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Title Effects of absorption and inhibition during grating formation in photopolymer materials
Authors(s) Gleeson, M. R.; Kelly, John V.; Close, Ciara E.; O'Neill, Feidhlim T.; Sheridan, John T.
Publication date 2006-10-01
Publication information Journal of the Optical Society of America B, 23 (10): 2079-2088
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Effects of absorption and inhibition during grating
formation in photopolymer materials
Michael R. Gleeson, John V. Kelly, Ciara E. Close, Feidhlim T. O’Neill, and John T. Sheridan
School of Electrical, Electronic and Mechanical Engineering, College of Engineering, Mathematical and Physical
Sciences, University College Dublin, Belfield, Dublin 4, Ireland
Received February 28, 2006; revised June 28, 2006; accepted June 29, 2006; posted July 6, 2006 (Doc. ID 68367)
Photopolymer materials are practical materials for use as holographic recording media, as they are inexpen-
sive and self-processing (dry processed). Understanding the photochemical mechanisms present during record-
ing in these materials is crucial to enable further development. One such mechanism is the existence of an
inhibition period at the start of grating growth during which the formation of polymer chains is suppressed.
Some previous studies have indicated possible explanations for this effect and approximate models have been
proposed to explain the observed behavior. We examine in detail the kinetic behavior involved within the pho-
topolymer material during recording to obtain a clearer picture of the photochemical processes present. Ex-
periments are reported and carried out with the specific aim of understanding these processes. The results
support our description of the inhibition process in an acrylamide-based photopolymer and can be used to pre-
dict behavior under certain conditions. © 2006 Optical Society of America
OCIS codes: 090.0090, 090.2900, 090.2890, 160.5470
.
1. INTRODUCTION
Photopolymer materials have the ability to optically
record high-diffraction-efficiency, low-loss, volume holo-
graphic gratings in self-processing materials and are of
ever-increasing commercial importance. Improving their
characteristics, to attain the full potential of these mate-
rials, requires the development of accurate models, vali-
dated using reproducible experimental data sets, which
can ultimately provide clear insight into the photochemi-
cal processes involved during recording. Exploration of
the kinetics involved in the recording of holographic grat-
ings is an integral part of this work. In this paper the pho-
tochemical kinetics involved during recording in our
acrylamide-based photopolymer
1
material is examined.
Specifically we aim to better understand what takes place
inside the material during exposure, i.e., to explain the ef-
fects of the variation of the absorbance of the photosensi-
tive dye with time and the suppression of radical produc-
tion due to the presence of inhibitors. By improving the
physical accuracy of our model, and understanding the re-
lationship between the rate of polymerization and the
concentrations of monomer, polymer, dye, initiator, and
inhibitor, we aim in this paper to improve the validity of
the one-dimensional polymerization-driven diffusion
(PPD) model.
2
Development of this model will facilitate
the eventual full nonlocal polymerization-driven diffusion
(NPDD) modeling of such materials.
3–6
We proceed as follows. First we examine the photo-
chemical processes involved during grating formation and
incorporate the suppression of radical production into the
rate equations, which form the basis of our models. We
then incorporate the effects of changes in the absorbance
of the photosensitive dye during exposure. Having devel-
oped appropriate rate equations, we derive and solve the
resulting coupled equations numerically.
3–6
A set of ex-
periments is carried out that, following comparison with
the model, support the assumptions made in deriving our
equations and illustrate the effects associated with the in-
hibition and bleaching processes.
2. PHOTOCHEMICAL PROCESSES
Assuming bimolecular termination,
5
we begin by discuss-
ing radical chain polymerization. In particular we exam-
ine polymerization due to photoinitiation and examine
the effect on excited dye molecules due to the presence of
inhibitors such as oxygen. Furthermore we include the ef-
fect of a time-dependent transmittance,
7–9
which de-
scribes the change in the material absorption during grat-
ing growth.
Free radical polymerization is a chain reaction involv-
ing three steps: initiation, propagation, and termination.
Initiation involves the production of free radicals,
I ——
→
k
d
R
•
, 共1兲
where k
d
is the radical generation rate constant. The rate
of this reaction is given by
R
d
= d关R
•
兴/dt = k
d
关I兴, 共2兲
where 关R
•
兴 is the free radical concentration and 关I兴 is the
initiator concentration. The free radicals then bind to a
monomer M to form the chain initiation species M
1
•
,
R
•
+ M ——
→
k
i
M
1
•
, 共3兲
where k
i
is the initiation rate constant. The radical M
1
•
then propagates by bonding with monomer molecules to
form long polymer chains with an active tip known as a
macroradical.
