
H
CH2
C
N
H
CH2
C
O
΂
΃
3
Sarcosyltriglycine, METG
O
O
C
ϩ
Ϫ
H N
CH2
C
N
H
CH2
O
Figure 1 Typical HPLC chromatographs for the solid-
state thermal degradation of TGME at the intermediate ex-
tent of reaction (a) and end of reaction (b). 1: TG; 2:
3
΂
΃
3
Tetraglycine, TG
METG; 3: TGME; 4: METGME; and 5: ME TG.
2
ϩ
H
O
O
C
Ϫ
CH3
N
CH2
C
N
H
CH2
O
΂
΃
3
the HPLC mobile phase, whereas in the second case
parts of the material from the ampul (20% to 70%)
were removed and dissolved separately for analysis.
Chemical Analysis. Analysis of TGME and vari-
ous reaction products was carried out by HPLC
CH
3
N,N-Dimethyl glycyl triglycine, ME TG
2
ϩ
H
N
H
O
C
O
C
CH3
CH2
N
H
CH2
O
CH3
(Spectra System Instrument with an integrator) on an
΂
΃
Scheme I
3
Altech Econosil C , 5 ␮ column, 250 mm by 2.6
1
8
mm. The separation of components was carried out
using an isocratic solvent system consisting of 1%
Sarcosyltriglycine methyl ester, MET GME
Scheme I
(v/v) of acetonitrile and 0.1% (v/v) trifluoroacetic
acid in water at a flow rate of 1 ml/min and UV de-
tection at 210 nm [9]. All solvents used were of
HPLC grade. Twenty (20) ␮L of solution, containing
The molar absorptivities of TG, TGME, and reacted
samples after exposure to 100ЊC and 115ЊC, were de-
termined at 210 nm in the HPLC mobile phase and
1
–2 mg/ml of a particular sample in the mobile
phase, were injected. Depending on the extent to
which the reaction had occurred, five or fewer main
peaks were observed on each chromatogram. Typical
chromatograms for an intermediate stage and the final
stage of the reaction are shown in Figures 1(a) and
3
Ϫ1
3
Ϫ1
found to be 1.91 ϫ 10 M , 1.92 ϫ 10 M , 2.03 ϫ
3
Ϫ1
3
Ϫ1
10 M , and 2.09 ϫ 10 M , respectively. Since
these values were all so close, it was possible to ob-
tain the various concentrations directly from the chro-
matograms without using different calibration coeffi-
cients for the different peaks. The concentration of
TGME also was determined by a method that utilized
an external standard of TGME. The values calculated
in this manner agreed within 5–7 relative percent
with those measured directly from the chro-
matograms. This indicates that the HPLC conditions
used allowed us to monitor all major products.
1
(b), respectively. The peaks for tetraglycine (TG),
TGME, and the main product, sarcosyltriglycine
METG) were identified and assigned by the addition
(
of TG, TGME, and METG to the solution being ana-
lyzed. Other peaks were assigned on the basis of ear-
lier work [8], which identified an intermediate sarco-
syltriglycine methyl ester (METGME) (HPLC peak
area increased to a maximum and then disappeared
by the end of the reaction) and another final product,
Differential Scanning Calorimetry. All DSC mea-
surements were carried out on a Seiko SSC/5200 dif-
ferential scanning calorimeter. Measurements were
carried out in the temperature interval of Ϫ20ЊC to
210ЊC at a heating rate of 10ЊC/min. The instrument
was calibrated using the melting points of indium,
gallium, tin, and water as temperature standards.
N,N-dimethylglycyltriglycine (ME TG) (HPLC peak
2
area increased during the reaction). The presence of
the dimethylated products ME TG and METGME
2
was confirmed by mass spectrometry, as described
below. Structural formulas for the reactant and prod-
ucts are given in Scheme I.

