Self-diffusion of hydrophilic poly(propyleneimine) dendrimers in poly(vinyl alcohol) solutions and gels by pulsed field gradient NMR spectroscopy

Article abstract of DOI:10.1021/ma025636k

Pulsed-field gradient NMR spectroscopy was used to study the diffusion of three different poly(propyleneimine) dendrimers with hydrophilic triethylenoxy methyl ether terminal groups (generations 2, 4, and 5) in poly(vinyl alcohol) aqueous solutions and gels. The effects of the diffusant size, polymer concentration (from 0 to 0.26 g/mL), and temperature on the self-diffusion coefficients have been studied, and the model of Petit et al. [Macromolecules 1996, 29, 6031] was used to fit the experimental data. The Stokes-Einstein hard-sphere radii were also calculated in the zero concentration limit and were compared with those of the linear poly(ethylene glycol)s under the same conditions. The proton NMR relaxation times (T₁ and T₂) were measured to study the mobility of the dendrimer core part and terminal group as a function of the dendrimer size.

Full text of DOI:10.1021/ma025636k

Macromolecules 2003, 36, 839-847
839
Self-Diffusion of Hydrophilic Poly(propyleneimine) Dendrimers in
Poly(vinyl alcohol) Solutions and Gels by Pulsed Field Gradient NMR
Spectroscopy
W. E. Ba ille, C. Ma lvea u , a n d X. X. Zh u *
D e´ partement de Chimie, Universit e´ de Montr e´ al, C.P. 6128, Succ. Centre-ville, Montr e´ al,
Qu e´ bec, Canada H3C 3J 7
Y. H. Kim a n d W. T. F or d
Department of Chemistry, Oklahoma State University, Stillwater, Oklahoma 74078
Received August 30, 2002; Revised Manuscript Received November 25, 2002
ABSTRACT: Pulsed-field gradient NMR spectroscopy was used to study the diffusion of three different
poly(propyleneimine) dendrimers with hydrophilic triethylenoxy methyl ether terminal groups (generations
2
, 4, and 5) in poly(vinyl alcohol) aqueous solutions and gels. The effects of the diffusant size, polymer
concentration (from 0 to 0.26 g/mL), and temperature on the self-diffusion coefficients have been studied,
and the model of Petit et al. [Macromolecules 1996, 29, 6031] was used to fit the experimental data. The
Stokes-Einstein hard-sphere radii were also calculated in the zero concentration limit and were compared
with those of the linear poly(ethylene glycol)s under the same conditions. The proton NMR relaxation
times (T
1 2
and T ) were measured to study the mobility of the dendrimer core part and terminal group as
a function of the dendrimer size.
In tr od u ction
molecular weight of the PVA matrix, and only a small
variation was observed with the degree of hydrolysis of
the PVA. The diffusion in hydrophilic polymers is mostly
affected by formation of the hydrogen bonds between
the solute and the polymer matrix. All these studies
have allowed us to test the applicability of a new
physical model developed by our group. This model
have been used successfully to describe the effect of the
polymer concentration, the temperature, and the diffu-
sant size on the diffusion process in polymer gels.
The diffusion of various solute molecules in polymer
solutions and gels is very important in the application
of polymer solutions and gels. Examples include per-
1
meation through polymer membranes, diffusion of
2
plasticizers in plastic materials, and drug release from
1
1
3
polymer gels. The study of self-diffusion of different
solutes and additives in polymers may help in the
elucidation of the effects of polymer concentration, size
and shape of the diffusant, temperature, and specific
interactions within the polymer network. The informa-
tion is critical in determining the applicability of
polymeric materials in industry. Various physical mod-
4
The PEG diffusants used in our studies so far differed
4
,27
in their molecular weights
and sometimes in their
2
8
chain-end groups, but they are all linear polymers. A
hydrodynamic radius can be estimated for such mol-
ecules in solution, but the exact process of diffusion of
these linear oligomers and polymers in a gel matrix
remains to be clarified. In addition, there is a problem
of polydispersity of the PEG samples even after frac-
tionation. It would be interesting to compare their
behavior with that of more spherical molecules with
similar molecular weights but different sizes of the cross
sections. Dendritic polymers are molecules of choice
because of the good control in their molecular size and
shape (more spherical). Moreover, dendritic polymers
have a more regular conformation and a lower degree
of polydispersity in comparison to highly branched
5
-13
els of diffusion have been proposed over the years.
These physical models of diffusion are essential for the
interpretation of the results. Pulsed-field gradient (PFG)
NMR techniques have been used successfully in the
study of self-diffusion of solute and solvent molecules
in polymer solutions and gels.4
We have previously studied the self-diffusion in
polymer solutions and gels of solute molecules ranging
from small molecules to polymers.4,11,25-28 The results
obtained by the study of the diffusion of small diffusants
with different functional groups (alcohol, amine, am-
monium salt, amide, and acid) in poly(vinyl alcohol)
,11,14-24
(PVA) solutions and gels show that the diffusion behav-
2
9-32
ior is primarily influenced by the size of the diffusant
and secondarily by the chemical interactions.25 For the
self-diffusion of linear oligo- and poly(ethylene glycol)s
polymers.
These characteristics make dendrimers
very interesting and useful as a model diffusion probes.
Also, they can be used in many different applications
(
PEGs) the molecular size of the diffusant plays the
such as molecular encapsulation for drug delivery,
4
,25,27
33,34
most important part in the diffusion process.
The
membrane transport, or molecular recognition.
There-
effect of the polymer matrices on the self-diffusion of
PEG with molecular weight 600 was also studied for
different ternary polymer-water-PEG systems. The
diffusion of the PEG did not vary significantly with the
fore, the self-diffusion process of three different poly-
(propyleneimine) (PPI) dendrimers with hydrophilic
triethylenoxy methyl ether (TEO) terminal groups in
PVA aqueous solutions and gels was studied by the PFG
NMR technique. The effects of polymer concentration,
temperature, and dendrimer size on the self-diffusion
coefficients were investigated. We have also studied the
2
6
*
To whom correspondence should be addressed. E-mail: julian.
zhu@umontreal.ca.