Gleeson et al. Vol. 23, No. 10 /October 2006/ J. Opt. Soc. Am. B 2079
0740-3224/06/102079-10/$15.00 © 2006 Optical Society of America
M
n
•
+ M ——
→
k
p
M
n+1
•
, 共4兲
where k
p
is the propagation rate constant and M
n
•
is a
macroradical of n monomeric units where a monomeric
unit is the largest constitutional unit contributed by a
single monomer molecule.
10
The initiator consists of a
photosensitive dye and a reducing agent. The dye can be-
come excited in the presence of a photon and when excited
can accept an electron from the reducing agent, i.e., a ter-
tiary amine (triethanolamine),
11
and can then produce a
free radical R
•
:
dye + h
→ dye
*
,
dye
*
+ M → R
•
. 共5兲
As the radicals are produced, some of the dye molecules,
which are excited, return to their ground state or move to
a triplet state:
12
dye
*
——
→
k
f
dye + h
,
dye
*
——
→
k
isc
dye
T
,
dye
T
+ C
I
——
→
k
R
dye
•−
+ C
I
•
+ H
+
. 共6兲
An excited dye molecule, which has been promoted to that
state due to a photon interaction, can either decay by
fluorescence
13
at a rate of k
f
back to its ground state or
can be excited to a triplet state dye
T
by intersystem cross-
ing at a rate of k
isc
. The excited triplet state can then un-
dergo an electron transfer reaction (rate constant k
R
) with
the coinitiator C
I
, which creates the primary radical C
I
•
,a
dye radical dye
•−
, and a free proton H
+
.
1,12
The dye radi-
cal eventually reacts with other constituents and forms a
bleached state.
12
This process results in dye molecules be-
ing consumed and can lead to cessation of polymerization
if the dye is consumed too quickly.
All of these effects tend to suppress the creation of radi-
cals and therefore slow the rate of polymerization (of
monomer). Other contributors to this radical suppression
are inhibitors, which are present within the chemicals in
our material, and the presence of oxygen, which also acts
as an inhibitor.
14–18
We assume that the effect of inhibi-
tion during the fabrication process is primarily due to the
oxygen present in our material. Therefore we suggest that
the oxygen reacts with the excited dye and deactivates it
into a passive state;
dye
*
+
——
→
k
Y +
•
, 共7兲
where Y is the photoreduction product and is assumed to
only participate once (i.e., its relaxation time is long com-
pared with the exposure time),
is the oxygen, and k
is
the deexcitation rate of the dye.
14
This effect is present
throughout exposure but is most noticeable at the start.
The equation governing the removal of excited dye is
R
=
d关Y兴
dt
= k
关dye
*
兴关
兴, 共8兲
where R
is the rate of removal of free radicals due to the
removal of excited dye molecules by oxygen. This process
reduces the concentration of excited dye molecules avail-
able for creating free radicals. Thus it ultimately reduces
the monomer polymerization rate.
The frequency of encounters between the free radicals
is another factor determining the rate of polymerization.
It can be accounted for using the cage effect.
19
It is as-
sumed that only some fraction f of the free radicals pro-
duced will react with the monomer (in the starting reac-
tion).
From Eqs. (2) and (8) we see that the rate of production
of free radicals R
r
that are responsible for the production
of monomer radicals, with the inclusion of the presence of
oxygen, can then be given by
R
r
= fk
d
关I兴 − R
, 共9兲
with the initiator concentration 关I兴. From relation (3) the
rate of production of monomer radicals R
i
can be written
as
R
i
=d关M
1
•
兴/dt = k
i
关R
•
兴关M兴, 共10兲
where 关M兴 is the monomer concentration and 关R
•
兴 is the
primary radical concentration. In general the rate of pro-
duction of monomer radicals R
i
is much greater than the
rate of production of free radicals R
r
; therefore initiator
radicals are consumed as fast as they are generated.