3
42
SHALAEV, BYRN, AND ZOGRAFI
Samples of TGME, from 8 to 15 mg, were placed
into sealed aluminum pans with pin holes.
variation is believed to be due mainly to possible in-
homogeneity in the preparation of physical mixtures.
Mass Spectrometry. Mass spectrometric analysis
was carried out with a VG Auto Spec M mass spec-
trometer in the Chemistry Department of the Univer-
sity of Wisconsin–Madison. The matrix used for the
samples was 3-nitro-benzyl alcohol and glycerol; 35
kV Csϩ ions were used to produce ionization, and
X-Ray Diffraction. Powder diffraction patterns
were obtained on a Siemens D-500 Diffraktometer-
Kristalloflex instrument at the following conditions:
CuK␣ radiation, 20 mA, 40 kV, x ϭ 1.5418 Å, scan
rate 6Њ/min, and on a Scintag PADV instrument under
the following conditions: CuK radiation, 30 mA, 45
␣
Ϫ1
kV, and scan rate 5Њ/min. In each case, about 15–30
mg samples were placed on a quartz sample holder
and the measurements were carried out at room tem-
perature.
The weight fraction of the reactant (TGME) crys-
talline phase during the reaction was followed by X-
ray diffraction and the use of an internal standard,
LiF, as described in detail previously [10]. In the
present case, physical mixtures of TGME and LiF in
a 1/1 weight ratio were prepared and areas-under-the
peak at 2␪ ϭ 5.7 degrees (for TGME) and at 2␪ ϭ
the scanning rate was 2 s decade . The purely crys-
talline TG, the initial sample of TGME and 8 reacted
samples, produced at 100ЊC and 115ЊC, and contain-
ing 78.7%, 63.9%, 60.8%, 53.2%, 45.5%, and 0%
TGME were analyzed. All samples but TG revealed a
line with a mass of 261, which could be assigned to
ϩ
ϩ
both H TGME and METGH. Reacted samples had
additional lines at 275 and 289. The line at 275 is
ϩ
consistent with the intermediate, METGME, or the
ϩ
final dimethylated product in the form ME TGH .
2
The line at 289 appears to be consistent with a possi-
3
8.8 degrees (for LiF) were determined using the
ble intermediate, ϩME TGMe. We were not able to
2
Origin computer software. The values reported repre-
sent the average of at least two diffractograms for
every physical mixture. The fraction of crystalline
TGME, X, was calculated with the following equa-
tion,
follow this substance by HPLC, but most likely its
concentration does not exceed 5%. We did not detect
any lines at 246 that could be assigned to TG, but in
this system analysis of purely crystalline TG, like-
wise, produced no lines other than those due to the
matrix.
(
1/K)STGME
X ϭ
(1)
SLiF (MLiF/Msample
)
RESULTS
where STMGE is the peak area for TGME at 2␪ ϭ 5.7Њ,
SLiF ϭ is the peak area for LiF at 2␪ ϭ 38.8Њ, Msample
and MLiF are the weight percents of sample and LiF
in the physical mixture, respectively, and K is a con-
stant of proportionality. K was calculated for 4 inde-
pendent samples of nonreacted TGME to be 2.35 Ϯ
In Table I we present the composition of final prod-
ucts in mole percent analyzed for reactions carried
out at the different temperatures. Also included in the
table are corresponding values reported in previous
work by Sluyterma and Veenedaal [8] using a differ-
ent analytical procedure for detecting products. As
can be seen, the agreement for various temperatures
in this study, and between this and the earlier work, is
good. It should be noted, however, that in ref. [8] it
was reported that 11% of a water-insoluble polymer
was formed. We attempted to check for the presence
0
.36. Such calibration was carried out with crystalline
samples that had been annealed in-vacuo for 1 h at
00ЊC; the fraction of crystalline TGME in these
1
samples was taken to be 1. Physical mixtures with
sample/LiF ratios of 4/1 and 1/4 gave the expected re-
sults with an experimental uncertainty of 10%. Such
Table I Final Composition of Reactant and Products of the Thermal Degradation of
Tetraglycine Methyl Ester
Composition (mol%)
Temperature, °C
TGME
TG
METG
ME TG
2
9
00
00
3
7.4
2.1
5
14.6
14.9
18
63.6
65.1
53
12.1
13.9
13
1
1
a
1
15
0
15.6
67.9
12.3
a
From ref. [8].