1
0.1021/ma025636k CCC: $25.00 © 2003 American Chemical Society
Published on Web 01/17/2003

8
40 Baille et al.
Macromolecules, Vol. 36, No. 3, 2003
mobility of the dendrimer core part and terminal group
as a function of generation by the measurement of H
NMR relaxation times (T1 and T2).
where M
0
and M are the magnetization at the equilibrium and
1
at the nth echo, respectively. τ is a fixed echo time (set at 2
ms), which allows the attenuation of diffusion and J -modula-
tion effects, and n is an even number of echoes (32 values
between 2 and 2400 echoes were chosen).
P u lsed F ield Gr a d ien t NMR Mea su r em en ts. The self-
diffusion coefficient (D) measurements were performed using
the stimulated echo (STE: 90°-t -90°-t -90°-t -echo) se-
Exp er im en ta l Section
2 n
Ma ter ia ls. DAB-dendr-(NH ) (n ) 8, 32, and 64) and all
other chemicals were purchased from Aldrich (Milwaukee, WI).
Triethylamine (TEA) was dried over anhydrous 3 Å molecular
sieves and freshly distilled. All other chemicals were used as
1
2
1
quence developed by Tanner37 on a Bruker DSX-300 NMR
spectrometer operating at a frequency of 300.18 MHz for
protons. A Bruker imaging probe (Micro2.5 probe) was used
with a three orthogonal field gradient coils system permitting
a maximum gradient of 100 G/cm. The self-diffusion coef-
ficients were obtained from the attenuation of the NMR signals
due to the application of the gradient pulses of various
strengths in a stimulated echo sequence, as given in the
following expression3
received. D
2
O was purchased from C.I.L. (Andover, MA).
2
-[2-(2-Meth oxyeth oxy)eth oxy]a cetyl Ch lor id e. A solu-
tion of 2-[2-(2-methoxyethoxy)ethoxy]acetic acid (5.34 g, 30.0
mmol) and oxalyl chloride (6.35 g, 50.0 mmol) in 3 mL of
toluene was stirred for 4 h at 65 °C. The solvent and excess
reagent were removed under reduced pressure, and the residue
was dried at 40 °C under vacuum to give a light yellow oil
8-40
(
5.34 g, 90%) which was used without further purification.
Mod ified P oly(p r op ylen eim in e) Den d r im er s.3
5,36
2-[2-
t2
2t1
I
I0
2
2
2
ln ) -
-
- ln 2 - γ δ G D(∆ - δ/3)
(3)
(
2-Methoxyethoxy)ethoxy]acetyl chloride (3.00 g, 15.3 mmol)
T1 T2
was added to a solution of PPI dendrimer DAB-dendr-(NH
1.00 g, 1.29 mmol), DMF (5.0 mL), and TEA (0.9 g, 8.89 mmol)
at 0 °C. The solution was stirred under nitrogen at 70 °C for
4 h. Water (5 mL) was added to hydrolyze the excess acid
chloride. The mixture was made basic to pH > 14 using 5 g
27 mmol) of tetramethylammonium hydroxide pentahydrate
and was extracted with CH Cl (4 times 10 mL). The combined
dichloromethane solution was dried over Na SO and evapo-
rated. The oily residue was dried at 40 °C under vacuum to
give a light yellow oil of PPI(TEO) (2.21 g, 83%). DAB-dendr-
NH (n ) 32 and 64) were also modified with triethylenoxy
methyl ether end groups by the same procedure to yield PPI-
2 8
)
(
where I and I
and in the absence of the gradient, respectively, t
interval between the first two 90° rf pulses and between the
third 90° rf pulse and the middle of the echo, t is the delay
0
are the NMR signal intensities in the presence
1
is the
2
2
(
between the second and the third 90° rf pulses, γ is the
gyromagnetic ratio of the nucleus under observation, δ and G
are the duration and the strength of the applied gradient pulse,
respectively, D is the self-diffusion coefficient, and ∆ is the
time interval between the two successive gradient pulses (also
called the diffusion time). The self-diffusion coefficient was
extracted from the slopes of the lines obtained from linear
regression of the logarithm of signal intensity as a function of
2
2
2
4
8
(
2 n
)
1
13
(TEO)32 and PPI(TEO)64. The H and C NMR spectra of
dendrimers made by a slightly different procedure were the
same as the spectra of those reported here.
Sa m p le P r ep a r a tion . Samples were prepared following
the method described previously.
degree of hydrolysis ) 99%) was added to a D
containing 1 wt % of dendrimer probes in 5 mm o.d. NMR
tubes. The NMR tubes were sealed to avoid solvent evapora-
tion and heated at temperature between 100 and 110 °C to
dissolve the PVA and to help in the mixing of the sample. The
heating helped also to prevent gelation effects. The concentra-
tion of PVA ranged from 0 to 0.26 g/mL.
2
3
5,36
G . Excellent linear relationships have been obtained in these
experiments with correlation coefficients of 0.996 and better.
The error of the measured D values was estimated to be e5%
by carrying out repeated experiments with selected samples.
The experiments were performed at different temperatures
from 5 to 45 °C (fluctuation of the temperature (0.3 °C), and
concentrations of PVA ranged from 0 to 0.26 g/mL. Temper-
ature calibration was done periodically on the NMR instru-
4
,11
PVA (M
n
) 52 000 with a
2
O solution
ment by measuring the chemical shift difference between the
H NMR signals of CH and OH groups of pure ethylene glycol
2
1
at various temperatures since this difference is sensitive to
1
Molecu la r Weigh t Deter m in a tion . H NMR experiments
temperature changes.41
were performed on a Bruker AMX-300 spectrometer operating
at a frequency of 300.13 MHz to determine the molecular
weights (M ) of the dendrimer, based on the ratio of the
n
integrated peak areas of the methylene protons at 4.05 ppm
to the methylene groups in the core.
