Modeling of low-temperature depolymerization of poly(methyl methacrylate) promoted by ion beam

Article abstract of DOI:10.1063/1.479432

The influence of ion beam in the low-temperature depolymerization of polymethyl methacrylate (PMMA) was studied. The diffusion coefficient of small molecules formed in PMMA at low temperature was analyzed by providing three analytical solutions, two continuous models and a discrete model. The pulsed ion beam technique was found to be useful in extracting diffusion coefficient values, as well as for measuring the depolymerization kinetics at a temperature well below that required for suitable thermal initiator formation.

Full text of DOI:10.1063/1.479432

Modeling of low-temperature depolymerization of poly (methyl methacrylate) promoted
by ion beam
A. Raudino, M. E. Fragalà, G. Compagnini, and O. Puglisi
Citation: The Journal of Chemical Physics 111, 1721 (1999); doi: 10.1063/1.479432
View online: http://dx.doi.org/10.1063/1.479432
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JOURNAL OF CHEMICAL PHYSICS
VOLUME 111, NUMBER 4
22 JULY 1999
Modeling of low-temperature depolymerization of poly „methyl methacrylate…
promoted by ion beam
a)
A. Raudino, M. E. Fragal a` , G. Compagnini, and O. Puglisi
Dipartimento di Scienze Chimiche dell’Universit a` di Catania, Viale Andrea Doria, 6-95125 Catania, Italy
͑
Received 14 January 1999; accepted 20 April 1999͒
The role played by the ion beam in the depolymerization of poly͑methyl methacrylate͒ ͑PMMA͒ is
the creation of initiators for the unzipping reaction at relatively low temperatures ͓lower than the
ceiling temperature of PMMA (T ϭ220 °C)͔. Monomer ͑MMA͒ evolution out of the polymeric
C
film is thus governed by the kinetics of the chemical reaction and also by the diffusion of monomer
inside the polymeric film. In this work this phenomenon has been described by using simple models
in which both one- and two-dimension monomer diffusion is considered. Exact analytical solutions
of the models are provided. The theory fit well the experimental data and allows for the
determination of important parameters such as the kinetics constant of the depolymerization and the
monomer diffusion coefficient, D, inside the polymer film. Thus, the pulsed ion beam technique,
here modeled, is useful not only for extracting D values of slow diffusants in polymers, but even for
measuring kinetics constants of depolymerization at a temperature well below that needed for the
formation of suitable thermal initiators. © 1999 American Institute of Physics.
͓
S0021-9606͑99͒50727-7͔
I. INTRODUCTION
used in order to get an easier interpretation of the results.
Among these, we mention the use of very thin polymeric
film ͑deposited on silicon substrate͒ as target ͑thickness Ͻ1
Current interest for radiation effects in polymers is due
to the fact that the most important applications of macromo-
lecular solids such as resist, imaging, coating, and aerospace
materials, require a detailed knowledge of the behavior of
these materials under a great variety of radiations, spanning
␮m with He ions at some hundreds keV energy͒ in order to
allow the projectile come to stop within the substrate. This in
turn means that most of the energy released by the bombard-
ing ion traveling through the film is used to excite and/or
ionize the electrons of the macromolecules while only a neg-
ligible amount of energy is lost via ballistic energy loss
from ultraviolet ͑UV͒, ␥-ray, electrons, high-energy ions,
and oxygen atom bombardment in the upper atmosphere.1
Understanding the fundamental aspects of these interactions
is significant from the basic point of view and plays, at the
same time, a critical role in the development of many mod-
ern technologies such as microelectronics and aerospace ap-
plications.
The role of the sample temperature in these studies is
crucial. We have recently identified two temperature regimes
in the behavior of Poly͑methylmethacrylate͒ ͑PMMA͒ under
ion beam irradiation:2 ͑i͒ A low-temperature regime ͑tem-
6
events. Moreover, in order to minimize track overlap, all the
experiments are performed at very low fluence (Ͻ5
12
Ϫ2
ϫ10 ions cm ). In such a way the probability that a given
ion hits a region of the target already bombarded by previous
ions is very low ͑single track regime͒.7 Figures 1͑a͒ and
1͑b͒ show a typical experiment during which a pulse of ions
is allowed to hit a PMMA thin film ͑8600 Å thick͒. A quad-
rupole mass spectrometer, tuned at the molecular weight
mass of the MMA monomer (m/Zϭ100) follows the mono-
mer evolution during the pulse. At a temperature lower than
Tg no monomer formation is detected ͓Fig. 1͑a͔͒, while at
higher temperature the beam induces a polymeric unzipping
͓Fig. 1͑b͔͒. In particular, during the ion pulse, the monomer
signal intensity first increases, reaches a maximum and then
slowly decreases. When the beam is switched off the signal
drops rapidly to zero. Clearly during the step increase of the
signal we have to consider that monomers are formed inside
the macromolecular film with a given kinetics and then dif-
fuse away from the sample surface; therefore, we can hope to
extract information on both kinetics and diffusion coeffi-
cients from these data. Of course, when the beam is switched
off, the decrease of the signal is only due to MMA diffusion.
From a careful observation of the data we note that at high
irradiation time (tϾ10 s) there is a slight decrease of the
signal instead of reaching a stable plateau. In this paper we
–9
,3
perature lower than the glass transition temperature, T , of
g
the macromolecular solid͒, where the chemical perturbation
induced by the energy release of ion is limited to the ester
pendent groups and ͑ii͒ a high-temperature regime where the
perturbation propagates through the macromolecular main
chain. In the first case one observes the formation of small
gaseous molecules, main chain scission, formation of double
4
,5
bonds and decrease of the molecular weight of the sample.
In these cases, even at the highest beam energy and fluence,
no trace of the methyl methacrylate monomer ͑MMA͒ could
be observed. On the contrary, under irradiation at high tem-
perature (TϾT ), one observes a catastrophic depolymeriza-
g
tion of the sample, yielding considerable amount of MMA
monomer ͑unzipping͒.
In such studies, typical experimental arrangements are
a͒Electronic mail: gbuemi@dipchi.unict.it
0021-9606/99/111(4)/1721/11/$15.00
1721
© 1999 American Institute of Physics
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1

1722
J. Chem. Phys., Vol. 111, No. 4, 22 July 1999
Raudino et al.
useful when dealing with very small D values (D
Ϫ10
2 Ϫ1
Ͻ10
cm s ).
In the present study we plan to build physico-chemical
models of MMA formation and diffusion during pulsed ion
beam irradiation of thin film of PMMA. The aim is that of
extracting information on both MMA diffusion coefficient
and kinetics constant from data. Exact analytical solutions
will be provided for all the presented cases.
