J . Phys. Chem. 1988, 92. 4922-4938
4922
extensive formation of styrene and a-methylstyrene, respectively, from our observations appears probable.
spectra in isopropyl and tert-butylbenzene pyrolysis in ref 4 was interpreted by the common product, in one case by C-H bond split, in the other by C-C bond split. Some of the arguments in favor of this interpretation do not apply any longer. Therefore, the stable end absorption of isopropylbenzene pyrolysis in Figure 6 can now equally well be interpreted by a primary C-C bond split followed by phenylethyl decomposition forming styrene, Le. PhCH(CH3)2 (+M) PhCHCH, (+M)
-
PhCHCH3
+ CH3 (+M)
PhCHCH2
+ H (+M)
Conclusions The present work has demonstrated that UV molecular absorption spectroscopy, in spite of overlapping absorption continua from several species, can be used for analyzing the high-temperature pyrolysis of ethylbenzene. A dominant C-C bond split was identified and its rate constant was measured. Inspection of the complex mechanism of secondary reactions reconfirmed the surprisingly fast rate of consumption of benzyl radicals. The observations of benzyl radical properties, which we have verified now with several different benzyl precursors, call for a reinterpretation of several shock wave studies using other detection techniques. A clarification of the benzyl fragmentation pathways and rates may be expected from separate laser photolysis studies of benzyl radicals. The high-temperature pyrolysis mechanism of aromatic hydrocarbons apparently involves a much more complicated mechanism of secondary reactions, as compared to that at moderate temperature conditions, that intense future work is required before a complete modelling is possible.
(23) (3)
The analogous C-C bond split in tert-butyl benzene with subsequent formation of a-methylstyrene, i.e. PhC(CH3)3 (+M)
-
-+
PhC(CH3)2 + CH3 (+M)
(24)
+ H (+M)
(25)
PhC(CH3)2 (+M) PhCCH3CH2
would lead to the product spectrum of a-methylstyrene. Our present investigations of genuine a-methylstyrene indicate very similar spectroscopic and kinetic properties of this molecule and styrene. Therefore, the identity of product spectra from isopropyl and tert-butylbenzene, as demonstrated in Figure 11, finds a rational explanation in terms of primary C-C bond splits as well. In our present work we have not attempted to analyze the details of the decomposition kinetics of these molecules. Various radical attacks of the parent molecules in analogy to the ethylbenzene mechanism are certainly important secondary processes following reactions 23 and 3, or 24 and 25. In this way, the high-temperature pyrolysis mechanisms soon become of similar or even larger complexity as for ethylbenzene and toluene. However, an
Acknowledgment. Financial support of this work by the Deutsche Forschungsgemeinschaft as well as discussions of laser photolysis experiments with L. Lindemann are gratefully acknowledged. Registry No. PhCH2CH,, 100-41-4;PhCHCH3, 2348-51-8; CH,, 2229-07-4; H, 12385-13-6; PhCHj, 108-88-3; PhCH2, 2154-56-5; PhCHBrCH,, 585-7 1-7; PhCH(CH3)2,98-82-8; PhC(CH3),, 98-06-6; styrene, 100-42-5;benzyl iodide, 620-05-3; a-methylstyrene,98-83-9.
Photochemistry In Strongly Absorblng Media James R. Sheats,* Hewlett- Packard Laboratories, 3500 Deer Creek Road, Palo Alto, California 94304
James J. Diamond,* and Jonathan M. Smitht Department of Chemistry, Bates College, Lewiston, Maine 04240 (Received: September 29, 1987; In Final Form: February 8, 1988) We report the solution to the coupled nonlinear first-order partial differential equations representing the dynamics of photochemical reactions in systems essentially free from diffusion. No limitation regarding absorption of light by products or inert medium is assumed. These equations have been solved with the boundary condition that the species be uniformly distributed in the material prior to irradiation; the incident light may have an arbitrary time dependence. This analytical method of solution has been applied to photobleaching reactions described by various rate laws, where both reactants and products can absorb light. In addition, model systems in which the products absorb more than the reactants (photoantibleaching reactions) are also studied. An empirical procedure is introduced that accounts for changes in surface reflectance during the photoreaction in a satisfactory manner. Finally, a comparison is made with the corresponding equations describing photochemistry in perfectly stirred media.