6
The
rate-determining step is thus the decomposition of the ini-
tiator. Initiator radicals are consequently formed with a
rate
d关R
•
兴/dt = R
r
− R
i
= 共fk
d
关I兴 − R
兲 − k
i
关R
•
兴关M兴 =0. 共11兲
Thus the rate of monomer radical generation R
i
is equal
to the chain initiation rate and
R
i
= k
i
关R
•
兴关M兴 = fk
d
关I兴 − R
. 共12兲
Chain growth would continue in this way until the supply
of monomer is exhausted were it not for the strong ten-
dency of radicals to react in pairs to form paired-electron
covalent bonds with the loss of radical activity. At suffi-
ciently low initiator concentrations, chain termination
will occur mainly by combination,
6,19
M
n
•
+ M
m
•
——
→
k
tc
M
n+m
, 共13兲
or by disproportionation,
6,19
M
n
•
+ M
m
•
——
→
k
td
M
n
+ M
m
. 共14兲
On the basis of these observations, the rate of termination
R
t
can then be given by
R
t
= k
t
关M
•
兴
2
, 共15兲
where k
t
=k
tc
+k
td
is the termination constant, and 关M
•
兴 is
the total concentration of all chain radicals of size M
1
•
and
larger.
19
For low-monomer conversions we assume that
the rate of radical formation equals the rate of radical dis-
appearance (Bodenstein steady-state principal
19
); this is
2080 J. Opt. Soc. Am. B/ Vol. 23, No. 10 / October 2006 Gleeson et al.
also the case at low-monomer conversions, i.e., when only
a little monomer is used up, 关I兴⬇关I
ic
兴, where 关I
ic
兴 is the
initial initiator concentration. Although we assume here
that the rate at which the radicals are suppressed, k
,is
constant, the rate may differ during exposure. Reaching
steady state in this process implies that the rate of initia-
tion R
i
and the rate of termination R
t
are equal, yielding
fk
d
关I兴 − R
= k
t
关M
•
兴
2
. 共16兲
Solving for 关M
•
兴 we obtain
关M
•
兴
stat
=
冉
fk
d
关I兴 − R
k
t
冊
1/2
, 共17兲
where 关M
•
兴
stat
is the total concentration of all chain radi-
cals of size M
1
•
and larger at steady state. Assuming that
much more monomer is consumed due to propagation po-
lymerization than in the initiation reaction, the propaga-
tion rate R
pg
[the rate of reaction of relation (4)] is ap-
proximately equal to the polymerization rate R
p
;
therefore
R
p
⬵ −d关M兴/dt = k
p
关M
•
兴关M兴. 共18兲
Substituting into Eq. (18) from Eq. (17) for 关M
•
兴 yields
R
p
= k
冉
fk
d
关I兴 − R
k
t
冊
1/2
关M兴. 共19兲
Examining the photochemical formation of free radicals,
there are a number of initiation mechanisms,
20
many in-
volving a photochemical electron transfer reaction. If we
reexamine the way in which the monomer radicals are
formed, we see that they are dependent on the quantity of
free radicals formed per photon absorbed and the inten-
sity of the light used for recording. As the inhibitor
present indirectly consumes some of the absorbed pho-
tons, there is a reduction in the number of free radicals
available for the initiation of monomer radicals and thus
for the formation of polymer chains. The rate of initiation
R
i
can therefore be given by
R
i
= f共⌽
⬘
− R
⬘
兲I
a
= ⌽I
a
− R
, 共20兲
where ⌽
⬘
is the total number of radicals produced per
photon absorbed and R
⬘
is the decrease due to the pres-
ence of oxygen. I
a
is the intensity of light absorbed in
moles of light quanta per liter per second and ⌽ is the to-
tal number of propagating chains that would be initiated
per light photon absorbed if no inhibition took place.
6,17
As only a fraction f of the free radicals, which are pro-
duced, reacts with the monomer in the starting reaction,
the cage effect has been included. R
⬘
indicates that the
process of production of radicals, which cannot produce
chains due to the cage effect, may still undergo inhibition.