THERMALLY INDUCED METHYL TRANSFER
343
Figure 3 Molar fraction of the products and intermediates
as a function of reduced time, t/t0.5 . (ᮀ) 83ЊC; (᭢) 93ЊC;
(᭹) 100ЊC; and (᭺) 115ЊC. a: TG; b: METGME; c:
Figure 2 Molar fraction of the reactant (TGME) as a
function of reduced time; t/t0.5 (t0.5 values are given in Table
II). (ᮀ) 83ЊC; (᭢) 93ЊC; (᭹) 100ЊC; and (᭺) 115ЊC.
METG; and d: Me TG.
2
of any insoluble matter by gravimetric analysis using
Pro-weight 47 mm glass fiber filters and found about
at most 0.8% insolubles in materials reacted at 100ЊC
and 115ЊC. At this point it is not clear why this and
the earlier work gave such different results in this re-
gard, while agreeing reasonably well otherwise.
Figure 2 presents the time profile for loss of
TGME at the four temperatures plotted in terms of
the reduced time, t/t0.5 , where t0.5 is the time required
for 50% of TGME to react. The kinetic results were
in good agreement for all initial samples of TGME
over the first 90% of the reaction, with some differ-
ences noted beyond this point. Table II provides the
values of t0.5 at the four temperatures. In Figure 3 are
depicted similar plots for the appearance of TG,
a single phase or heterophase environment. To answer
this we attempted to detect the appearance of any new
crystalline or amorphous phases using powder X-ray
diffraction and DSC. X-ray diffraction patterns at dif-
ferent extents of reaction are shown in Figure 4 for
representative samples. Here, we find that at all four
temperatures up to 22 mol% conversion of TGME the
reaction proceeds without the appearance of new
phases, as would be reflected in any new diffraction
peaks. According to the quantitative x-ray diffraction
analysis, the amount of crystalline phase of TGME
METG, ME TG, and METGME. Note that the con-
2
tent of METGME passes through a maximum at
about t/t0.5 ϭ 1 and drops to zero, while that of TG
also shows a maximum at t/t0.5 ϭ 1, but then de-
creases to a final concentration of about 15%, as
shown in Table I. The contents of the other main
products remain very low up to about t/t0.5 ϭ 1 and
then abruptly increase. The superimposition of all
data obtained at different temperatures would indicate
that the underlying mechanisms involved in this reac-
tion are the same over this temperature range.
A critical question to be answered in any solid-
state reaction study is whether the reaction occurs in
Table II Time for 50% of Reaction (t ) as a Function
0
.5
of Temperature
Temperature (°C)
t0.5 , (h)
8
9
00
15
3
3
1890
575
300
88
Figure 4 X-ray powder diffraction patterns. The numbers
indicate the fraction of conversion, ␣. *Marks the peaks of
the reactant remaining at the end of the reaction.
1
1

3
44
SHALAEV, BYRN, AND ZOGRAFI
Table III Molar Fraction of TGME Determined by
HPLC and Weight Fraction of TGME Crystalline Phase
Determined by PXRD
tion of the TGME X-ray peaks during this period. Be-
yond ␣ ϭ 0.35–0.40 a new crystalline phase appears
that may influence the kinetics of this reaction.