Introduction Photochemical reactions in condensed media are of interest in a wide variety of applications; examples range from lithography for microcircuit fabrication’ and optical data recordingz4 to polymer degradation5v6 and solar energy capture.’-1° In such systems the resultant chemical transformation is often governed by a pair of coupled partial differential equations:
--ar(zJ) - (a$, + apcp + (Y,C,)Z
az
f Current address: Department of Chemistry, Wesleyan University, Middletown, CT 06457.
0022-365418812092-4922.$01.50/0
Here C, denotes the concentrations of reactants, C, that of products, and C, that of an inert medium; ai the respective (base e) molar extinction coefficients, and t and z refer to time and position (along the beam direction), respectively; Z(z,t) represents (1) Thompson, L. F.; Willson, C. G.; Bowden, M. J. Introduction to Microlithography; ACS Symposium Series 219; American Chemical Society: Washington, DC, 1983. (2) Janai, M.; Moser, F. J. Appl. Phys. 1982,53, 1385-1386. (3) Morinaka, A,; Oikawa, S.; Yamazaki, H. Appl. Phys. Lett. 1983.43, 524-526. (4) Moerner, W. E.; Levenson, M. D. J. Opt. SOC.B 1985, 2, 915-924. (5) Gooden, R.; Hellman, M. Y.; Hutton, R. S.; Winslow, F. H. Macromolecules 1984, 17, 2830-2837. (6) Marinero, E. E. Chem. Phys. Lett. 1985, 115, 501-506. (7) Renschler, C. L.; Faulkner, L. R. J. Am. Chem. SOC.1982, 204, 33 15-3 3 20. (8) Hargreaves, J. S.; Webber, S.E. Macromolecules 1982, 15, 424-429. (9) Kim, N.; Webber, S. E. Macromolecules 1982, 15, 430-435. (IO) Ng, D.; Guillet, J. E. Macromolecules 1982, 15, 724-727.
0 1988 American Chemical Society
The Journal of Physical Chemistry, Vol. 92, No. 17, 1988 4923
Photochemistry in Strongly Absorbing Media the light intensity at a depth z below the irradiated surface zo at time t ; F(CJ is a rate function that can take a variety of forms. Energy transfer and concentration-dependent spectra may add further complications that will not be considered here. If the sample of interest is sufficiently dilute, the light intensity I will be approximately independent of z , and the rate equation (eq 1) will then reduce to an ordinary differential equation that can be solved by standard methods. In many applications of practical importance, however, this is not the case, and one is left with a mathematical problem that has generally been attacked by numerical We have obtained solutions to these equations for essentially arbitrary kinetic functions F(C,), in the presence of arbitrary amounts of absorbance by products and medium. In this paper we consider only media in which mass transport is negligible within the reaction relaxation time and over the optical extinction length. These restrictions are most readily satisfied by solids, but many photoprocesses in fluids will also be accommodated. For example, the root mean square displacement of a typical fluid molecule ( D = cm*/s) along the beam axis is 0.045 cm in 1 s; thus, our treatment would provide a useful description of a photoreaction with a relaxation time of 1 s or less if the optical extinction length were about 1 cm or greater. The solutions are obtained as the zeros of a nonlinear algebraic function that is monotonically increasing and whose roots thus can be easily computed by a stable numerical procedure; the expression for this algebraic function includes an integral that in general cannot be expressed in terms of elementary functions. Thus the mathematical problem of solving a pair of coupled nonlinear partial differential equations (for which there is no guarantee that a given discretization represents the true solution) is reduced to a far simpler numerical problem whose convergence and correctness are guaranteed. These algebraic solutions are also amenable to further analysis (for example, obtaining extrema with respect to a parameter13,14). In its general form, the analysis is limited to systems in which the concentrations of all species are initially uniformly distributed. For the case of a reaction first order in both C,and I , it is possible to obtain the general solution for arbitrary boundary conditions. In either case, the incident light may have an arbitrary temporal variation, and concentration-dependent surface reflectivity is accounted for in an experimentally satisfactory manner. Finally, the analysis assumes that a single rate law F(C,)applies throughout the entire reaction. This condition may not be satisfied by some photoreactions in crystalline media that are strongly influenced by local packing effects;I5 as the reaction progresses, the crystal structure may change and thereby change the rate constants or even the rate law. In amorphous solids such structural changes are likely to be insignificant unless the photoreaction causes a change of state of the medium. These limitations must be borne in mind in any specific application.
I! INCREASING DEPTH BELOW SURFACE + Figure 1. Diagram of experimental setup described by the equations in this article. The region between Zkc(t) and I(zo,t) is infinitely small. In general I(z,,t) < linc(f) due to reflections.