The paper is organized as follows. In Sec. III A we try to
build a very simple model where it is assumed that MMA
formation can be adequately described by one ͑overall͒ ki-
netics constant. Moreover the hypothesis is made that mono-
mer is formed everywhere inside the sample, i.e., a spatially
uniform source is assumed. This model has been improved
later by considering that the overall chemical reaction is split
into two steps: ͑i͒ The formation of suitable initiators and ͑ii͒
the depolymerization reaction propagation step. In Sec. III B
we consider that the ion beam induced chemical reactions do
really occur within some physically well restricted regions of
the sample: the ion ͑latent͒ tracks. All the above models will
be compared with experimental data at short irradiation
times. Finally, in the last Sec. III C we try to compare the
model predictions with the data collected at longer times,
where the film thickness changes and track overlap are no
more negligible. As mentioned before, exact analytical solu-
tions for all models will be provided, allowing to extract
kinetics and diffusional informations from pulsed ion experi-
ments where gaseous products are formed.
II. EXPERIMENT
Thin films of PMMA (M ϭ30 000 Dalton͒ are obtained
w
by spin coating on silicon substrate of chloroform solutions
͑7% in weight͒. The thickness of the films are measured by
ellipsometry or by a stylus profilometer. The micrometric
scan performed on the film surface by the profilometer stylus
controls the roughness of the pristine sample detecting an
2
almost flat surface. 4 cm squared samples ͑8600 Å thick͒
ϩ
have been irradiated at different temperature with He ions
having an energy of 300 keV using an ion current of 0.01
␮
A. An in situ mass analysis has been performed in order to
FIG. 1. Monomer ͑MMA͒ evolution during an ion pulse of 20 s at ͑a͒ T
ϭ110 °C and ͑b͒ Tϭ170 °C.
study the monomer, MMA ͑molecular peak at m/Zϭ100),
evolution during the irradiation.11 The mass spectrometer has
an electron impact energy of 50 keV and works in a mass
range of 1–300 amu with a resolution of ϳ1 amu. The work-
Ϫ7
will show that this is due, at least in part, to the fact that a
reduced film thickness and other effects essentially ascribed
to polymer ion beam induced damage, decrease the monomer
production rate.
ing pressure in the irradiation chamber was of ϳ10 Torr.
The substrate temperature was controlled by a thermocouple
placed close to the sample. The selected temperatures are
higher than the glass transition temperature of PMMA (Tg
ϭ105 °C) but lower than its ceiling temperature (TC
ϭ220 °C). A calibration procedure was used for a quantita-
tive determination of the monomer evolution after each treat-
ment. Further details on the experimental procedure can be
found elsewhere.12 A measure of the residual thickness after
irradiation has been performed by using the stylus profilo-
meter. Depending on the ion fluence and the temperature
regime small changes in the film thickness have been de-
tected only in the high fluence regime ͑see Sec. III C͒ with
no significant variations of the sample roughness.
It is to note that pulsed beam experiments similar to
those here studied have been already reported by Venkatesan
1
0
et al. In that study the technique was employed to extract
the diffusion coefficient of small molecules formed in
PMMA at low-temperature (TϽT ) by fitting the experi-
g
mental data to a simple numerical model that takes into ac-
count the diffusion coefficient of the species produced by the
beam. This technique is very useful for measuring diffusion
coefficients of slow diffusants in polymers, and, though
it could appear rather ‘‘sophisticated,’’ it is extremely
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J. Chem. Phys., Vol. 111, No. 4, 22 July 1999
Depolymerization promoted by ion beam
1723
III. RESULTS
In previous works2,3 we have studied the effects of ion
beam on promoting PMMA depolymerization in a tempera-
ture range between T (105 °C) and T (220 °C). Typically,
g
C
a quadrupole mass spectrometer is employed to perform the
mass spectrum of the gaseous products emitted by the poly-
mer during ion bombardment. The resulting mass spectrum
is a complex one and it is composed by a series of peaks
attributable to the molecular peak and to the relative frag-
mentations of the MMA monomer and other small molecules
emitted by PMMA during bombardment, such as
4
H , CH , CO, CO , HCOOCH .
2
4
2
3
The peak at m/Zϭ100 corresponds to the molecular
peak of MMA monomer. Its intensity can be easily corre-
lated to the MMA mole evolution rate by means of a cali-
bration procedure.12 The detected monomer evolution rate
must be ascribed both to the depolymerization reaction, ini-
tiated by the ion beam, and to the MMA diffusion through
the polymeric matrix. In the first section of this work we
want to present two reaction-diffusion models in which one-
dimensional MMA diffusion is hypothesized. In these mod-
els we do not consider the ion track presence and a homoge-
neous monomer concentration is supposed inside the whole
film thickness. The difference between these two models is
that in the first one we do not distinguish between the differ-
13,14
ent depolymerization reaction steps,
while in the second
we assume a slow initiation step of reaction followed by a
faster propagation one. From the solutions of the reaction-
diffusion equations it is possible to determine the kinetics
constants and the diffusion coefficient of the studied diffu-
sant molecule.
FIG. 2. Schematic drawing of the proposed theoretical models: ͑a͒ Model 1;
b͒ Model 2; ͑c͒ Model 3.
͑
which neglects the diffusion arising from the monomer con-
centration gradient between the ion tracks and the surround-
ing polymer matrix.
A. Continuum models
Assuming the Fick’s law for the one-dimensional diffu-
sion process along z, we can describe the monomer inside the
polymer film like a gas which is formed at a constant rate
and then diffuses through a homogeneous matrix with an
appropriate diffusion coefficient, D ͓Fig. 2͑a͔͒, which is as-
sumed to be constant during the monomer evolution time
Initial and boundary conditions are
c
͉zϭz0ϭ0,
͑3a͒
͑3b͒
ץc
ץz
ͯ
ϭ0,
zϭ0
2
ץc
ץt
ץ c
c
͉
ϭ0,
͑3c͒
͑3d͒
ϭDͩ ͪϩKf͑t͒,
͑1͒
tϭ0
2
ץz
lim cϭ0,
t→ϱ
where K is the rate of monomer formation and f(t) is a
function of the irradiation time which describes the depoly-
merization reaction induced by the ion beam. This step func-
tion suddenly goes to zero when the beam is turned off ͑at
tϾt , with t the irradiation time͒, and is unitary for tϽt
where Eq. ͑3a͒ describes the complete evaporation of the
monomer at the film–vacuum interface, zϭz ͑no substantial
0
retrapping of the escaped monomers reaction is allowed͒,
while Eq. ͑3b͒ imposes monomer reflection at the polymer
0
0
0
film-inert support (SiO ) interface. Conditions ͑3c͒ and ͑3d͒
f͑t͒ϭ1ϪH͑tϪt ͒,
͑2͒
2
0
ensure vanishing monomer concentration inside the polymer
film before and after long time following the irradiation, re-
spectively. Solution to Eq. ͑1͒ describes the depolymeriza-
tion kinetics in the presence of the ion beam and is devel-
oped in the appendix. The final equations for the flux of
monomer J(t) escaping from the film–vacuum interface are
for tϽt0
H(t) being the Heaviside step function.