A nonlinear least-squares fitting method was used in all
cases to fit the experimental diffusion data to the model of
Petit et al.11 The correlation coefficients r yielded were in the
range 0.997-0.999. The quality of the fitting to the experi-
mental data was also indicated by the good fits shown in all
the figures. The same sets of fitting parameters were obtained
regardless of the initial values used in the procedure.
2
Rela xa tion Tim e Mea su r em en ts. 1H spin-lattice (T
)
1
and spin-spin (T
Bruker AMX-300 spectrometer operating at 300.13 MHz.
Experiments were conducted at 25 °C. T was obtained with
the classical inversion-recovery pulse sequences. All measure-
ments were performed using eight scans with a repetition time
of 30 s. A total of 24 increments of the recovery time between
2
) relaxation times were measured on a
1
Resu lts a n d Discu ssion
NMR Ch a r a cter iza tion . 1H NMR spectra of the
synthesized products were done in D O at 1 wt % of the
dendrimers. The reaction of the amine groups on the
dendrimer surface with TEO is evidenced by the high
field shift of the methylene protons from 4.75 ppm (for
TEO before reaction) to 4.05 ppm (peak labeled f in
Figure 1). These spectra also confirm the structure of
dendrimers. All the proton signals of the dendrimers
are assigned (Figure 1). The NMR spectra were also
used to determine the molecular weight of these den-
drimers (Table 1). The molecular weight was obtained
from the integrated area ratio of methylene protons at
.05 ppm (peak f) and methylene protons at 1.73 ppm.
The comparison between the values expected for an
ideal dendrimer growth and the values determined
experimentally by H NMR spectroscopy allows the
determination of the molecular weight. This ratio also
helps to determine the completeness of the end groups
2
0
.01 and 25 s were used. The T
1
values were extracted by the
use of
M
M0
τ
T1
)
1 - k exp -
(1)
[
(
)
]
where M
0
and M are the magnetization at equilibrium and
1
for a recovery time τ, respectively; T and k can be calculated
by least mean square method. The variable k is ideally equal
to 2, but for all experiments k obtained are between 1.8 and
2
2
.0. T measurements were performed with the classical Carr-
4
Purcell-Meiboom-Gill (CPMG) pulse sequences. The T
ues were extracted from
2
val-
1
M
M0
2τn
T2
)
exp -
(2)
(
)

Macromolecules, Vol. 36, No. 3, 2003
Poly(propyleneimine) Dendrimers 841
F igu r e 2. Semilogarithmic plot of the attenuation of the
dendrimer NMR signals in a 1 wt % aqueous solution (without
2
PVA) at 25 °C as a function of (Gγδ) (∆ - δ/3). δ ) 1.0 ms, ∆
)
8
400 ms. Squares, PPI(TEO) ; circles, PPI(TEO)32; and
triangles, PPI(TEO)64
.
the three distinct zones. This result indicates that the
mobility increases from the inner core of the dendrimers
to the outer part. Table 2 also shows a decrease in T2
at higher generations of the dendrimers. A decrease by
a factor of ca. 2 was observed between the PPI(TEO)8
and the PPI(TEO)64 for all peaks, indicating a decrease
in the mobility of the dendrimers. In the measurements
F igu r e 1. 1H NMR spectra of three different poly(propylene-
imine) dendrimers with hydrophilic triethylenoxy methyl ether
terminal groups at 1 wt % in D
B) PPI(TEO)32, (C) PPI(TEO)64
2 8
O at 25 °C: (A) PPI(TEO) ,
(
.
Ta ble 1. Som e Ch a r a cter istics of th e Mod ified
P oly(p r op ylen eim in e) Den d r im er s
1
of H T1, for the terminal groups a decrease in the T1
no. of
generations
terminal
groups
Ma (g/mol) Mnb (g/mol)
was observed with increasing dendrimer generation, but
an increase in the T1 was observed for core protons. This
may be related to the relative mobility of the different
parts of the dendrimers. The terminal protons are more
mobile and still lie in the extreme narrowing region,
corresponding to short correlation times, while the
protons in the core of the dendrimers are less mobile
and correspond to longer correlation times lying on the
right side of the T1 minimum on the curve of T1 as a
function of the correlation time. Therefore, there is no
contradiction with the T2 results since T2 generally
decreases as a function of molecular correlation time.
Diffu sion Mea su r em en ts. Figure 2 is a semiloga-
rithmic plot of the attenuation of the NMR signals as a
samples
PPI(TEO)8
PPI(TEO)32
PPI(TEO)64
2
4
5
8
32
64
2055
8639
17396
2000
8600
17000
a
Molecular weight calculated for ideal dendrimer growth.
b
1
Number-average molecular weight determined by H NMR
spectroscopy.