To analyze these observations more quantitatively,
we calculated the peak areas in the powder X-ray dif-
fractograms (PXRD) for TGME and estimated any
change in the fraction of this crystalline phase during
the overall reaction, as followed by HPLC. As shown
in Table III, we can distinguish three regions, as also
depicted in Figure 2. In the first region, ␣ ϭ 0 to
HPLC
PXRD
1.0
0.97(0.06)
0.99(0.18)
1.07(0.17)
0.68(0.15)
0.57(0.1)
0.36(0.03)
0.38(0.05)
0.41(0.04)
0.28(0.12)
0.24(0.03)
1
0
0
0
0
0
0
0
0
0
0
.0
.94(0.03)
.85(0.02)
.78(0.03)
.63(0.03)
.56(0.04)
.36(0.02)
.24(0.01)
.16(0.01)
.09(0.01)
.02(0.01)
0
.22, the reaction proceeds without any apparent de-
crease in the amount of reactant crystalline phase de-
spite the fact that the amount of reactant is decreasing
chemically. This would be consistent with a bulk re-
action taking place through the formation of a crys-
talline solid solution of products in reactant. In the
second region, ␣ ϭ 0.36 to 0.64, the reactant crys-
talline phase decreases with time and the fraction of
this phase remaining agrees reasonably well with the
extent of reaction that has taken place. Correspond-
ingly, a new crystalline phase appears. This would
support the suggestion that in this region the reactant
phase consists of pure TGME crystals and the prod-
uct crystalline phase consists of pure product. Thus,
the reaction in this region proceeds through a phase
transformation from reactant to product. In other
words, in this region the reaction mechanism has
shifted from one of bulk control to interface control.
In the third region, ␣ Ͼ 0.76, the changes in crys-
talline TGME (PXRD) appear to be slower than the
changes in overall TGME concentration (HPLC), in-
dicating that in this region both bulk control and in-
terface control may be involved in the chemical trans-
formation. Since most of the reaction occurs in the
first two regions, subsequent kinetic analyses will be
restricted to ␣ ϭ 0 to 0.76. Since region 1 appears to
be one involving a single phase reaction and region 2,
a heterophase reaction, different kinetic models will
be used for each of these regions.
There are three general types of kinetic models
that can be applied to solid-state reactions [4]: mod-
els describing moving boundaries (Avrami–Erofeev
equation, contracting geometry); models for diffu-
sion-controlled processes; and models based on the
law of mass action. The first two models are not
likely to be applicable to the methyl transfer of
TGME at ␣ Ͻ 0.25, since there is no indication of
the appearance of a new phase nor the presence of
moving boundaries. Diffusion normally can play a
critical role in reactions involving two solid phases or
involving a solid interaction with a liquid or gas.
However, this particular reaction starts with 100% re-
actant and involves the interaction of two molecules
in close proximity to one-another. Moreover, diffu-
sion-controlled reactions usually have deceleratory
Note: Numbers in parentheses represent the standard deviation.
does not change in this portion of the reaction (Table
III). Observation of reacted samples with an extent of
reaction of ␣ ϭ 0.15 and 0.40 by optical microscope
under a 200x magnification (at room temperature)
did not reveal any apparent phase separation or in-
homogeneity. From about 35% to 70% reaction a new
broad peak occurred at 2␪ equal to 30Њ, and some of
the original small peaks disappeared. At this point it
appears that a certain amount of the phase containing
crystallinE TGME starts to decrease (Table III).
Above 70% of the reaction very strong new lines ap-
pear, indicating a new crystalline phase(s) while the
peaks related to TGME essentially disappear.
DSC measurements were used to monitor the pos-
sible appearance of a liquid phase due to a low-tem-
perature eutectic for samples that had undergone ex-
tents of reactivity ranging from ␣ ϭ 0.05 to 0.97, and
it appeared that no apparent eutectic melting occurred
over the range of 0–150ЊC (data not shown). More-
over, in such samples there were no indications of a
glass transition that would be associated with an
amorphous phase, as had been suggested in ref. [8],
based on limited X-ray diffraction data.