Strictly speaking we should consider the lateral compo-
nent to the diffusion due to the concentration gradient of
monomer ͑MMA͒ inside the ion tracks. In a first approach
we neglect the lateral diffusion and work with a one-
dimensional model ͑a more refined model that accounts for
the lateral diffusion is developed in Sec. III B͒. Other
ϱ
1
0
8
1
authors have proposed a numerical model to calculate the
diffusion coefficient of small molecules in thin polymer films
Jϩ͑t͒ϭAKz0 1Ϫ
exp͑Ϫ␣ t͒ , ͑4a͒
ͫ
␲2 nϭ1 ͑2nϪ1͒2
͚
n
ͬ
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1724
J. Chem. Phys., Vol. 111, No. 4, 22 July 1999
Raudino et al.
while for tϾt0
ϱ
8
␲2
1
JϪ͑t͒ϭAKz
0
ͫ
͚
nϭ1 ͑2nϪ1͒2
ϫexp͑Ϫ␣ t͒͑exp͑␣ t ͒Ϫ1͒ ,
͑4b͒
͑5͒
n
n 0
ͬ
where
␣nϭ͑2nϪ1͒2
␲2
D
2
0
.
4
z
A is the surface of the polymeric film, z is its initial thick-
0
ness and t is the irradiation time.
0
The value of K is derived considering that the total
amount of emitted monomers ͑i.e., the area S of the MMA
signal ͓see Fig. 1͑b͔͒ can be calculated as
ϱ
Sϭ
͵
J͑t͒dt.
͑6͒
0
Inserting Eq. ͑4͒ into the above expression we obtain after
some algebra the exact expression
t
ϱ
0
Sϭ
͵
Jϩ͑t͒dtϩ
͵
J ͑t͒dtϭAKz t ,
͑7͒
Ϫ
0 0
0
t
0
from which the kinetics constant K can be immediately ob-
tained. As intuitively expected, the amount of produced
monomers, S, is independent of the diffusion coefficient D.
Even if we have used a simple but exactly solvable
model ͑which neglects the lateral diffusion contribution and
considers only a global K that does not distinguish between
the initiation and propagation stages͒, the agreement between
the theoretical and experimental data is quite good, as it can
be seen from Figs. 3͑a͒–3͑c͒ where is reported the monomer
ϩ
mass signal of samples irradiated with He 300 keV for 20 s
at 135 °C, for 15 s at 150 °C and for 15 s at 170 °C. In order
to propose a more realistic model which better mimics the
rather complex depolymerization kinetics a second model
has been developed.
FIG. 3. Experimental ͑thin line͒ and theoretical ͑thick line͒ monomer evo-
lution rate calculated by using the Model 1 for: ͑a͒ 20 s ion irradiation at
Tϭ135 °C, ͑b͒ 15 s ion irradiation at Tϭ150 °C, and ͑c͒ 15 s ion irradia-
tion at Tϭ170 °C.
Here we consider the role played by the initiators pro-
duced inside the ion tracks. To do this we assume that free
radicals are formed inside the film during bombardment and
that they initiate the unzipping process
ץcI
ϭK f͑t͒ϪK c ,
͑8a͒
͑8b͒
i
p I
ץt
Irradiation
2
ץcM
ץ c
M
ץz2
͓
͓
Polymer͔N ——→ ͓Polymer͔NϪmϩ͓Radical͔m ,
ϭD
ͩ
ͪ
ϩ␤K c ,
p I
ץt
Unzipping
Radical͔m ——→ Monomerϩ͓Radical͔mϪ1
.
where c and c are the concentrations of the initiators and
1
M
of monomers, D the monomer diffusion coefficient, ␤ the
number of monomeric units produced by a single initiator,
and f(t) is the step function defined through Eq. ͑2͒ which
describes the formation of initiators under ion beam. Initial
and boundary conditions read
N is the initial number of monomeric units and m is the
polymerization degree of the polymer radical formed by the
initiators (mϽN). In this way we can distinguish between
initiation kinetics constant, K , and propagation kinetics
i
constant, K . Even in this second model the monomer for-
p
mation is homogeneous throughout the film, hence the diffu-
sion motion occurs only along the z axis ͓Fig. 2͑b͔͒. Since
the polymer fraction is practically unitary, the initiators sud-
denly react with the surrounding polymer chains, allowing
one to neglect the diffusion of the initiators in comparison
with that of the products. The constitutive equations are
c
͉
ϭ0,
͑9a͒
M zϭz
0
ץcM
ץz
ͯ
ϭ0,
͑9b͒
͑9c͒
zϭ0
c
͉
ϭ0,
M tϭ0
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J. Chem. Phys., Vol. 111, No. 4, 22 July 1999
Depolymerization promoted by ion beam
1725
lim c ϭ0,
͑9d͒
͑9e͒
M
t→ϱ
lim c ϭ0,
I
t→ϱ
which are an obvious generalization of Eqs. ͑3͒ of Model 1.
The final equations for the monomer flux across the
polymer–vacuum interface have been derived in the appen-
dix and are
ϱ
8
␲2
1
1
J ͑t͒ϭA␤K z 1Ϫ
ϩ
i
0
ͫ
͚
2
nϭ1 ͑2nϪ1͒ ͑1Ϫ␣ /K ͒
n
p
␣n
ϫ
ͫ
exp͑Ϫ␣ t͒Ϫ
exp͑ϪK t͒
ͬ
,
͑10a͒
n
p
ͬ
Kp
for tϽt0
while
ϱ
8
␲2
1
1
J ͑t͒ϭA␤K z
Ϫ
i
0
ͫ
͚
2
nϭ1 ͑2nϪ1͒ ͑1Ϫ␣ /K ͒
n
p
␣n
ϫ
ͫ
exp͑Ϫ␣ t͒͑exp͑␣ t ͒Ϫ1͒Ϫ
n
n 0
Kp
ϫexp͑ϪK t͒͑exp͑K t ͒Ϫ1͒
ͬ
,
͑10b͒
p
p 0
ͬ
for tϾt0
␣n being defined by Eq. ͑5͒. Following the same procedure
developed for the previous model, the total amount of
formed monomers ͑proportional to the area of the signal͒
turns out to be
t
ϱ
0
Sϭ
͵
Jϩ͑t͒dtϩ
͵
J ͑t͒dtϭA␤K z t ,
͑11͒
Ϫ
i 0 0
0
t
0
which does not depend any longer on the diffusion rate D
FIG. 4. Experimental ͑thin line͒ and theoretical ͑thick line͒ monomer evo-
lution rate calculated by using the Model 2 for: ͑a͒ 20 s ion irradiation at
Tϭ135 °C, ͑b͒ 15 s ion irradiation at Tϭ150 °C, and ͑c͒ 15 s ion irradia-
tion at Tϭ170 °C.
and propagation constant K , but is proportional to the ini-
p
tiation rate K and to the efficiency of chain unzipping, ␤.
i
Thus ␤ represents the average number of monomer units that
can be produced by unzipping of a polymer chain. So, if we
assume that the efficiency of chain unzipping is maximum, it
follows that ␤ equates the polymerization degree of the poly-
mer.