Ta ble 2. NMR Rela xa tion Tim es for th e Differ en t 1
H
Sign a ls of th e Den d r im er s a t 25 °C
NMR peaksa
sample
a
b
f
h
i
spin-spin relaxation times, T1 (s)
PPI(TEO)8
PPI(TEO)32
PPI(TEO)64
2.48
1.88
1.64
0.64
0.52
0.49
0.50
0.46
0.45
0.24
0.28
0.31
0.21
0.28
0.31
2
2
2
function of γ δ G (∆ - δ/3) with δ and ∆ equal to 1.0
and 400 ms, respectively. According to eq 3, this
relationship is linear if the diffusion is isotropic and the
slope of the line equals -D. All self-diffusion coefficient
measurements presented in this paper were performed
on the methyl protons labeled a (see Figure 1). However,
we have verified that the self-diffusion coefficient is the
same for all protons. Moreover, the self-diffusion coef-
ficient was determined on all directions (x, y, and z) of
the applied gradient pulse to verify whether the diffu-
sion is isotropic. The results obtained show that D is
independent of the direction. Attempts were made to
detect restricted diffusion of the dendrimers by measur-
ing their self-diffusion coefficient as a function of
diffusion time ∆ from 20 to 600 ms for the dendrimers
with two different PVA concentrations (0.12 and 0.26
g/mL). At each of the PVA concentrations, the self-
diffusion coefficients obtained as a function of diffusion
time are identical, indicating that no restricted diffusion
is present. The excellent linearity observed in Figure 2
indicates that the self-diffusion coefficients are mono-
disperse, and the D values obtained decrease with the
increase in the generation or hence the molecular weight
of the dendrimers. The decrease in D values can be
spin-lattice relaxation times, T2 (s)
PPI(TEO)8
PPI(TEO)32
PPI(TEO)64
2.02
1.21
0.94
0.42
0.29
0.22
0.26
0.16
0.11
0.04
0.03
0.02
0.04
0.03
0.02
a
Peaks a, b, f, h, and i correspond to the proton signals as
identified in Figure 1.
amidation. The experimental intensity ratio obtained
for each dendrimer generation shows that the modified
poly(propyleneimine) dendrimers synthesized are prac-
tically ideal.
1
Rela xa tion Tim es Mea su r em en ts. H NMR relax-
ation times (T1 and T2) were measured to study the
motions of three different parts of the dendrimers
(methyl and methylene protons of the terminal groups
and methylene protons of the core) as a function of
generation. The results obtained are listed in Table 2.
All dendrimers showed a sharp decrease (ca. 50 times)
in the T2 from the methyl protons of the terminal groups
(peak a) to the methylene protons of the core (peaks h
and i). Moreover, a large difference is observed among

8
42 Baille et al.
Macromolecules, Vol. 36, No. 3, 2003
with increasing polymer concentration (c). Thus, the
self-diffusion coefficient of the diffusant decreases when
the polymer concentration increases. However, in the
viscous gel regime, the correlation length decreases
more slowly. Thus, the effect of the polymer concentra-
tion on the self-diffusion coefficient tends to be less
significant, as illustrated in Figure 3. The viscous gel
regime is determined by many factors such as the
molecular interactions and the molecular weight of the
polymer. In the case of PVA, the degree of hydrolysis is
also an important factor. Figure 3 also shows a depen-
dence of the self-diffusion coefficients on the molecular
weight or the molecular size. However, this molecular
weight effect becomes less significant for dendrimers
with higher molecular weights. This trend is more
obvious at higher PVA concentrations for PPI(TEO)32
and PPI(TEO)64 dendrimers.
F igu r e 3. Dendrimers self-diffusion coefficients as a function
of PVA concentration at 25 °C. Dashed lines are fits to eq 7.
Squares, PPI(TEO)
TEO)64
8
; circles, PPI(TEO)32; and triangles, PPI-
Dashed lines in Figure 3 represent the result of the
(
.
1
1
fit to the model of Petit et al.:
described by the following expression in the case of an
uncharged spherical particle at infinite dilution moving
in laminar flow:
D0
D )
(7)
4
2
2ν
1
+ ac
RT
2
D )
(4)
where a ) D0/kâ , ν is a constant, which is characteristic
of the polymer-solvent system, k represents the jump
frequency over the energy barriers, which is expected
to depend on temperature and size of the diffusant, and
c is the polymer concentration. This jump frequency can
be written in an Arrhenius form:
1
/3 4/3 4/3 2/3 1/3 1/3
2
3
π
N
νj M η
where D is the self-diffusion coefficient, R is the gas
constant, T is the absolute temperature, η is the
viscosity of the solvent (D2O), N is the Avogadro’s
number, νj is the experimentally determinable partial
specific volume of the diffusing molecule, and M is the
molecular weight of the diffusant. Equation 4 is ob-
tained from the Stokes-Einstein equation
-
k T
∆E
k ) F exp
(8)
p
(
)
B
where Fp is a frequency prefactor and ∆E is the height
of the potential barrier. The model of Petit et al. treats
the polymer solution as a transient statistical network,
through which the diffusant diffuses in a series of jumps
over a potential barrier determined by the correlation
length. Thus, when ê decreases, there is an increase in
the energy barrier that the diffusant molecule has to
overcome to diffuse in the polymeric network.
k T
B
D )
(5)
0
6
πηRh
where D0 is the self-diffusion coefficient at zero polymer
concentration, kB is the Boltzmann constant, and Rh is
the effective hydrodynamic radius of the diffusing
molecule, which is related to νj and M by the following
4
2
expression:
The model was used to fit the variation of the self-
diffusion coefficient as a function of PVA concentration
for all dendrimers (and at different temperatures as
3
3
Mνj
R )
(6)
h
2
x
4πN
well). The values of the parameters D0, kâ , and ν
obtained from the fit with eq 7 are listed in Table 3. It
shows that excellent agreement of the D0 values ob-
tained from fitting to the ones measured by the NMR
experiments. The margin of errors for all the parameters
is very small, indicating the high quality of the fittings.
The D0 values in the table decrease with the molecular
weight. This result agrees well with eq 4, which links
the self-diffusion coefficient with the inverse of the cubic
root of the molecular weight. When the strength of the
interactions between the diffusing molecules and the
polymer matrix is strong enough, this trend could be
reversed. Moreover, we have observed that the kâ2
parameter decreases with increasing molecular size of
the diffusing dendrimers (see Figure 4). The dashed line
By substituting eq 6 into eq 5, the relationship between
the diffusion coefficient and the molecular weight (eq
4
) can be obtained. From eq 4 and without any specific
interaction process between the diffusant and the
polymer, the self-diffusion coefficient is inversely pro-
portional to the cubic root of the molecular weight,
which explains the decrease of D with the increase of
dendrimer size.