DISCUSSION
From the results of the experiments described above
using HPLC, X-ray diffraction and DSC measure-
ments, we would conclude that about the first 25% of
the reaction proceeds in a single phase where crys-
talline TGME reacts to form a solid-solution with the
closely related products, TG and METGME and
small amounts of METG. Support for this comes
from the lack of any new X-ray peaks and the reten-

THERMALLY INDUCED METHYL TRANSFER
345
kinetic curves, whereas the reaction under study re-
veals acceleratory character. Thus, considering the
single-phase character of this reaction for ␣ Ͻ 0.25,
we will assume that the use of equations based on the
law of mass action would be most appropriate. On the
other hand, the reaction at ␣ Ͼ 0.35 in the het-
erophase, as will be shown below, is most likely con-
trolled by phase transformation and hence equations
for a moving phase boundary or for diffusion-con-
trolled processes should be used for these conditions.
It is important to recognize that simply fitting ki-
netic data to a particular model or a particular equa-
tion does not necessarily establish the reaction mech-
anism. The major intent of the analyses that follow,
therefore, is to determine rate constants based on the
most physically reasonable models of the particular
solid-state conditions that exist, and to obtain some
measure of the activation energies from temperature
dependence of these rate constants and application of
the Arrhenius equation.
sumption of the reactant, A, as a function of time, t,
can be written as
dA
Ϫ
ϭ k(A)A
(2)
dt
where k(A) is the rate constant at concentration A. To
integrate eq. (2), it is necessary to know the exact
form of k(A). One approach is to assign a concentra-
tion dependence of rate constants to either the activa-
tion energy (E ) or the preexponential frequency fac-
a
tor (B). It has been suggested by Luty and Eckhardt
[12], for example, on the basis of the “chemical pres-
sure” theory, that the activation energy is reversibly
proportional to some function of the extent of reac-
tion. A particular equation for a concentration-depen-
dent E has been suggested [5] for the polymerization
a
of diacetylene. The equation operates with several pa-
rameters which can be determined independently
from single-crystal studies and gives a very good de-
scription of actual kinetic data. The preexponential
factor also may be concentration-dependent, although
to our knowledge this has not been discussed in the
literature. The basis for suggesting this comes from a
recognition that the preexponential factor for solid-
state reactions is identified in terms of specific vibra-
tions in the reaction coordinate [4]. The phonon-
assistance model suggests, for example, that certain
vibrations (phonons) in the solid-state serve the same
function as molecular collisions in the gas phase in
initiating chemical reactions [13,14]. It has been fur-
ther shown that concentration-dependent frequencies
are exhibited by binary organic solutions, as seen
with the 1,4-dihalonaphthalenes [15].
Single-phase Region. The key feature of the
methyl transfer reaction of TGME for ␣ Ͻ 0.25, is its
significant acceleratory character. Complex homoge-
neous reactions often exhibit such acceleratory be-
havior if they involve autocatalytic or chain processes
[11]. In this reaction it is possible that either of the
intermediates, TG or METGME (Scheme I), can re-
act with TGME to produce a true chemical autocat-
alytic character. Another possibility is to treat these
acceleratory processes in terms of mechanical
stresses created by product molecules in the crystal
lattice that give rise to a constantly increasing rate
constant; the more the product accumulates the more
significant will be the increase in rate constant
[
12,13]. Recent kinetic results reported for prepara-
The choice between these two modes of concen-
tration-dependence could be made on the basis of in-
dependent measurements of cell dimensions and vi-
brational frequencies as a function of the extent of
reaction, or by using curve fitting with different mod-
els describing concentration-dependence of B and Ea
in the Arrhenius equation. Independent experimental
information required for the first approach, however,
is not available in the case of TGME since, so far, we
have been unable to grow single crystals and so to ob-
tain their crystal structure. In order to use the second
approach (curve fitting procedure) it is necessary to
compare fitting results for several different models.