Model 2 ͑Model 1 is obtained as the limit of Model 2 when
the propagation rate K goes to infinity͒ and assume that at
p
tϭ␶ (0Ͻ␶Ͻt ) a few not overlapping cylindrical tracks of
0
The better correlation of the experimental data with the
second model is clearly evidenced in Figs. 4͑a͒–4͑c͒ where
for the sake of comparison are reported the same experimen-
tal data shown in Fig. 3. The models described so far refer to
a homogeneous formation of monomer inside the film fol-
lowed by a one dimensional diffusion of the monomer to-
ward the film surface. In reality the initiators are created only
inside the ion track giving rise to a monomer concentration
gradient around the ion track. In the following third model
we take into account this effect by considering a two dimen-
sional diffusion equation.
radius a and height z are randomly formed within the film
0
͓see Figs. 2͑c͒ and 5͔. Inside the cylinder the initial concen-
tration of initiators c is Q. They rapidly react with the poly-
1
mer producing monomers which diffuse both laterally ͑inside
the film͒ and perpendicularly ͑toward the surface͒. After the
track formation (tу␶) the monomer evolution can be de-
scribed by the two-dimensional reaction-diffusion equation.
2
2
ץcM
ץt
ץ c
1 ץcM ץ c
M
M
ϭD
ͩ
ͭ
ϩ
ϩ
ͪ
ץr2
r ץr
ץz2
␤
KpcI 0ϽrϽ␣
aϽrϽR
ϩ
,
͑12͒
B. Discrete model
0
Both the previous models can be improved accounting
for the lateral diffusion of the monomers. We start from the
where the last term implies substantial monomer formation
only inside the track (0ϽrϽa) and R is the ‘‘diffusional
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1726
J. Chem. Phys., Vol. 111, No. 4, 22 July 1999
Raudino et al.
R from the track origin the lateral outgoing flux of monomers
must be balanced ͑on the average͒ by the ingoing flux of
monomers coming from the surrounding tracks ͑see Fig. 5͒.
The general solutions to Eq. ͑12͒ which satisfy the boundary
conditions ͑15a͒ and ͑15b͒ are
ϪK tЈ
p
c ϭ␤K Qe
,
͑16a͒
I
p
ϱ
ϱ
␦
c ϵc Ϫc ϭ
Anm͑tЈ͒cos͑␭ z͒J ͑␤ r͒,
n 0 m
M
M
M
͚ ͚
nϭ1 mϭ1
͑
16b͒
where
tЈϭtϪ␶,
FIG. 5. Ion track coordinates.
␲
␭n
ϭ
͑2nϪ1͒,
2
z0
and J (␤ r) is a Bessel’s function of zeroth order. By ap-
plying the boundary condition ͑15c͒ it follows that the dif-
ferent ␤m are the roots of the equation
size’’ of the track which is proportional to the track forma-
tion rate per unit time ͑see below͒. R can be estimated by
imposing that the number of ions per unit time, N, crossing
0
m
2
the polymer surface times the area ␲R around a single track
ץJ ͑␤ r͒
0
m
must be equal to the total surface area A of the polymer film:
ͯ
ϭϪ␤mJ1͑␤mR͒ϭ0.
͑17͒
2
ץr
N•␲R ϭA ͑see Fig. 5͒. Therefore, the basic assumption is
rϭR
that the lateral monomer concentration coming from the
tracks formed at early times, tϽ␶, is uniformly smeared out
over the sample and therefore does not contribute to the lat-
eral flux.
Finally the coefficients Anm(tЈ) are to be determined by im-
posing the initial conditions ͓Eq. ͑14͔͒. The calculation of
Anm(tЈ) is attained as follows. We first insert Eqs. ͑16a͒ and
͑16b͒ into Eq. ͑12͒ obtaining
The initiator concentration c is ruled by the equations
1
ϱ
ϱ
ץcI
ץAnm͑tЈ͒
ץtЈ
2
2
ϭϪK c , 0ϽrϽa, tϾ␶,
͑13a͒ ͚ ͚
ͫ
ϩD͑␭ ϩ␤ ͒Anm͑tЈ͒
ͬ
p
I
n
m
ץt
nϭ1 mϭ1
cIϭ0, aϽrϽR, tϾ␶.
Initial conditions at tϭ␶ are
͑13b͒
ϪKptЈ
␤K Qe
0ϽrϽa
p
ϫcos͑␭ z͒J ͑␤ r͒ϭ
.
͑18͒
n
0
m
ͭ
0
aϽrϽR
c
͉
ϭcM,
͑14a͒
͑14b͒
M tϭ␶
Multiplying both sides of Eq. ͑18͒ by J (␤ r)cos(␭ z),
integrating over the film volume element rdrdzd␾ in the
range 0ϽrϽR, 0ϽzϽz0 , and 0Ͻ␾Ͻ2␲ and making use
0
m
n
Ј
Ј
Q
0ϽrϽa
aϽrϽR
c
͉
ϭ
,
I tϭ␶
ͭ
0
15
of the integral identities
where c ϵc (z,t) is the average monomer concentration
M
M
R2
R
coming from the surrounding tracks which, as already said,
is assumed to be independent of r and ␾. Equation ͑14b͒
imposes that the initiators are confined within a cylinder of
radius a just after the track formation at tϭ␶. Boundary
conditions for cM are
2
͵
J0͑␤mr͒J0͑␤m r͒rdrϭ
J1͑␤mR͒␦m,m ,
͑19a͒
͑19b͒
Ј
Ј
2
0
a
a
͵
J0͑␤mr͒rdrϭ
J1͑␤ma͒,
␤m
0
c
͉
ϭ0,
͑15a͒
M zϭz
we find
0
ץcM
ץz
ץAnm͑tЈ͒
2
2
ϪK tЈ
p
ͯ
ͯ
ϭ0,
͑15b͒
ϩD͑␭ ϩ␤ ͒Anm͑tЈ͒ϭQnm
e
.