(
1) Effect of P olym er Con cen tr a tion . The self-
diffusion coefficients of these dendrimers as a function
of PVA concentration are shown in Figure 3. This figure
shows a decrease in the D values with the PVA
concentration, from 0 to 0.26 g/mL. The concentration
range includes the dilute and the viscous gel regimes.
The decrease of the self-diffusion coefficient with PVA
concentration can be described by the correlation length
ê, which represents the mesh size of a transient statisti-
cal network. As illustrated by de Gennes’s scaling
theory43 (ê ) âc , where â should be a constant and
does not vary as a function of polymer concentration or
of the molecular weight of the polymer), this ê decreases
2
serves as visual guide of the decrease in kâ and has no
physical meaning. Since â remains constant for a given
1
1
polymer-solvent system, the results obtained confirm
that an increase in the molecular size leads to a lower
4
,11,27
jump frequency.
Consistent with the results re-
-ν
ported by Masaro et al. for PEGs as diffusant, the kâ2
parameter decreases with the molecular size of the
13
2
diffusant. For example, the kâ parameter for PEGs

Macromolecules, Vol. 36, No. 3, 2003
Poly(propyleneimine) Dendrimers 843
2
Ta ble 3. Self-Diffu sion Coefficien ts (D0), Hyd r od yn a m ic Ra d ii (Rh ), a n d F ittin g kâ a n d ν P a r a m eter s Obta in ed for th e
Den d r im er s in Aqu eou s P VA System s
D0 (10- m /s)
11
2
calcda
kâ2 a (10-11
)
νa
Rhb (nm)
rms error
c
sample
exptl
PPI(TEO)8
PPI(TEO)32
PPI(TEO)64
16.4 ( 0.2
9.1 ( 0.1
7.0 ( 0.3
16.4 ( 0.3
9.1 ( 0.1
6.9 ( 0.2
0.79 ( 0.09
0.16 ( 0.01
0.10 ( 0.02
0.59 ( 0.03
0.68 ( 0.02
0.69 ( 0.04
1.21
2.18
2.86
0.06
0.06
0.06
a
Obtained as a fitting parameter from eq 7. b Calculated from eq 5. c Root-mean-square errors of the nonlinear least-squares fitting to
eq 7.
2
F igu r e 4. Variation of the kâ parameter as a function of the
hydrodynamic radius of the dendrimers at 25 °C. The dashed
line is drawn only as visual guide.
with molecular weights of 600, 2000, and 4000 de-
-
11
-11
creased from 1.20 × 10
to 0.47 × 10 . To compare
2
two diffusants with similar molecular weights, the kâ
-11
parameter obtained for PPI(TEO)8 is 0.79 × 10
while
-
11
that for PEG-2000 is 0.53 × 10 , indicating that the
jump frequency for the dendrimer is somewhat higher
than that of the linear PEG.
The values of the ν parameter in Table 3 are in the
same range of the values (0.49-0.76 with an average
of ca. 0.58) reported by Masaro et al. for other diffusants
in the same polymer-solvent system under the same
0 h
F igu r e 5. Dependence of D and R of dendrimers (A) and
PEG (B) on their molecular weights at 25 °C.
4
,26,27
conditions.
This ν parameter seems to be charac-
teristic of a given polymer-solvent system and is
independent of the diffusing molecule.
The model describes well the variation of the self-
diffusion coefficient of the dendrimers in the PVA-
water-dendrimer ternary system. Theoretically, the
model of Petit et al. should be valid only when c is
superior or equal to the critical overlap concentration
and Rh as a function of the molecular weight. The values
of the slope obtained for both variations are identical
but have opposite signs (slope ) 0.40). This result
indicates that the density distribution is between a
fractal structure (slope ) 0.50) and a uniform density
4
4
distribution (slope ) 0.33). Apparently, the slope of
0.40 differs from that indicated in eq 4 (0.33), but it is
important to note that the dendrimers are neither
spherical in shape nor uniform in density. It would be
interesting to compare the result with that of linear
(
c*), the concentration at which intermolecular entangle-
ments between polymer chains can occur. As shown in
Figure 3, the obtained fitting curves with the model of
Petit et al. are continuous for the entire concentration
range of PVA, including the dilute regime.
2
2
PEGs from the literature. The data in Figure 5B show
the dependence of D0 and Rh on the molecular weight
for PEGs (M ) 600-10 000). The slope obtained for
these linear polymers is equal to 0.49, which indicates
that the linear diffusants such the PEGs have a fractal
density distribution. Understandably, the PEGs as
diffusants are less uniform in density than the den-
drimers used in this study.
Table 3 reports the effective hydrodynamic radius (Rh)
obtained for each dendrimer from the Stokes-Einstein
equation (eq 5). The hydrodynamic radii for our modified
dendrimers are somewhat larger than those obtained
by Rietveld et al.44 for PPI dendrimers with primary
amine end groups. The difference should correspond to
the extra length of the added TEO groups on the PPI
dendrimers.
The density distribution of dendrimers and linear
polymers coils depends on many factors including the
quality of the solvent, the temperature, the size of the
dendrimer, and the molecular weight of the polymer.
To obtain a qualitative evaluation of the density distri-
bution, D0 and Rh were plotted as a function of the
molecular weight. Figure 5A is a logarithmic plot of D0
(2) Effect of Tem p er a tu r e. The self-diffusion coef-
ficients of the modified poly(propyleneimine) dendrimers
were determined over a temperature range from 5 to
45 °C for eight different PVA concentrations. An ex-
ample of the results is shown in Figure 6A for the
variation of the self-diffusion coefficient of PPI(TEO)8
dendrimer with PVA concentration (from 0 to 0.26
g/mL). As expected, D increases with increasing tem-
perature, but the effect of temperature becomes less

8
44 Baille et al.
Macromolecules, Vol. 36, No. 3, 2003
0
F igu r e 7. Dependence of D of dendrimers on molecular
weight at three different temperatures. Squares, circles, and
triangles represent the values obtained at 5, 25, and 45 °C,
respectively.