However, there is only one exact function suggested
tions of TGME that were caused to become less crys-
talline and more disordered by freeze drying [9] indi-
cate much faster rates and the lack of a slow phase
followed by an acceleratory period, as would be ex-
pected for an autocatalytic process. Indeed, the
curves exhibit a deceleratory process instead [9]. In
preliminary unpublished studies with freeze-dried
and milled samples, we have observed also that in all
other respects the reaction mechanisms seem to be
the same, i.e., the same amounts of products and in-
termediates are accumulated at comparable extents of
reaction. This would suggest that the kinetics ob-
served for crystalline TGME are most likely due to a
concentration-dependent rate constant caused by
greater disorder in the crystal and not a true chemical
autocatalysis. Thus we will choose to use the most
likely physically significant kinetic model, one that
involves a simple first-order reaction model with a
concentration-dependent rate constant.
for the concentration-dependence of E [5], and there
a
are no equations available for a concentration-depen-
dent preexponential factor. In view of these limita-
tions in our ability to determine the exact basis for a
concentration-dependent rate constant, below we sim-
ply present a general approach that does not separate
the dependency on the basis of either activation ener-
The first-order differential equation for the con-

3
46
SHALAEV, BYRN, AND ZOGRAFI
gies or preexponential factors. In this work, therefore,
we assume that the rate constant is a linear function
of reaction extent at ␣ ϭ 0–0.25, as given in eq. (3).
k ϭ k ϩ a(A Ϫ A)
(3)
o
o
where k is the rate constant for the reaction when
o
1
00% reactant is present.
Equation (2) can now be integrated using eq. (3)
for the function k(A). Integrating from A to A gives
0
the following expression,
1
A (k ϩ a(A ؋ A))
o o o
t ϭ
log
(4)
(5)
Figure 6 Arrhenius plots using the rate constants obtained
from the fitting of experimental data in the single-phase re-
gion to eq. (5), and in the heterophase region to eq. (9). (᭿)
represents concentration-independent part of rate constants
(
k ϩ aA )
Ako
o
o
or at A ϭ 1
o
Ϫ1
k0 in h for the single-phase region; (ᮀ) represents coeffi-
Ϫ2
(
k ϩ a) и exp(t(k ϩ a))
o o
cients of proportionality, a, in h , for concentration-depen-
A ϭ
k ϩ a и exp(t(k ϩ a))
dent parts of rate constants in the single-phase region; and
o
o
(
᭡) represents rate constants, k, in hϪ1, for the heterophase
In Figure 5 the fit of experimental data in the single-
region.
phase region to eq. (5) is given with a and k used as
o
adjustable parameters. Figure 6 and Table IV give the
temperature dependence of the first-order rate con-
stants, as expressed through the Arrhenius equation.
The temperature coefficients for the parameter a are
also given in Table IV. The apparent activation energy
rectly to chemical transformations only in particular
cases when the chemical reaction is governed by
phase transformations, or, in other words, when the
original phase consists of pure reactant and the new
phase consists of pure product (to be discussed in
more detail below).
Usually in following solid-state chemical reactiv-
ity, as in the present study, the concentration of reac-
tants and products in a heterophase system is the av-
erage value, whereas in such systems a meaningful
kinetic analyses should be evaluating local concentra-
tions in a particular phase. In the case of a two-phase
system, we can relate the average reactant concentra-
appears to be similar for k and a.
o
Heterophase Region. Usually solid-state chemical
kinetics in a heterogeneous system are treated with
the aid of a nucleation/growth model, such as that of
Avrami–Erofeev or the contracting geometry model
[4]. Such models, as initially developed, relate di-
tion A (t) to local concentrations by
av
A (t) ϭ X(t)A (t) ϩ (1 Ϫ X(t)) A (t)
(6)
av
1
2
where A (t) and A (t) are the local concentrations in
1
2
phases 1 and 2, respectively, and X is the weight frac-
tion of the first phase. If the reactant is insoluble in
the product phase and the product is insoluble in the
reactant phase A ϭ 0, A ϭ constant ϭ A , and eq.