͑20͒
͑21͒
n
m
ץtЈ
Qnm being defined as
͑Ϫ1͒n
zϭ0
rϭR
ץcM
ץr
ϭ0.
͑15c͒
aJ ͑␤ a͒
1
m
Qnmϵ4␤K Q
.
i
2
2
Finally, the long time behavior is: lim c ϭ0, lim c ϭ0.
␭n z R ␤ J ͑␤ R͒
0
0 m m
M
I
t→ϱ
t→ϱ
Equations ͑15a͒ and ͑15b͒ are the reflection and adsorp-
tion condition at the polymer–substrate and polymer–
vacuum, respectively ͑already described in Models 1 and 2͒
while Eq. ͑15c͒ accounts for the fact that at a certain distance
Equation ͑20͒ has to be solved for Anm(tЈ). By employing
the initial condition ͑14a͒ we get Amn(tЈ)→0 for tЈ→0, then
we may easily integrate Eq. ͑20͒ yielding after simple rear-
rangements
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J. Chem. Phys., Vol. 111, No. 4, 22 July 1999
Depolymerization promoted by ion beam
1727
Ϭt , then suddenly drops to zero at ␶ϭt . Once the analyti-
Qnm
0
0
Anm͑tЈ͒ϭ
2
n
2
cal expression for P(␶) is known, we easily integrate Eq.
27͒ yielding
D͑␭ ϩ␤ ͒ϪK
p
m
͑
2
n
2
m
ϫ͓exp͑ϪK tЈ͒Ϫexp͑ϪD͑␭ ϩ␤ ͒tЈ͔͒,
p
2
JTOT͑t͒ϭ␲a QP A␤z
0
0
͑
22͒
ϱ
8
␲2
1
1
which, inserted back into Eq. ͑16b͒, gives the final expres-
sion for the excess monomer concentration ␦cM
ϭc (r,z,t)Ϫc (z,t) inside the polymer film. This result
allows one to calculate the mean flux of monomers crossing
the vacuum–polymer interface around a single track by av-
eraging the local flux over r and ␾ ͑see Fig. 5͒
ϫ 1Ϫ
ͫ
͚
2
2
n
nϭ1 ͑2nϪ1͒ ͑1ϪD␭ /K ͒
p
2
M
M
D␭n
2
ϫ exp ϪD␭ t͒Ϫ
ͫ
͑
exp ϪK t͒
͑
ͬ
,
͑28a͒
n
p
ͬ
Kp
for tϽt0
R
ϱ
ץ␦cM
8
1
1
J͑tЈ͒ϭϪ2␲D
͵
ͯ
rdr,
͑23͒
2
JTOT͑t͒ϭ␲a QP A␤z
ץz
0
0
2
␲2 nϭ1 ͑2nϪ1͒ ͑1ϪD␭ /Kp͒
͚
2
2
n
0
zϭz
0
2
and using Eqs. ͑21͒ and ͑22͒ we obtain
D␭n
2
ϫ
ͫ
exp͑ϪD␭ t͒͑exp͑D␭ t ͒Ϫ1͒Ϫ
n
n 0
ϱ
ϱ
Kp
aJ ͑␤ a͒J ͑␤ R͒
1
m
1
m
J͑tЈ͒ϭ8␲D␤K Q
p
͚ ͚
2
m
2
0
nϭ1 mϭ1 z0R␤ J ͑␤mR͒
ϫexp͑ϪK t͒͑exp͑K t ͒Ϫ1͒
ͬ
,
͑28b͒
p
p 0
1
ϫ
͓exp͑ϪK tЈ͒
p
2
2
m
for tϾt0 .
D͑␭ ϩ␤ ͒ϪK
p
n
2
n
Finally, relating the eigenvalues ␭ and ␣ (␣ ϭD␭ )
n
n
n
2
n
2
m
Ϫexp͑ϪD͑␭ ϩ␤ ͒tЈ͔͒.
͑24͒
we obtain the striking result that Eqs. ͑28a͒ and ͑28b͒ are all
but nothing that Eqs. ͑10a͒ and ͑10b͒ of the continuous
Model 2 previously described! ͑provided we identify the
macroscopic kinetics constant K with the microscopic one:
Ki ϵ␲a QP ). In other words, the local flux of mono-
mer escaping from the polymer–vacuum interface is a local
property which is greater near the tracks and rapidly decreas-
ing with the distance, but the mean flux of monomers is
identical to that calculated by assuming a uniform production
of monomers inside the polymer instead of a discrete one
͓compare Eqs. ͑10͒ and ͑28͔͒. This property is likely related
to the linear nature of our reaction-diffusion equation ͓Eqs.
But, following Eq. ͑17͒: J (␤ R)ϭ0, so, apparently, the
1
m
above expression for the flux J(tЈ) goes to zero. However,
according to the theory of Bessel’s function,15 the first root
i
micro
2
0
to J (␤ R)ϭ0
is ␤ ϭ0, therefore the ratio
J (␤ a)J (␤ R)/␤ contained in Eq. ͑24͒ is undetermined.
1
m
1
2
1
1
1
1
1
Since J (x)ϭx/2 as x→0, we easily obtain
1
1
J ͑␤ a͒J ͑␤ R͒
aR mϭ1
mϾ1
1
m
1
m
4
0
ϭ
.
͑25͒
2
ͭ
␤
m
͑
12͒ and ͑13͔͒ for which the results are independent of the
This result, together with the identity J (x)
simplifies Eq. ͑24͒
ϭ1, greatly
0
x→0
stage of the averaging process. In other words, pre-averaging
the kinetics equations over the variables r and ␾ and succes-
sively solving the diffusion equation yields the same final
formulas obtained by averaging the whole reaction-diffusion
equation.
ϱ
a2
1
2
n
J͑tЈ͒ϭ2␲D␤K Q
p
͚
z0 nϭ1 D␭ ϪK
p
2
n
ϫ͓exp͑ϪK tЈ͒Ϫexp͑ϪD␭ tЈ͔͒.