7
. Similar results are obtained with the other dendrim-
ers used in this study.
Dashed lines in Figure 6 are the result of the fit with
the model of Petit et al. (eq 7). Excellent fitting is
obtained for the variation of the self-diffusion coefficient
with PVA concentration for all temperatures. The self-
diffusion coefficient at zero polymer concentration (D0)
was used to show the effect of the temperature on the
density distribution of these three dendrimers (Figure
F igu r e 6. (A) Self-diffusion coefficients and (B) normalized
self-diffusion coefficients (D/ D ) of the PPI(TEO) dendrimer
as a function of PVA concentration at five different tempera-
tures. Dashed lines are fits to eq 7.
0
8
7
). Figure 7 shows the variation of the logarithm of D0
as a function of the logarithm of molecular weight at
three different temperatures (5, 25, and 45 °C). The
values of the slope obtained are -0.39, -0.40, and -0.39
for 5, 25, and 45 °C, respectively. These results show
no effect of the temperature on the density distribution
of the dendrimers within this temperature range. From
the D0 values determined with the model of Petit et al.,
the hydrodynamic radius was calculated with the
Stokes-Einstein equation (eq 5) at different tempera-
significant with increasing PVA concentration. This
decrease is due to the intermolecular entanglements and
hydrogen bonding between polymer chains near and
above the critical overlap concentration (c*). In the case
of PVA-water system, the c* is mostly affected by the
molecular weight and the degree of hydrolysis, and its
value is between 1 and 5% (using the definition c* )
5-47
tures. The results in Figure 8A show a decrease of Rh
1
/[η]).4
It can also be viewed as the concentration of
with increasing temperature for these three dendrimers.
However, the decrease is more obvious for dendrimers
with higher molecular weight (PPI(TEO)32 and PPI-
the transition from the dilute regime, where the coils
are isolated, to a more concentrated state, where the
coils entangle. Above this c*, a fundamental change in
the physical properties of the polymer solutions occurs
(TEO)64). This may be due to the effect of the quality of
4
8,49
the solvent, which decreases with increasing tempera-
(
e.g., the viscosity behavior).
As shown previ-
5
1
4
5,50
ture. When the solvent quality decreases, the den-
drimers shrink, which leads to a decrease in Rh. This
phenomenon affects higher molecular weights first
because they are more sensitive to the solvent quality.
ously,
no critical effects have been observed by self-
diffusion measurements. The same behavior (i.e., the
reduction of the temperature effect on the self-diffusion
coefficient at higher PVA concentrations) was observed
for the two other dendrimers (PPI(TEO)32 and PPI-
(This phenomenon is used in the fractionation of poly-
mers.) Stechemesser et al. obtained the same result in
their study of the effect of solvent quality on the self-
diffusion coefficient of tetrafunctional poly(amidoamine)
dendrimers by holographic relaxation spectroscopy.51
However, this decrease with temperature is small
(
TEO)64). Moreover, the effect of the temperature on the
self-diffusion coefficient is less significant with increas-
ing molecular weight of the dendrimers (data not
shown). This result agrees well with eq 4, which links
the self-diffusion coefficient with temperature and the
inverse of the cube root of the molecular weight. It is
obvious from this equation that D should be less affected
by the same temperature variation when the molecular
weight is higher.
Another interesting observation is that the depen-
dence of the self-diffusion coefficients of the dendrimer
on temperature may be largely contained in the self-
diffusion coefficient of the diffusant in the pure solvent
(inferior or equal to 14%). The result in Figure 8A
indicates that the properties of these dendrimers may
be sensitive to temperature. Figure 8B shows the
variation of the ν parameter with temperature. This
solvent quality is slightly affected as shown by the small
decrease in the value of ν at a higher temperature, but
in general, the variation of ν with temperature and with
the diffusant is small as shown in Figure 8B, indicating
the parameter is characteristic of a given polymer-
solvent system.
(
D0). Figure 6B shows the normalized diffusion coef-
ficients (D/ D0) for the same PPI(TEO)8 dendrimer as a
function of polymer concentration at various tempera-
tures. The data points follow the same trend and are
seen to fall more or less on the same line fitted with eq
The variation of the self-diffusion coefficients with the
temperature can be used to determine the diffusional
activation energy (Ea) with an equation similar to the

Macromolecules, Vol. 36, No. 3, 2003
Poly(propyleneimine) Dendrimers 845
F igu r e 10. Variation of the diffusional activation energy with
the PVA concentration for three dendrimers: Squares, PPI-
(
TEO)
8
; circles, PPI(TEO)32; and triangles, PPI(TEO)64. The
dashed line is drawn only as visual guide.
h
F igu r e 8. Variation of the hydrodynamic radius (R ) (A) and
the ν parameter in the model of Petit et al. (B) with the
temperature for the three dendrimers. Squares, PPI(TEO)
circles, PPI(TEO)32; and triangles, PPI(TEO)64
8
;
.
2
F igu r e 11. Semilogarithmic plot of the kâ parameter as a
function of reciprocal temperature for the three dendrimers.
8
Squares, PPI(TEO) ; circles, PPI(TEO)32; and triangles, PPI-
(TEO)64
.
of the slope with the polymer concentration, which
implies an increase of the diffusional activation energy.
This increase is clearly visible in Figure 10, which shows
the rapid increase in the diffusional activation energy
with the polymer concentration. We can also see an
increase of this energy with the molecular weight for
each PVA concentration. The dashed line is drawn as
visual guides. All these results indicate that the diffus-
ing molecule requires more energy at higher molecular
weight and at higher polymer concentration to escape
its present surrounding and move into an adjacent
environment. As for the determination of diffusional
activation energy by the variation of the self-diffusion
coefficient with temperature, the same Arrhenius ap-
F igu r e 9. Semilogarithmic plot of the self-diffusion coef-
ficients of the PPI(TEO)32 dendrimer as a function of reciprocal
temperature for four PVA concentrations. Squares, 0.00 g/mL;
circles, 0.06 g/mL; upward triangles, 0.16 g/mL; and downward
triangles, 0.26 g/mL.
