J . Phys. Chem. 1992,96,7462-7465
7462
(32) Satterfield, C. N.Mass Transfcr in Heterogeneous Catalysis; MIT Press: Cambridge, MA, 1970; p 157. (33) Pusey, P.H.; Tough, R. H. A. In Dynamic Light ScafterinE A p plication8 ofPhoron CorrekafionSpectroscopy; Pecora, R.,Ed.; Plenum Pres:
New York, 1985; pp 85-179. (34) Janson, J.-C.; Hedman, P. Biofechnol.Prog. 1987, 3 ( I ) , 9. (35) Athalye, A.; Gibbs, S.G.; Lightfoot, E. N. J. Chromatogr. 1992,589 (1 &2), 71.
Laser Interferometry-Fluorescence Quenching Study of the Solvent-Swollen Gel Layer of Thin Poly(methyl methacrylate) Films Undergoing Dissolution Tbierry Nivaggioli, Fei Wang, and Mitchell A. Winnik* Department of Chemistry and Erindale College, University of Toronto, Toronto, Ontario M5S I A l , Canada (Received: April 7 , 1992)
A new method to analyze experimental results based on laser interferometry and fluorescence quenching (LIFQ) is reported. The method was applied to study the dissolution of thin poly(methy1methacrylate)(PMMA) films. Two fundamental parameters of the solvent-swollen gel layer at the solvent-polymer interface were taken into account: its thickness and its refractive index profile. Good agreement with the experimental results was obtained when using a simple linear profile. We were able to describe the evolution of the gel layer thickness as the dissolution takes place.
I. Introduction The dissolution of glassy polymer films is conveniently divided into three steps. The first is the diffusion of the solvent molecules into the rigid polymer matrix. The solvent molecules then promote the relaxation of the polymer chains and the formation of a solvent-swollen gel layer or transition layer. The last step is the diffusion of the polymer chains from the gel to the pure solvent. In case I1 diffusion,Ia the second step is taken to be the ratelimiting step. The case I1 diffusion model predicts a linear dependence on time of the change in thickness of the film, whereas a t ’ / 2 dependence would be anticipated if the initial Fickian diffusion (step 1) were the slow step in the dissolution process. Our concern is with the solvent-swollen gel layer at the polymer-solvent interface. Even when this layer is too thin to be observed by optical microscopy, its presence can be infered through optical interferometry studies. Aside from the fundamental scientific concern about the mechanism of dissolution of thin polymer films, the characteristics of the gel layer are of practical interest to the microelectronics industry. With the drive for ever-increasing resolution in microelectronic device fabrication is the recognition that, in solvent-processed photoresists, swelling at the solvent-polymer interfacecan limit the attainable resolution. During the last decade, much effort has been made in describing, fundamentally and experimentally, the mechanism of diffusion in polymers and their dissolution,’ with particular emphasis on the transition l a ~ e r . ~ The , ~ ultimate objective is to be able to determine the thickness of the gel layer and its solvent concentration profile. We report here a new method to analyze experimental results based on laser interferometry and fluorescence quenching (LIFQ) experiments. The analysis of the interferometry results is based on the description of the optical properties of a statified medium by a characteristic matrix. We will show how these results, associated with simultaneous fluorescence quenching experiments, give valuable information to describe the dissolution process of a thin poly(methy1 methacrylate) (PMMA) film. We will also compare our method with a more simple approach originally developed by Rodrigueza2
the reflection coefficientof the system is related to the elements of the matrix by
r=
- (m21 + nsm22)
(mil
+ Wl2)nl
( 4 1
+ nsml2)nl + (m21 + nm22)
(1)
where nl is the refractive index of the first layer and n, that of the final layer. r can be a complex number, and the reflectivity of the system is given by R = rr* = lrI2
(2)
II.1. Application to a Thin PMMA FUm Undergoing Dissolution. The system of interest to us is shown in Figure 1. It is composed of a polymer film and two semiinfinitemedia, solvent and substrate, characterized by two fixed refractive indices (nl and n3). The polymer film is divided into two regions. One is the glassy part, characterized by the uniform refractive index of the polymer (n2). The other is the solvent-swollen gel layer (at the polymer-solvent interface) and is characterized by a refractive index that is a function of z (n(z)). This system is a stratified medium along the z axis. We have shown that associated with this system is the following characteristic m a t r i ~ : ~ r
M=
I
cos (a)- kgyIL\,sin ( a )
-i[k,,A, cos ( a ) +
1 sin ( a ) ]
1
“2
-i[/6A cos ( a ) + % sin ( a ) ] cos ( a )-
&?sin ( a ) n2
J
where a = kohn2;k,, the wave vector; and h, the thickness of the dry PMMA, A = J’n2(z) dz and B = J’p dz = Ag (nonmagnetic materials). Here 4 represents the total thickness of the solvent-swollen gel layer and n(z) its refractive index profile. Substituting the coefficients of matrix M into eq 1, one obtains, after some simplifications (3)
11. OpticsTbeory
Abelk4 showed that the optical properties of a stratifed medium can be fully described by a 2 X 2 matrix,called the characteristic matrix of the medium. If one obtains
0022-3654/92/2096-7462$03.00/0
where rI2and 123 are the classical Fresnel coefficients. The terms $, 4, and f are dependent on A and 4.’Equation 3 leads directly to
0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 18, 1992 1463
Study of the Gel Layer of Thin PMMA Films
Xf
2.2
a,
c.
2.1
E
r
c
U
!!!
2
a u) U
1.9
0
Figure 1. Schematic representation of a PMMA film undergoing dissolution. b ..->
60
80
100
Gel Layer Thickness (8.u.)
Figure 3. Squared refractive index for two different hypothetical profiles: linear and cosine. The left-hand intercept refers to n2 of pure solvent; the right-hand intercept refers to that of PMMA.
f < 1 ( with gel layer )
5
40
20
profile), Krasicky and Rodriguez2 use the following expression, which they say is based upon an optical approximation
f
sin ( a / 2 ) fK,R
r
Time
Figure 2. Reflectivity versus time for the dissolution process with and without a gel layer. Amp represents the amplitude of oscillations and s is the 'offset".
A good approximation for transparent films (consideringthe small values of the Fresnel coefficients) is R ==j%122 + 2fr12r23cos
+ r232
(5)
Heavens has shown6that this result is only valid for a certain range of refractive index differences (rV