͑26͒
Although the solution of Model 3 does not contain any
mathematical approximation, the model however describes
only approximately the interactions among the tracks. In-
deed, we made two basic assumptions: ͑i͒ around a track
there is a radial distribution of monomers coming from the
surrounding tracks formed at the same time tϭ␶ ͓boundary
condition ͑15c͔͒ and; ͑ii͒ the monomers formed at tϽ␶ have
been smeared out over the film. In other words, we adopted
a mean-field picture which neglects the monomer concentra-
tion fluctuations related to the random nature of tracks for-
mation. Obviously, the above assumptions are correct in the
low fluence regime where the interaction among the tracks is
small. Apart from the good agreement between the models
and the experimental results, the coincident results obtained
for the averaged monomer fluxes by continuum and discrete
models prove the conceptual validity of a continuum picture,
despite the production of chemical species under irradiation
is a strongly nonlocal process which, apparently, cannot be
p
Coming back to the original variable tЈϭtϪ␶, one may cal-
culate by Eq. ͑26͒ the average flux of monomers produced by
all the tracks formed in the time interval ␶Ϭ␶ϩd␶, where
obviously ␶ is a continuous variable spanning from 0 to the
final irradiation time t . Letting P(␶) be the tracks forma-
0
tion rate per unit area and unit time, the overall flux of mono-
mers is
t
JTOT͑t͒ϭA
͵
P͑␶͒h͑␶͒J͑tϪ␶͒d␶,
͑27͒
0
where A is the polymer–vacuum interfacial area and the
function h(␶) is defined as h(␶)ϭ1 for tϾ␶ and h(␶)ϭ0
for tϽ␶. The function h(␶) accounts for the fact that no
monomers are produced before the formation of a track at
time tϭ␶. In our experiments the irradiating beam intensity
P(␶) was kept constant, P0 say, over the whole range 0
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1728
J. Chem. Phys., Vol. 111, No. 4, 22 July 1999
Raudino et al.
FIG. 6. Experimental ͑thin line͒ and theoretical ͑thick line͒ monomer evolution for 20 s ion irradiation at Tϭ170 °C, calculated by using the Model 2. The
insert reports the experimental ͑dotted line͒ and the theoretical curves ͑Model 2͒ at high irradiation time calculated without ͑continuous line͒ and with ͑dashed
line͒ considering the thickness variation during the experiment due to monomer evolution.
correctly described by a continuum model. However, as pre-
viously said, the coincidence between the two models holds
unless the chemical kinetics is linear ͓i.e., ͑pseudo͒ first or-
der͔.
monomers corrected for the thickness variation accompany-
ing the monomer evaporation.16 It is evident from the insert
of Fig. 6 that thickness variation cannot completely explain
the discrepancy between experimental and theoretical curves
at long irradiation times. On the other hand we do not be-
lieve that variations of surface film morphology, due to the
C. Anomalous behavior at high irradiation time
fact that we are working at TϾT , play some role because
profilometric analysis of irradiated samples excludes perma-
nent ion beam induced morphology changes. Therefore,
g
In this last section secondary effects on the MMA evo-
lution in presence of the ion beam are considered. In Fig. 6 is
shown the monomer signal during an ion irradiation at T
ϭ170 °C for 20 s. As one can see the theoretical curve does
not fit well the experimental one as the irradiation time in-
creases over 15 s. A decrease in the MMA evolution rate is
evident. Two different effects could explain this phenom-
enon. First of all we can suppose that during the experiment
the polymer film reduces its original thickness due to mono-
mer evolution. In Fig. 7 is reported how the film thickness
decreases on increasing the irradiation time, working at dif-
ferent temperatures. As one can see, for the irradiation time
used in our experiment ͑less than 1 min͒ only a 2% thickness
reduction has been detected.
A comparable reduction can be theoretically calculated
by assuming that the thickness, z , changes with time be-
0
cause of the loss of monomers in the vacuum.
t
0
tЈ
z ͑t͒ϭz ͑t͉͒tϭ0ϪC
͵
Jϩ͑t͒dtϩ
͵
J ͑t͒dt ,
0
0
ͫ
Ϫ
ͬ
0
t
0
l3
Cϭ N ,
͑29͒
A
A
where l3 is the monomer volume, NA is the Avogadro’s
number and A the surface of the polymer film. Inserting Eq.
FIG. 7. Percentage of the initial film thickness removed during the ion beam
irradiation as a function of the irradiation time.
͑29͒ into Eqs. ͑10a͒ and ͑10b͒ we may calculate the flux of
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1

J. Chem. Phys., Vol. 111, No. 4, 22 July 1999
Depolymerization promoted by ion beam
1729
TABLE I. Kinetics constants of initiation and propagation (K ␤ and K )
and diffusion coefficients ͑D͒ values calculated by the proposed reaction-
diffusion Model 2.
other ion beam effects are responsible for such discrepancy,
in particular there is the possibility that kinetics and diffusion
coefficients change when the overlap among the tracks be-
comes more important. Therefore, the models we have pre-
sented are representative of the experimental data only at
short irradiation time.
i
p
8
K ␤ ͑sϪ1͒
Kp ͑sϪ1
D ͑cm2 sϪ1
2ϫ10Ϫ10
3.5ϫ10
1.1ϫ10
1.2ϫ10
2.1ϫ10
Temperature ͑°C͒
͒
͒
i
135
150
160
7ϫ10Ϫ4
90
100
100
120
300
Ϫ3
Ϫ10
Ϫ9
Ϫ9
Ϫ9
1.85ϫ10
Ϫ3
1.6ϫ10
3.2ϫ10
3.1ϫ10
Ϫ3
Ϫ3
IV. CONCLUSIONS
170
85
1
In this paper we provided three exact analytical solutions
of a reaction-diffusion problem regarding ion beam induced
monomer ͑MMA͒ production in polymer thin films. Continu-
ous ͑Models 1 and 2͒ and discrete models ͑Model 3͒ have
been developed and critically discussed. In particular al-
though the local flux of monomer escaping from the
polymer–vacuum interface varies along the interface, the
mean monomer flux of the discrete model is identical to that
calculated by assuming uniform monomer production inside
the polymer instead of a local one. Since almost all the ex-
perimental techniques measure mean fluxes, the validity of
rather simple continuous models for describing an ensemble
of local kinetics looks quite interesting, provided the density
of local kinetic events is not too high.
solute diffusion in polymer matrices. To our knowledge no
literature data refer to diffusion of MMA in thin film of
PMMA under ion beam irradiation, even if many workers
have reported the diffusion coefficients of small molecules in
polymer membranes, measured by permeation or sorption
techniques,18 that range from 10
Ϫ5
to 10
Ϫ10
cm s . For
2 Ϫ1
comparison, diffusion of argon through a Poly ͑pentaerythri-
toltribenzoate acrylate͒ ͑PPTBA͒ membrane ͑400 ␮m thick͒
Ϫ6
2
Ϫ1
19
is ϳ10 cm s at 100 °C while the diffusion coefficient
values of oxygen through a PMMA film ͑60 ␮m thick͒ is
Ϫ7
2
Ϫ1
20
ϳ5ϫ10 cm s at Tϭ180 °C. Instead, by using the
pulsed ion procedure the diffusion coefficient value of me-
10
In addition to MMA production and diffusion in PMMA,
the proposed equations are useful for the general problem of
measuring the diffusion coefficient of slow diffusants in
polymers. In these cases, when one wants to measure very
thyl formiate in PMMA thin films ͑Ͻ 1 ␮m͒ turns out ϳ1
Ϫ10
2 Ϫ1
ϫ10
cm s at room temperature. This value is in good
agreement with the ones we have found for MMA. The po-
tentiality and the limitations of ion beam technique for mea-
suring D values have been described in detail in the original
reference10 and do not need to be repeated here.