2
proach was used with the kâ parameter to obtain the
Arrhenius equation:
apparent activation energy (Figure 11), since we assume
that â is a constant within the temperature range
studied here. A good linear relationship is obtained for
all dendrimers. The apparent activation energies ob-
tained are 28.0, 36.1, and 40.9 kJ /mol for the PPI(TEO)8,
PPI(TEO)32, and PPI(TEO)64 dendrimers, respectively.
D(T) ) D exp(-E /RT)
(9)
∞
a
where D∞ is the self-diffusion coefficient at infinite
temperature. Figure 9 shows the temperature depen-
dence of the PPI(TEO)32 dendrimer as a function of the
inverse of the temperature for four different PVA
concentrations. The Arrhenius equation allows a good
fit to the variation of D, observed for all regimes
1
1
According to the model of Petit et al., these results
confirm that the larger diffusing molecule has a lower
jump frequency and thus needs an higher energy to
escape its present surrounding to diffuse in the poly-
meric network. In comparison to the results obtained
(
straight lines in Figure 9), and shows the accuracy of
this approach. Figure 9 also shows a continuous increase
4
by Masaro et al. with PEG diffusants, the energy

8
46 Baille et al.
Macromolecules, Vol. 36, No. 3, 2003
barriers of the dendrimers can be related to that
obtained for ethylene glycol (Rh ) 0.24 nm) and two
PEG samples with molecular weights of 600 and 2000
(3) Gao, P.; Fagerness, P. E. Pharm. Res. 1995, 12, 955-
9
64.
(
4) Masaro, L.; Zhu, X. X.; Macdonald, P. M. Macromolecules
998, 31, 3880-3885.
1
(Rh ) 1.25 and 2.27 nm, respectively), which have Ea
(
5) Fujita, H. Adv. Polym. Sci. 1961, 3, 1-47.
6) Yasuda, H.; Lamaze, C. E.; Ikenberry, L. D. Makromol. Chem.
1968, 118, 19-35.
values of 30.0, 36.5, and 39.0 kJ /mol, respectively.
Although the dendrimers used here generally possess
higher molecular weights than the linear PEGs studied,
the energy barriers for the dendrimers are much lower
in comparison with the linear PEGs. For the one that
is comparable in molecular weight, PPI(TEO)8 has a
similar molecular weight as PEG-2000, but its energy
barrier is much lower (28 kJ /mol for PPI(TEO)8 vs 39
kJ /mol for PEG-2000). It is possible that ethylene glycol
and PEG can form hydrogen bonds more easily with the
polymer matrix and with the aqueous surrounding in
comparison to the dendrimers. For each dendrimer, the
activation energy value obtained with the model of Petit
et al. (Figure 11) is higher than the diffusional activa-
tion energy (Figure 10). The difference is not yet well
understood. However, it seems that the apparent acti-
(
(7) Vrentas, J . S.; Duda, J . L. J . Polym. Sci., Polym. Phys. Ed.
977, 15, 403-416.
8) Vrentas, J . S.; Duda, J . L. J . Polym. Sci., Polym. Phys. Ed.
977, 15, 417-439.
9) Phillies, G. D. J . Macromolecules 1986, 19, 2367-2376.
1
(
(
1
(
10) Vrentas, J . S.; Vrentas, C. M. Macromolecules 1994, 27,
4684-4690.
11) Petit, J .-M.; Roux, B.; Zhu, X. X.; Macdonald, P. M. Macro-
molecules 1996, 29, 6031-6036.
(
(
(
12) Amsden, B. Macromolecules 1998, 31, 8382-8395.
13) Masaro, L.; Zhu, X. X. Prog. Polym. Sci. 1999, 24, 731-
7
75.
(14) Nesmelova, I. V.; Skirda, V. D.; Fedotov, V. D. Biopolymers
2002, 63, 132-140.
15) Seland, J . G.; Ottaviani, M.; Hafskjold, Bjørn J . Colloid
(
(
(
Interface Sci. 2001, 239, 168-177.
16) Derrick, T. S.; Larive, C. K. Appl. Spectrosc. 1999, 53, 1595-
2
vation energy obtained from k can be related to the
1
600.
maximum energy that the diffusing molecule has to
overcome to diffuse in the polymer matrix at gel polymer
concentration.
17) Penke, B.; Kinsey, S.; Gibbs, S. J .; Moerland, T. S.; Locke, B.
R. J . Magn. Reson. 1998, 132, 240-254.
(18) Manz, B.; Callaghan, P. T. Macromolecules 1997, 30, 3309-
3
316.
(
19) Yokoyama, H.; Kramer, E. J .; Fredrickson, G. H. Macromol-
Con clu sion
ecules 2000, 33, 2249-2257.
It is clearly shown that the self-diffusion coefficients
of the dendrimers decrease with increasing molecular
size of the diffusant, with increasing PVA concentration
but with a decreasing temperature. The physical model
of Petit et al. has been used successfully to describe the
variation of the self-diffusion coefficient with the mo-
lecular size of the dendrimer, the polymer concentration,
and the temperature. These results show that the larger
diffusing molecule needs higher activation energy to
escape its present surrounding and to move into an
adjacent environment. It is also the case when the
polymer concentration increases. The apparent activa-
tion energy of diffusion varies in the range 28.0-40.9
kJ /mol from PPI(TEO)8 to PPI(TEO)64 dendrimers. The
study shows that the dendrimers have a density distri-
bution between a fractal structure and a uniform
density distribution and that the temperature in the
range 5-45 °C has no effect on the density distribution.