In addition, a nonreactive (inert) medium may be present with time-dependent concentration Cm(z,f) = Cmo(z). The concentrations of these species determine the attenuation of light along the beam path according to the Beer-Lambert equation (eq 2). Note that the molar extinction coefficients ai used here are related to the usual decadic molar extinction coefficients q through the relation ai = ei In 10. In general, the quantities C,,C,, and I depend on both time and position, leading to the coupling of the differential equations. The basic equations can be expressed as
(5)
These are subject to the boundary conditions
C,(z,to) = c,O(z)
(7)
I(zo,t) = Zo(4 (8) where Zo(t) is the light intensity in the medium inside the surface zo at time t (this is not necessarily the same as the incident light intensity, because of surface reflections, see Figure 1). Equations 5 and 6 (and associated boundary conditions 7 and 8) can be taken as the basic coupled differential equations of interest, which describe the photoreaction in the absence of change in surface reflectivity, nonreactive scattering, or diffraction. First-Order Case. Let us consider the simplest case, for which the reaction rate is first order in both intensity and concentration, and only the reactant absorbs light. The coupled equations (eq 5 and 6) then become
Theory Differential Equations. We consider reactions of the form
R
hv
XP (3) where R and P represent the photoreactant and photoproduct, and X is the ratio of the stoichiometric coefficients in the reaction. The rate equation (eq 1) for this reaction is assumed to be first order in the light intensity. The initial concentrations of reactant and product are C>(z) and C t ( z ) ; the time-dependent concentration of photoproduct is determined by the stoichiometry:
~~
These equations are exactly solvable, and their solutions have been reported by several authors.I6-*l The general solutions subject to the boundary conditions (eq 7 and 8) are &')
Cr(z,t) = C,O(Z)e&)
+ ,v(f) - 1
(1 1)
~
(1 1) Oldham, W. G.; Nandgaonkar, S. N.; Neureuther, A. R.; O'Toole, M. M. ZEEE Trans. Electron Devices 1979, ED-26, 717-722. (12) Neureuther, A. R.; Oldham, W. G. Solid State Technol. 1985,28 (5). 139-144. (13) Diamond, J. J.; Sheats, J. R. ZEEE Electron Deuice Lett. 1986, EDL-7, 383-386. (14) Sheats, J. R.; O'Toole, M. M.; Hargreaves, J. S. SPIE Proc. 1986. 631, 171-177. (15) Heller, E.; Schmidt, G. M. J. Zsr. J . Chem. 1971, 9, 449-462.
(16) (17) (18) (19) (20) 569. (21)
Wegscheider, R. Z . Phys. Chem. 1922, 103, 273-306. Herrick, C. E., Jr. J . Opt. SOC.Am. 1952, 42, 904-910. Herrick, C. E., Jr. ZBM J . 1966, 2-5. Kessler, H. C., Jr. J . Phys. Chem. 1967, 71, 2736-2737. Mauser, H. Z . Naturforsch. B: Anorg. Chem., Org. Chem. 1967, 22, Simmons, E. L. J . Phys. Chem. 1971, 75, 588-590.
4924
Sheats et al.