Ϫ10
2
small diffusion coefficient (DϽ10
cm /s), the usual
‘‘membrane’’ technique, where a gas diffuses from one side
to the other of a self supporting and defect-free polymer film,
cannot be used. Indeed with typical film thickness, which is
of the order of 0.1 mm, the measurement time becomes very
long ͑ϳ some weeks͒ and in such cases the only way to
We limit our attention to the most limiting feature of the
technique that is the problem of ion damage ͑see Sec. III C͒.
In order to avoid this limitation low fluence must be used
perform it is that of making use of polymer powders,17
a
(Ͻ10 ions cm ), but even in this case this does not seem
a serious limitation since a few seconds of bombardment are
enough to collect all the necessary data.
13
Ϫ2
choice which makes it difficult the interpretation of the mea-
sured diffusion coefficients.
Venkatesen et al.10 have shown that pulsed ion beam
technique can be used in such cases by studying the evolu-
tion of gaseous products from thin polymer films and extract-
ing the D value from the data. In that case, uniform
In principle another limitation could be that the molecule
of which one wants to measure the diffusion coefficient must
be produced by the ion beam. As already pointed out, one
can circumvent this problem by using a bi-layer arrangement
with a thin layer of a ‘‘source polymer’’ ͑which, under ion
beam, produces the molecule of interest͒ and a thicker film
of polymer in which the D value is to be measured.
‘‘source’’ concentration was assumed and the model was
solved by means of finite-difference partial-differential equa-
tion technique. The models presented in this paper represent
a significant extension of the Venkatesan’s approach, not
only in terms of a more detailed modelling but even because
exact analytical solutions are provided. In this sense we be-
lieve that Eqs. ͑10a͒ and ͑10b͒ furnish a great impulse to the
use of pulsed ion beam technique for measuring kinetics con-
stants and diffusion coefficients of slow diffusants in poly-
By means of the method presented in this paper we can
extract also the kinetics constant values for initiation and
propagation steps of the depolymerization reaction ͑see
Table I͒. The kinetics constants values of K and K reported
i
p
in literature refer only to a conventional thermal depolymer-
ization of PMMA and have an order of magnitude of
13
Ϫ2
Ϫ5 Ϫ1
5
Ϫ1
21–24
mers. At low fluences (р10 ions cm ) the values found in
this paper are totally independent of the ion fluence as well
as monomer concentration inside the polymer film. This jus-
tified the assumption of a constant diffusion coefficient made
in developing the theoretical model. The data are shown in
Table I.
ϳ10
s
and ϳ10 s , respectively,
at temperatures
above 300 °C. These values generally increase on increasing
temperature, following Arrhenius plots.
At a first sight the kinetics constant values we have
found for ion beam induced thermal depolymerization of
PMMA are lower than those reported in the literature. How-
ever, kinetics constants reported in literature have been mea-
sured at much higher temperatures ͑300–500 °C͒, while be-
As one can see, the diffusion coefficients of MMA here
reported increase with increasing temperature. In particular
the plot of Ϫlog D vs (TϪT ) shows the downward devia-
low the Ceiling Temperature, T , the depolymerization
g
C
tion predicted by the Williams–Landel–Ferry ͑WLF͒
theory ͑or related models͒ for a free volume picture of
kinetics cannot take place. The difference between our and
previous measurements lies in the fact that we are studying a
1
8
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129.120.242.61 On: Thu, 27 Nov 2014 04:20:01

1730
J. Chem. Phys., Vol. 111, No. 4, 22 July 1999
Raudino et al.
process where the depolymerization initiators are formed by
the ion beam, so that we are exploring temperature ranges
well below TC .
This is, we believe, the most striking feature of the pro-
posed technique, i.e., that of allowing to measure depolymer-
ization kinetics at temperature well below that needed for the
production of suitable thermal initiators.
ץ
˜
c ͑s͒
M
˜
J͑s͒ϭϪDA
ͯ
ץz
zϭz
Ϫst
0
0
D 1/2
1/2
1
Ϫe
s
ϭ␤AK Kp
ͩ ͪ
tgh
ͫ
ͩ ͪ
z . ͑A4͒
ͬ
i
0
s͑sϩKp͒
s
D
The flux in the real space, J(t), can be obtained by inversion
Ϫ1
˜
˜
of Laplace transform J(s): J(t)ϭL J(s) Inversion can be
made by first expanding the hyperbolic tangent of Eq. ͑4A͒
2
6
as
ACKNOWLEDGMENTS
ϱ
1/2
1/2
D
s
2D
1
Financial support of CNR ͑Rome͒ and MURST are
gratefully acknowledged. The Authors wish to thank the col-
leagues, Mircea Chipara for useful discussion on WLF equa-
tion and Lorenzo Torrisi for the assistance in the use of the
mass quadrupole and of the stylus profilometer.
ͩ ͪ tgh
ͫ
ͩ ͪ z0
ͬ
ϭ
͑A5͒
͚
s
D
z0 kϭ1 ␣ ϩs
k
2
2
2
where (␣kϵ(2kϪ1) ␲ D/4z0) and then inverting each
25
term of the series with the aid of the relationships
1
1
LϪ1
ϭ
͑eϪatϪeϪbt͒,
͑A5a͒
ͫ
ͬ
͑
sϩa͒͑sϩb͒
bϪa
and
APPENDIX
1
1
ϪeϪst
s
0
t
LϪ1
ϭ
F͑tϪ␶͒d␶ tϽt0 ,
˜
F͑s͒
ͫ
ͫ
ͬ
͵
Since it can be easily shown that Model 1 is obtained
from Model 2 when the propagation rate of monomer unzip-
0
ϪeϪst0
͑A5b͒
t
0
ping K tends to infinity, we develop only mathematical de-
LϪ1
˜
p
F s ϭ
͑
F tϪ␶ d␶ tϾt0 ,
͑ ͒
ͬ
͵
͒
s
tails of Model 2 and, successively, we perform the limit Kp
0
→
ϱ.