2
1
o
(6) reduces to
A (t) ϭ X(t) A
(7)
(8)
av
o
Figure 5 Kinetic curves for TGME degradation at four
temperatures. The symbols represent the experimental
points, solid lines are the result of fitting experimental data
in the single-phase region to the first-order kinetic model
with a concentration-dependence rate constant (eq. (5)),
and broken lines are the result of fitting experimental data
in the heterophase region to the contracting volume kinetic
model (eq. (9)).
or in terms of the extent of reaction ␣ ϭ A/Ao ,
(t) ϭ X(t)
␣
where ␣(t) is now directly connected with phase
transformation and can be described in a number of

THERMALLY INDUCED METHYL TRANSFER
347
Table IV Parameters of the Arrhenius Equation for the Temperature Dependence of Rate
Constants for Single-Phase and Heterophase Reaction Regions
Kinetic Scheme
Ea (kJ/mole)
B
R
Single-phase Region
ko (eq.(5) in text)
a (eq.(5) in text)
107.5
110.1
25.1
31.2
0.962
0.995
Two-phase Region
126.6
k (eq.(9) in text)
35.3
0.995
Ea is the energy of activation, B, the preexponential factor, and R, the correlation coefficient for linear fit of
the data.
ways (e.g., Avrami–Erofeev equations, contracting or
extending geometry) depending on the nature of the
newly formed phase (crystalline or amorphous) and
the conditions of nucleation/growth in any particular
system. The powder X-ray diffraction data obtained
in this study allows us to assume that this is the likely
case for methyl transfer in TGME. Table III indicates
that the TGME content determined by HPLC is the
same as the weight fraction of crystalline TGME de-
termined by X-ray measurement at ␣ ϭ 0.36–0.64.
This indicates that in this reaction interval, the new
crystalline phase consists only of products and inter-
mediates, while the original crystalline phase contin-
ues to contain only TGME. Thus we assume that
once the new phase starts to crystallize the reaction
mechanism changes, i.e., the process no longer oc-
curs as a bulk reaction with the formation of a solid
solution of products in the reactant. Rather, beyond
Figure 5 represent the best fit to kinetic data using
such an equation. Figure 6 and Table IV give the tem-
perature-dependence of the rate constant from eq. (9).
The reported activation energy is similar to but
slightly higher than those obtained for the reaction in
the single-phase region.
CONCLUSIONS
The kinetics for thermally-induced methyl transfer in
polycrystalline samples of tetraglycine methyl ester
have been studied as a function of temperature by fol-
lowing the change in the concentration of reactant
and various intermediates and products by HPLC.
Corresponding phase changes during the reaction
were followed by X-ray diffraction. The reaction ap-
pears to consist of two major parts, one involving a
single-phase reaction through about 25% of conver-
sion, and the other a heterogeneous reaction above
about 30–35%. It has been shown that the reaction in
the single-phase region proceeds through formation
of a crystalline solid solution of products in the reac-
tant. The acceleratory character of kinetic curves in
the single-phase region is most likely due to a con-
centration-dependent rate constant rather than to au-
tocatalysis by the reaction products. Kinetic curves in
the single-phase region have been treated by a simple
first-order kinetic model assuming a linear-concentra-
tion dependence of the rate constant. It is suggested
that the acceleratory effect most likely arises because
of the creation of disorder in the parent crystal by
product molecules. The reaction in the heterogeneous
region exhibits further acceleration wherein the reac-
tion mechanism switches to a process controlled by
phase transformation. Apparent activation energies
obtained for the single-phase and heterophase stages
in the temperature range of 83–115ЊC are 100–130
kJ/mole.
␣
ϭ 0.35 it proceeds as an interfacially controlled re-
action. In this context, therefore, we would conclude
that the chemical transformation becomes directly
coupled with the phase transformation, so that the
rate of loss of the TGME crystalline phase should
also describe the rate of the chemical reaction.