The Journal of Physical Chemistry, Vol. 92, No. 17, 1988
where the functions p ( z ) and v ( t ) are determined by the initial conditions: p ( z ) = a r l z C , O ( z ’ )dz’
(13)
In b,+l(p,v)= e-(r’’)6 In b ( p , v ) , v > v’ (26) Then, in the limit of small 6, the resulting error in the next iterate bi+l is, for v > v’
20
6 In b,+, = In bi+l - In b = [e-(rru‘)6 - I] In b
v ( t ) = k l ‘ I o ( t ’ ) dt’
(14)
6bi+I lim - = (v
‘0
We will later demonstrate that these are the correct solutions. It is sufficient at this point to note that these equations (eq 11-14) represent functions that do in fact satisfy the coupled differential equations with the appropriate boundary conditions. In addition, their form suggests how the more general case might be solved. First, we transform the equations to a dimensionless form. We define two new dependent variables a and b in terms of the dimensionless independent variables p and u: a ( ~ , v=) Cr(zJ)/C,O(z)
(15)
b(g,V) = I(z,t)/Io(t)
(16)
The coupled partial differential equations and associated boundary conditions (eq 11-14) now become
b(0,v) = 1
6
6-4
(27)
- v’)b In ( l / b ) > 0
so the propagated error increases when this is larger than unity. If v’ is small, this expression becomes When p = v, then b = 6bi+,/6 = (v/2) In 2 = v/3, so the propagated error will become larger in the portion of the photobleaching curve where b is changing most rapidly. This certainly decreases one’s confidence in the algorithm. More sophisticated methods can avoid the above flaw in the naive algorithm, but this is a clear example of the difficulty present in solving nonlinear equations by numerical methods. This instability is also manifested in a noticeable but harmless manner in our own algorithm. Our approach exploits a simpler reformulation of the same equations into a new problem that can be addressed by stable numerical methods. General Rate Equations. We now transform the more general coupled partial differential equations (eq 5 and 6). We define a and b as in eq 15 and 16 but suppress the specification of the independent variables: The differential equation describing the reaction rate is then
with solutions or
The simplicity of this pair of equations suggests that a similar transformation in the coupled equations (eq 5 and 6) may make the essential mathematical problem more transparent. Before we proceed to the more general rate equations, we examine the nature of the instability present in approximate methods of solution of these coupled nonlinear equations. Suppose that we attempt a naive recursive method of solution:
In bi+l = -J”ai+l dp’
(22)
where the iterates, ai and b,, converge to the functions 2 and
ii = lim
w-
a,;
6 = lim
b,
6:
where
In general, v(t) depends explicitly on t through the arbitrary time dependence of the surface radiation Io(t),and it depends implicitly on z through the spatial dependence of the initial reactant concentration C,O(z). We can transform the Beer-Lambert equation in a similar fashion. The gradient of the (base e ) absorbance is
(23)
n-oJ
(33)
There is no guarantee that the sequence has converged to the exact solutions, a and b. In fact, the approximate solutions can diverge from the true solutions in the limit of large arguments, for example, v a. Suppose that, in the ith iteration, bi is too large; then too much reactant is consumed, and ai+l is too small. This means too little light is absorbed, and bj+l is also too large. For large enough initial errors, the error increases as the iteration
-
where b = Z(z,t)/Zo(t)= e-r, with a, and C,(z,t) as previously defined. The stoichiometry of the reaction (eq 3) allows this to be simplified:
a
_ - rz az
=a
+ ~apcpo ~ + a , ~ , -~ - c,(z,t)/C;] = r: + ( r i- r,z)(i- a ) ,
(34)
progresses.
( a , - xap)c:[l
Let us be more precise. Suppose that there is a small error 6 in a computed value of an iterate: bi(p,u) = b ( p , u ) , if v Iv‘
where l?,is the initial absorbance, ri’ is its derivative, rris the final absorbance when the photoreactant has been completely consumed, r; is its derivative, and a = C,/C:. Again, in general both r: and r f will depend on z. We now impose the initial condition that the reactants, products, and medium be uniformly distributed in space prior to irradiation; this implies that v(t), r:, and r; are space independent. This assumption considerably simplifies the mathematics; solutions of the general differential equations (eq 5 and 6) appear intractable without this assumption (vide infra). We can rewrite the rate equation as
= b(p,v)
+ 6,
otherwise
(24)
Then, using the rate equation (eq 21), we can compute the propagated error in a,+l: ai+l(p,v)= a(p,v), if v Iv’
= a(p,v)e-(“-d)6,otherwise (25) We then compute the next iterate, bi+l,focusing on points succeeding the introduction of error:
(35)
The Journal of Physical Chemistry, Vol. 92, No. 17, 1988 4925
Photochemistry in Strongly Absorbing Media or
(45)
aa
-- = f ( a ) b ill,
We now redefine the independent variables p and v: K(Z)