Laplace transforming Eqs. ͑8a͒ and ͑8b͒ of the main text
where F(t) stands for any arbitrary well-behaved function,
we obtain after some algebra
and using the initial conditions ͑9c͒ we find
ϱ
1
1
Ϫ␣ t
ϪK t
p
k
Jϩ͑t͒ϭ
Bk
ͫ
ͫ
͑1Ϫe
͒Ϫ
͑1Ϫe
͒
ͬ
tϽt0
Ki
͚
͑1ϪeϪst0͒ϪKp
kϭ1
␣k
Kp
s
˜
c ͑s͒ϭ
˜
c ͑s͒,
͑A1a͒
͑A1b͒
I
I
s
͑
A6a͒
ϱ
ץ2
Ϫ␣ t
ϪK t
k
p
˜
c ͑s͒
M
e
e
␣
t
͑eKpt0Ϫ1͒
k 0
s˜
c ͑s͒ϭD
ϩ␤Kp ˜c ͑s͒,
JϪ͑t͒ϭ
Bk
͑e Ϫ1͒Ϫ
M
ץz2
I
͚
ͬ
kϭ1
␣k
Kp
where
tϾt0
͑A6b͒
ϱ
where we have defined Bkϵ͓2A␤KiKpD͔/͓z0(KpϪ␣k)͔.
By using the identity
eϪstci͑t͒dt ͑iϭI or M͒,
26
˜
c ͑s͒ϭ
͵
i
0
ϱ
2
ϱ
2
0
1
4z0
2
1
z
and in deriving Eq. ͑A1͒ we have used the relationship25
ϭ
ϭ
2
,
͑A7͒
͚
͚
kϭ1 ␣k
␲ D kϭ1 ͑2kϪ1͒
2D
ϱ
1
eϪst͑1ϪH͑tϪt ͒͒dsϭ ͑1ϪeϪst0͒.
eventually we obtain the final formulas reported by Eqs.
˜
f͑s͒ϭ
͵
0
s
0
͑
10a͒ and ͑10b͒ of the main text. In the limit K →ϱ we
p
recover the result of Model 1 ͑Eqs. ͑4a͒ and ͑4b͒͒.
Solving Eq. ͑A1a͒ for ˜c 1(s) and combining the results with
Eq. ͑A1b͒ we get after integration
1
H. O’Donnell, The Effects of Radiation on High-Technology Polymers ͑E.
Reichmanis and J. O’Donnell Eds., ACS Symposium Series, 381, 1989͒.
1/2
0
2
1
ϪeϪst
s
`
M. E. Fragala, G. Compagnini, L. Torrisi, and O. Puglisi, Nucl. Instrum.
˜
c ͑s͒ϭ␤K K
ϩA sinh
ͫ
ͩ ͪ
z
ͬ
M
i
p
2
1
Methods Phys. Res. B 141, 169 ͑1998͒.
M. E. Fragal a` , G. Compagnini, and O. Puglisi, J. Mater. Res. 14, 228
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M. E. Fragal a` , G. Compagnini, A. Licciardello, and O. Puglisi, Ber. Bun-
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B 19/20, 865 ͑1987͒.
A. Licciardello and O. Puglisi, Nucl. Instrum. Methods Phys. Res. B 91,
s ͑sϩK ͒
D
p
3
1/2
s
4
ϩA cosh
ͫ
ͩ ͪ
z
ͬ
,
͑A2͒
2
D
5
the coefficients A and A are to be calculated by applying
the boundary conditions ͑9a͒ and ͑9b͒. Simple algebra yields
1
2
6
7
A ϭ0,
͑A3a͒
1
436 ͑1994͒.
ϪeϪst0
1/2
8
9
1
s
P. Thevenard, G. Guiraud, C. H. S. Dupuy, B. Delaunay, C. Riedel, and R.
Spohr, Radiat. Eff. 42, 69 ͑1979͒.
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Methods Phys. Res. B 32, 131 ͑1988͒.
T. Venkatesan, D. Edelson, and W. L. Brown, Appl. Phys. Lett. 43, 364
͑1983͒.
A ϭϪ␤K K
sec h
ͫ
ͩ ͪ z0
ͬ
.
͑A3b͒
2
i
p
2
s ͑sϩK ͒
D
p
˜
10
The Laplace transformed flux J(s) across the polymer–
vacuum interface of area A is easily calculated as
11
A few experiments were performed by using bigger currents ͑0.03, 0.05,
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J. Chem. Phys., Vol. 111, No. 4, 22 July 1999
Depolymerization promoted by ion beam
1731
z
2
2
0
and 0.1 ␮A, respectively͒. Even using these larger currents, the total
amount of produced monomers depends only on the product of the ion
1
z
z
0
¯
tϭ
dzϭ
.
͵
z0
D
3D
0
beam flux and the exposure time (t ).
0
12
¯
L. Torrisi, G. Ciavola, R. Percolla, and F. Benyaich, Nucl. Instrum. Meth-
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R. H. Boyd, J. Chem. Phys. 31, 321 ͑1959͒.
Replacing the appropriate values for D and z0 we find tϳ1 s. During that
time the thickness variation ͓calculated by Eq. ͑29͔͒ is about 0.1%. There-
fore, the diffusing monomers see a practically immobile boundary during
their diffusion toward the interface.
A. R. Berens and H. B. Hopfenberg, J. Membr. Sci. 10, 283 ͑1982͒.
D. Ehlich and H. Sillescu, Macromolecules 23, 1600 ͑1990͒.
M. F. Laguna, J. Guzman, E. Riande, and E. Saiz, Macromolecules 31,
7488 ͑1998͒.
1
1
1
3
4
5
17
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions
18
͑Dover, NY, 1972͒.
16
We are aware that in the case of moving boundary z ϭz (t) we cannot
19
0
0
solve the diffusion equation with a fixed boundary and then replace in the
final equation z0 by z (t) ͑J. Crank, The Mathematics of Diffusion, Clar-
20
0
J. R. MacCallum and A. L. Rudkin, Eur. Polym. J. 14, 655 ͑1979͒.
R. M. Simha and L. A. Wall, J. Phys. Chem. 56, 707 ͑1952͒.
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2
2
1
2
endon Press, Oxford, 1970͒. However, in the present case such an ap-
proximation is permissible. This can be proved by comparing the mono-
mer mean life time inside the polymer film with the corresponding film
thickness variation during that time. The monomer life time can be calcu-
lated as the time required to cross a film thickness z0 starting at tϭ0 from
͑
1989͒.
23
T. Kashiwagi, T. Hirata, and J. E. Brown, Macromolecules 18, 131
͑1985͒.
L. Manring, Macromolecules 24, 3304 ͑1991͒.
M. R. Spiegel, Laplace Transforms ͑McGraw-Hill, NY, 1965͒.
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products
͑Academic, NY, 1980͒.
a generic point z randomly distributed in the interval 0ϽzϽz . At z
24
25
0
ϭz0 the monomer escapes from the film. From the basic relationship z2
ϭDt we find the following expression for the mean residence time inside
the film:
26
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1