Since our kinetic analysis in this heterophase re-
gion is limited by the number of kinetic data points
obtained and by some scatter, a detailed kinetic
analysis that can differentiate mechanisms is not pos-
sible. However, we can gain some insight into this
system by choosing a simple model to estimate rate
constants and the temperature coefficient for this por-
tion of the reaction. Kinetic data for ␣ ϭ 0.36–0.76
for three temperatures (only one data point at 83ЊC
could be obtained in the heterophase region) are pre-
sented in Figure 5. These rate constants were fit to a
contracting geometry eq. [4],
Ϫ (1 Ϫ A)1/n ϭ k(t Ϫ ti)
1
(9)
where n ϭ 3, k is a fitting parameter, and t is an in-
i
duction period for crystallization. The dotted lines in

3
48
SHALAEV, BYRN, AND ZOGRAFI
This research was supported by the Joint Purdue/Wisconsin
Industrial Program on Molecular Mobility in Solids. The
authors express their gratitude to Dr. J. LePree and Dr.
K. A. Connors for help in carrying out the HPLC measure-
ments and Ms. J. M. Milkaupt for technical assistance.
Evgenyi Shalaev thanks the administration of SRC VB
Vector (Koltsovo, Russia) for a leave of absence which en-
abled him to carry out these studies.
Molecular Crystals, Y. Ohashi, Ed., Kodansha, Tokyo,
VCH, New York, 1993, pp. 115–153.
7. E. Yu. Shalaev and G. Zografi, J. Phys. Org. Chem., ac-
cepted.
8. (a) L. A. A. Sluyterma and H. J. Veenendall, Rec.
Travaux Chim. des Pays-Bas, 71, 137 (1952); (b)
L. A. A. Sluyterma and M. Koostra, Rec. Travaux
Chim. des Pays-Bas, 71, 277 (1952).
9
. K.-S. Huang, P. Johnson, and S. R. Byrn, J. Pharm.
Sci., submitted.
1
1
0. A. Saleki-Gerhardt, C. Ahlneck, and G. Zografi, Int. J.
Pharm., 101, 237 (1994).
1. V. N. Kondratiev, Chain Reactions in Comprehensive
Chemical Kinetics, C. H. Bamford and C. F. H. Tipper,
Eds., Elsevier, Amsterdam, 1969, Vol. 2, pp. 81–188.
BIBLIOGRAPHY
1
2
3
4
. G. R. Desiraju, Crystal Engineering, The Design of Or-
ganic Solids, Elsevier, Amsterdam, 1989.
. S. R. Byrn, R. R. Pfeiffer, and J. G. Stowell, Solid State
Chemistry of Drugs, 2nd ed., in press.
. I. C. Paul and D. Y. Curtin, Acc. Chem. Research, 6,
12. T. Luty and C. J. Eckhardt, J. Am. Chem. Soc., 117,
2441 (1995).
13. N. M. Peachey and C. J. Eckhardt, J. Phys. Chem., 98,
7106 (1994).
14. P. N. Prasad, Dynamics of Solid-State Polymerization,
in Crystallographically Ordered Polymers, D. J. Sand-
man, Ed., ACS Symposium Series, 1987, pp. 106–116.
15. (a) J. C. Bellows and P. N. Prasad, J. Chem. Phys., 66,
625 (1977); (b) J. C. Bellows and P. N. Prasad, J.
Chem. Phys., 67, 5802 (1977).
2
17 (1973).
. W. E. Brown, D. Dollimore, and A. K. Galwey, Reac-
tions in the Solid State. Comprehensive Chemical Ki-
netics, C. H. Bamford and C. F. H. Tipper, Eds., Else-
vier, Amsterdam, 1980, Vol. 22.
5
6
. R. H. Baughman, J. Chem. Phys., 68, 3110 (1978).
. Y. Ohashi, A. Uchida, and A. Sekine, in Reactivity in