= C;(CX, - X C X J ( Z - z0) =
(ri.- r ( ) ( z - z0)
(37)
provided that f(y) does not vanish in some neighborhood of y = 1 ; this is assured here since the rate of reaction is never zero in the presence of light and reactant. We can then define a function g(x) that is uniquely determined by the integral equation = x
In addition, we have a parameter {:
This is equivalent to solving the differential equation dg dx
(39)
g(0) = 1
- = -Ag(x)l;
(47)
The latter relation holds since ri and rf are proportional to ( z - zo). At a given depth z below the surface zo, we can relate p and { to ri and rr:
Since g(x) is the unique solution to the differential equation (eq 47), we have obtained our solutions for a and b:
ri = (1 + { ) P ; rf = b
a(z,v) = g(ve-r); b ( z ) = e-r
(40)
(48)
The amount of light absorbed by the sample, AE, is
or
p = r i - r f ;{ = -rir-f rr
AE = Jol[Zo(t’)- Z(z,t’)]dt’ -
We can interpret p as the change in the dimensionless optical depth r and v as the dimensionless surface fluence. In addition, since r is always nonnegative, {cannot lie in the half-open interval [-1,O). The proof of this follows a simple argument. First consider the case where neither rinor rfvanishes. From eq 40, we observe that if { is negative, then p must be negative as well; likewise, it then follows that the quantity (1 + {) must also be negative; hence {cannot lie in the open interval (-1,O). The case where only the photoreactant absorbs (rf= 0) corresponds to { = 0. This is clearly an idealization, but it is useful as a model. The opposite limit where only the photoproduct absorbs (ri= 0) corresponds to { = -1; this makes no physical sense since the photoreactant must absorb some light to react. Hence, we impose the restriction { # -1. The case where r is constant corresponds physically to a time-independent attenuation of light; this case is represented 0, {p = r. mathematically by the limit { - m, p The coupled differential equations and boundary conditions describing both the reaction rate and attenuation of light are now
-
aa
a v = f ( a ) b ; a(p,O) = 1 --ab = ({ + a)b; b(0,v) = 1 ap --
ACr = s z ( C : - C,) dz‘-= C: 20
The quantum yield for the reaction, Q = ACJAE, is then
The quantum yield Qo of an infinitely thin sample of depth dz exposed to an infinitesimal dose of radiation for a duration dt is
kF(C,O) 1 - g(v) kF(C,O) lim -= -(52) lim lim Q(z,v) =I C 0 2-zo r ,-o r z
(43)
The dimensionless concentration of reactant a now depends on depth below the surface z and the surface fluence v. The equation describing the reaction rate (eq 42) can be formally cast as an integral equation:
v
and the amount of reactant consumed, AC,, is
(42)
If the independent variable v(t) were space dependent, the transformation of eq 34 to eq 43 would not be possible, since then one would have a/& = ( a p / a z ) a / a p + (a v / d z ) a / a V. We have imposed the boundary condition that all species are uniformly distributed prior to illumination so that this transformation is possible. If the rate law is first order, then the independent variable v cannot depend on space, and the imposed boundary condition is no longer necessary. The solution to the general first-order equations is discussed in Appendix 11. The singular case where r is constant requires only the solution of an ordinary first-order nonlinear differential equation. Since some of the techniques used to develop the solution are applicable to the general solution of these equations, we will describe the case where I’ is constant first. Time-Independent Absorption. When there is no change in the absorbance of radiation (that is, ri = rf = r),the light intensity then follows Beer’s law: b ( z ) = e-rz(z-zo) = e-r (44)
c,o
The order in which the limits are taken in the previous expression is irrelevant. So we obtain the expression for the quantum yield:
(53) where
(54) For example, in a first-order reaction, f(y) = y and Ax) = e-x, so that
j(x) =
1’7 1
- e-Y
dy = E , ( x ) + In x
+y
(56)
where y = 0.57721 ... is Euler’s constant, and E , ( x ) is the exponential integral. These equations and their solutions are familiar objects in kinetics. Solution to the Coupled Equations. The pair of equations (42) and (43), although simple in appearance, still represent a formidable mathematical problem when attacked by conventional
4926
The Journal of Physical Chemistry, Vol. 92, No. 17, 1988
means. We now show how this task can be reduced to a problem no more difficult than the time-independent case, coupled with the problem of finding the zeros of a nonlinear function. Let us make the integral form of the rate equation our starting point:
Sheats et al. Therefore, we have obtained a compact expression for the needed integral:
(57) We now define an auxiliary function, d ( p , v ) , which describes the dimensionless local fluence:
W[4(MJ)l ( 7 1 ) and the differential equation (eq 63) determining 4 is now seen to be the integrable first-order partial differential equation
a4/acl = -K4 + w(4)l
(72)