Let us assume cosinusoidal spatially modulated illumi-
nation, i.e., I共x兲 =I
0
关1+V cos共Kx兲兴 where V is the fringe
visibility; K=2
/⌳, the grating vector magnitude, and ⌳
is the grating period. The concentration of photosensitiz-
ers is related to the absorbed intensity by Beer’s law:
19
I
a
共x,t兲 = I共x兲兵1 − exp关−
⑀
Z共t兲d兴其 = I共x兲关1−T共t兲兴, 共21兲
where
⑀
is the molar absorptivity, Z共t兲 is the time-
dependent concentration of the photosensitizers (initia-
tors), and d is the photopolymer layer thickness. Since the
concentration of photosensitizers is a function of time, the
transmittance of the layer T共t兲 (Refs. 7 and 8) also de-
pends on time and is discussed below. The concentration
of free radicals given in Eq. (17) can now be written as
关M
•
兴 =
再
f共⌽
⬘
− R
⬘
兲I共x兲关1−T共t兲兴
k
t
冎
1/2
. 共22兲
Therefore the polymerization rate from Eq. (19) is given
by
R
p
= k
p
关M兴
再
f共⌽
⬘
− R
⬘
兲I共x兲关1−T共t兲兴
k
t
冎
1/2
=
共t兲关M兴关I共x兲兴
1/2
,
共23兲
where
共t兲=
0
共t兲A共t兲
1/2
, A共t兲=1−T共t兲,
0
共t兲=k
p
关⌽共t兲/k
t
兴
1/2
,
and ⌽共t兲 =f共⌽
⬘
−R
⬘
兲.
We can summarize the above as follows: A共t兲 tells us
the fraction of the incident light absorbed, ⌽
⬘
tells us the
fraction of this absorbed light that results in initiation,
and finally ⌽共t兲 tells us the fraction of photons that lead
to polymerization.
The inclusion of the transmittance T共t兲 allows a time-
varying model for the polymerization rate R
p
to be ob-
tained. At the start of exposure the transmittance of the
exposing light will be small due to the high absorbance of
the photosensitive dye. This high absorption results in
large values of I
a
共x ,t兲 and of the polymerization rate fac-
tor
共t兲, and therefore a high rate of polymerization R
p
.
Similarly, as the transmittance of the layer increases dur-
ing the exposure, both the light absorbed by the photo-
polymer material layer and the polymerization rate con-
stant decrease, resulting in a lower rate of polymerization
R
p
.
To estimate T共t兲, several transmittance curves were ex-
perimentally obtained for differing exposure intensities in
our standard material layer.
1,7
There is an initial nonzero
transmittance at the beginning of exposure as the mate-
rial is never completely opaque. The transmittance func-
tion T共t兲 was then determined based on fits to the experi-
mental data. In Section 3 we introduce a loss fraction B to
allow for nonabsorptive losses, i.e., Fresnel boundary re-
flections. T共t兲 is estimated by fitting with the function
T共t兲 = E + G关1 − exp共− a
0
t + a
1
t
2
兲兴, 共24兲
where E, G, a
0
, and a
1
are constant parameters related to
the exposure intensity and the initial transmittance of
the layer. Equation (24) is used in our numerical simula-
tions, and some typical parameter results (and fitting
errors) for different exposing intensities are given in
Table 1.
In this section we have derived, for the first time to our
knowledge, the rate equations governing the photochemi-
cal processes involved in grating formation that include a
term to explicitly account for radical suppression due to
the effect of inhibition, which we have assumed to be
caused mainly by oxygen. Furthermore, following the
work of Blaya et al.,
8
time dependence for the absorbance
of the photosensitive dye and photoinitiators is included
in our rate equations.
Gleeson et al. Vol. 23, No. 10 /October 2006/ J. Opt. Soc. Am. B 2081
3. POLYMERIZATION-DRIVEN DIFFUSION
In this section the basic theory behind grating growth will
be presented. Following Galstyan et al.,
14
we incorporate
the presence of inhibition and time-varying absorption ef-
fects in the material, as discussed in Section 2, into a local
PDD model.
2
A dry photopolymer layer typically consists of a mono-
mer, binder, cross-linker, an electron donor, and a photo-
initiator. As the material is exposed to the recording
beams, the monomer is polymerized, and the amount of
polymerized monomer increases with the exposure. In our
material more monomer is polymerized in the bright
fringes of the interference pattern than in the dark
fringes. This results in a higher concentration of mono-
mer in the dark regions than in the bright, and therefore
a spatial monomer concentration gradient. The excess
monomer will tend to diffuse into the bright regions.