In comparison to the fractal structure obtained for linear
PEGs, dendrimers have a more uniform density distri-
bution. The Stokes-Einstein hard-sphere radii have
also been calculated at the zero concentration limit, and
we have observed an increase of Rh with the dendrimer
generation. However, the results show a slow decrease
in Rh with the temperature for these three dendrimers.
The study of the motion of different parts of the
dendrimers by NMR relaxation time measurements
shows that the terminal protons are more mobile than
the core protons for all dendrimers. The relaxation time
measurements also show a decrease in the mobility for
all protons along with increasing dendrimer generation.
(20) Nyd e´ n, M.; S o¨ derman, O. Macromolecules 1998, 31, 4990-
5002.
(
(
(
21) Waggoner, R. A.; Blum, F. D.; MacElroy, J . M. D. Macromol-
ecules 1993, 26, 6841-6848.
22) Baille, W. E.; Malveau, C.; Zhu, X. X.; Marchessault, R. H.
Biomacromolecules 2002, 3, 214-218.
23) Malveau, C.; Beaume, F.; Germain, Y.; Canet, D. J . Polym.
Sci., Part B: Polym. Phys. 2001, 39, 2781-2792.
(24) Rollet, A.-L.; Simonin, J .-P.; Turq, P.; Gebel, G.; Kahn, R.;
Vandais, A.; No e¨ l, J .-P.; Malveau, C.; Canet, D. J . Phys.
Chem. B 2001, 105, 4503-4509.
(25) Petit, J .-M.; Zhu, X. X.; Macdonald, P. M. Macromolecules
1
996, 29, 70-76.
(26) Masaro, L.; Ousalem, M.; Baille, W. E.; Lessard, D.; Zhu, X.
X. Macromolecules 1999, 32, 4375-4382.
(
(
(
27) Masaro, L.; Zhu, X. X.; Macdonald, P. M. J . Polym. Sci., Part
B: Polym. Phys. 1999, 37, 2396-2403.
28) Masaro, L.; Zhu, X. X. Macromolecules 1999, 32, 5383-
5
390.
29) Yonetake, K.; Masuko, T.; Morishita, T.; Suzuki, K.; Ueda,
M.; Nagahata, R. Macromolecules 1999, 32, 6578-6586.
(30) Uppuluri, S.; Keinath, S. E.; Tomalia, D. A.; Dvornic, P. R.
Macromolecules 1998, 31, 4498-4510.
(31) Scherrenberg, R.; Coussens, B.; van Vliet, P.; Edouard, G.;
Brackman, J .; de Brabander, E. Macromolecules 1998, 31,
4
56-461.
(32) Roover, J .; Zhou, L.-L.; Toporowski, P. M.; van der Zwan, M.;
Iatrou, H.; Hadjichristidis, N. Macromolecules 1993, 26,
4
324-4331.
33) Pan, Y.; Ford, W. T. Macromolecules 1999, 32, 5468-
470.
(
(
5
34) Chai, M.; Niu, Y.; Youngs, W. J .; Rinaldi, P. L. J . Am. Chem.
Soc. 2001, 123, 4670-4678.
(35) Pan, Y.; Ford, W. T. Macromolecules 2000, 33, 3731-
3
738.
(36) Kreider, J . L.; Ford, W. T. J . Polym. Sci., Part A: Polym.
Chem. 2001, 39, 821-832.
(
(
37) Tanner, J . E. J . Chem. Phys. 1970, 52, 2523-2526.
38) Callaghan, P. T.; Trotter, C. M.; J olley, K. W. J . Magn. Reson.
1980, 37, 247-259.
Ack n ow led gm en t. The financial support from the
Natural Sciences and Engineering Research Council
(
NSERC) of Canada and the National Science Founda-
(39) Stilbs, P. Prog. Nucl. Magn. Reson. Spectrosc. 1987, 19, 1-45.
(40) Price, W. S. Concepts Magn. Reson. 1997, 9, 299-336.
tion (USA) is gratefully acknowledged.
(41) Amman, C.; Meier, P.; Merbach, A. E. J . Magn. Reson. 1982,
4
6, 319-322.
Refer en ces a n d Notes
(
42) Waldeck, A. R.; Kuchel, P. W.; Lennon, A. J .; Chapman, B.
(
1) Hariharam, D.; Peppas, N. A. J . Controlled Release 1993, 23,
E. Prog. Nucl. Magn. Reson. Spectrosc. 1997, 30, 39-
1
23-136.
68.
(2) Clericuzio, M.; Parker, W. O.; Soprani, M.; Andrei, M. Solid
State Ionics 1995, 82, 179-192.
(43) De Gennes, P. G. Scaling Concepts in Polymer Physics;
Cornell University Press: Ithaca, NY, 1979.

Macromolecules, Vol. 36, No. 3, 2003
Poly(propyleneimine) Dendrimers 847
(
(
(
44) Rietveld, I. B.; Bedeaux, D. Macromolecules 2000, 33, 7912-
917.
45) Cosgrove, T.; Griffiths, P. C. Polymer 1994, 35, 509-
13.
46) Fang, L.; Brown, W. Macromolecules 1990, 23, 3284-
290.
(49) Flory, P. J . Principles of Polymer Chemistry, Cornell Univer-
sity Press: Ithaca, NY, 1953.
(50) Skirda, V. D.; Sundukov, V. I.; Maklakov, A. I.; Zgadzai, O.
E.; Gaturov, R. R.; Vasiljiev, G. I. Polymer 1988, 29, 1294-
1300.
7
5
3
(
51) Stechemesser, S.; Eimer, W. Macromolecules 1997, 30,
204-2206.
(
47) Te Nijenhuis, K. Adv. Polym. Sci. 1997, 130, 36-52.
48) Ferry, J . D. Viscoelastic Properties of Polymers; Wiley: New
York, 1980.
2
(
MA025636K