2–6,19
The governing one-dimensional diffusion equation is
u共x,t兲
t
=
x
冋
D共x,t兲
u共x,t兲
t
册
− F共x,t兲u共x,t兲, 共25兲
where u共x ,t兲 is the monomer concentration, D共x,t兲 is the
diffusion constant, and F共x ,t兲 is the polymerization rate.
Equation (25) is Fick’s law with the addition of a driving
function representing the physical effects of the photopo-
lymerization. The light intensity in the material is as-
sumed periodic and is described by
I共x,t兲 = I
0
关1+V cos共Kx兲兴, 共26兲
with I
0
the average irradiance. In Section 2 we derived an
equation governing the resulting polymerization rate. It
is proportional to the exposure irradiance raised to
␥
=1/2 (Ref. 21):
F共x,t兲 = F
0
共t兲关1+V cos共Kx兲兴
␥
A
␥
共t兲, 共27兲
where A关t兴, the time-varying absorbance of the
material,
8,9,22
has been included in the expression for the
polymerization rate to account for the change in absorbed
intensity during exposure. To account for the nonabsorp-
tive losses present, a loss fraction B has been included by
defining the polymerization rate parameter as F
0
共t兲
=
0
共t兲I
0
␥
共1−B兲. The value of the loss fraction B is empiri-
cally obtained by repeated measurement of the losses in
the layers and plates when all the monomer is polymer-
ized and the dye is bleached, assuming that the nonab-
sorptive losses can be obtained. The expression for the po-
lymerization rate parameter also includes the time-
varying function
0
共t兲, which we introduce to model the
inhibition period present at the start of exposure.
16–18,22,23
For the I共x,t兲 in Eq. (26), the monomer concentration
can be written as a cosine series
u共x,t兲 =
兺
i=0
⬁
u
i
共t兲cos共iKx兲. 共28兲
This is substituted into Eq. (25) with the initial condition
that u共x ,0兲=U
0
, where U
0
is the initial uniform monomer
concentration in the material. In our analysis we assume
that harmonics of an order greater than 2, i⬎ 2, can be
neglected, i.e., their contributions are assumed negligible
in comparison with that of the first three terms.
2
A set of
first-order coupled differential equations, in terms of the
monomer concentration harmonic amplitudes, is
obtained.
2,14
We now examine the F
0
共t兲 term appearing in Eq. (27)
using the result from Section 2. From repeated observa-
tions of our experimental results, we know that grating
growth is negligible for some period of time at the start of
exposure. This appears to be primarily due to the action
of inhibitors, such as oxygen,
14
which suppress the cre-
ation of free radicals. The process is as follows:
(i) the photosensitive dye is excited by the exposing
photons;
(ii) the excited dye then reacts with the monomer gen-
erating free radicals;
(iii) this leads to the formation of polymer chains,
14
and
then
(iv) the inhibitor acts to deactivate the dye from its ex-
cited state, stopping radical creation.
Although this process continually takes place during
grating formation, it is most obvious (and most easily ob-
served) at the beginning of exposure due to the high in-
hibitor concentration and the low concentration of excited
dye.
23
As exposure proceeds, the suppression of radicals
becomes less visible. We model this sharp temporal-state
transition using a step function
14
⌰关x兴, where
Table 1. Extracted Physical Parameters Obtained
from Fits to the Data
a
Parameter Case 1 Case 2 Case 3
I
0
共mW/ cm
2
兲
3.5 4.5 5.0
E
共mW/ cm
2
兲
0.857 1.124 1.636
G
共mW/ cm
2
兲
1.517 1.831 2.143
a
0
共s
−1
兲
共⫻10
−2
兲
6.238 7.321 8.397
a
1
共s
−2
兲
共⫻10
−4
兲
0.891 1.211 3.623
MSE
b
共⫻10
−3
兲
4.48 5.09 5.15
t
i
(s)
0.41 0.21 0.13
共cm
2
mWs
−1
兲
0.109 0.027 0.039
D
共cm
2
/s兲
共⫻10
−11
兲
3.0 2.5 3.5
C
共cm
3
/mol兲
共⫻10
−6
兲
3.1 5.5 6.1
MSE
共⫻10
−10
兲
0.32 1.72 1.26
a
See also Fig. 3.
b
Mean square error.
2082 J. Opt. Soc. Am. B/ Vol. 23, No. 10 / October 2006 Gleeson et al.