We again rewrite this as an integral equation: The function 4 satisfies the boundary conditions
+(O,v) = J”b(0,v’) dv’ = v;
$(p,O) = 0
(73) (59)
As previously noted, the function a(p,u) is uniquely determined by + ( p , v ) and g(x), the solution to the integral equation (eq 46): a(cl3v)
= g[4(cL,v)I
(60)
The function 4 satisfies a pair of partial differential equations: a w v = wp,v)
Partial differentiation of both sides of this equation with respect to v leads to an expression for the dimensionless light intensity b, where d+/dv = b:
(61)
and
(74) Since a ( w ) = g(+), once we have determined the local fluence 4, we can compute both a and b, and the pair of partial differential equations (eq 42 and 43) have then been solved under the assumed boundary conditions. The integrand of eq 73 is singular at the origin and so cannot be used in a direct numerical attempt to invert (and thus solve) the equation in terms of 4. The singular behavior can be isolated by noting that d / d x [Tx + w(x)l =
The problem at this point is the evaluation of the integral Jgab dv’, which is a key element of this derivation. We can rewrite the dimensionless rate equation (eq 42) as
We can write the integrand of eq 63 in a similar fashion:
r + g(x)
Multiplying each side of eq 73 by (1 relationship yield upon rearrangement
+ [)
(75)
and using this last
where the integral H(x) is
so that the desired integral is
The integrand in eq 77 is nonsingular at the origin, so the resulting integral can easily be evaluated numerically. We can find H(x) for small x:
This integral can be considerably simplified by noting that all dependence on p and v in the above equation arises from the function ~ $ ( p , v ) .We can associate a unique function w(x) with the integral
so that as x
-
0, we have
1
w(x) =
J-&)
dY
(67)
The first-order differential equation for w(x) is then
so that w(x) is also represented by the integral
H(x) =
X 0, u > 0. From eq 83, we note that if x is positive, then aG/dx is also positive (that is, G(x;p,v) is monotonically increasing for x > 0). We also observe for p,v # 0, that G(v;p,u) = (1 + {)p > 0 and also that limA+ C(x;p,u) = -OD, so that there is exactly one zero of the function G(x;p,u) in the interval (O,u]. In addition, +(p,O) = 0 and ~ ( O , U = ) u, so the set of solutions is completely and uniquely specified for all possible values of p and v. From the relation between b and 4 (eq 74) and the definition of G (eq 82), we see that we can also express the optical absorbance as r(p,v) =
- In b(p,u) = H(4)- H ( u ) + ri
(84)
The amount of light absorbed by the sample, AE, is (cf. eq 49)
m = - c:
(85)
kF( C;) (’ -
and the base 10 absorbance A is r / l n 10. This is equivalent to
(94) From eq 61 and 74, the partial derivative of b with respect to is
1 ab - g(4) - g ( 4 b au iV + w ( v )
1 - a)dz’ =
(95)
so the expression for the contrast is y = -u
g(4) - g(v) b + w(v)
(96)
As an example, the contrast for a first-order rate law where only the reactant absorbs may be shown to be e” - 1 (97) e@ ey- 1 Effects of Changing Surface Reflectance. The reflectivity, R , at the interface between a crystal surface and vacuum is, for normal incident light y = -v
and the amount of reactant consumed, AC,, can be shown to be
AC, = C:s’(
u
R =
+
+
+ nz2 (nl + + nz2 (nl -
where n = nl in2 is the complex index of refraction of the crystal. The gradient of the optical depth, rz,is proportional to the imaginary part of the index of refraction, nz:
’0
rz= 2wn2/c
The quantum yield for the reaction, Q = ACJAE, is then
(99)
We have already assumed that changes in the optical depth are linear in the change in concentrations of each species (Beer’s law), and it has been observed that changes in surface reflectivity are in some instances proportional to the amount of photoproduct produced in a reaction.24 We therefore linearize the dependence of the surface reflectivity upon chemical composition. It is more convenient for us to discuss the change in surface albedo in terms of the surface transmittivity. Let us define the surface transmission coefficient as
or
where Qo,the quantum yield for the photoreaction of an infinitely thin sample with the same composition, is now given by Qo = kF(C:)/(I’:). We note that the expression for the quantum yield of a photoreaction with time-independent absorption (eq 53) may 0, {p = be obtained from eq 90 by taking the limit {- m, p r, since
-
lim (1
r--
+ {) H ( x ) = r--lim ( 1 + {)
1-gb) dy = Cy w(y)
+
In addition, we may also calculate the contrast y, which is defined as1’sZ3 dA Y=d log10 E where the exposure E is proportional to u: (22) See any standard text on real analysis, for example: Buck, R. C. Aduamed Calculus; McGraw-Hill: New York, 1965; pp 284-285. (23) Babu, S.V.; Barouch, E. IEEE Electron Deuice Lert. 1986, EDL-7, 252-253.
where Zhc is the incident light intensity. We have used the symbol S for the surface transmission coefficient rather than the symbol T, to distinguish between the transmission of the surface and the transmission of the sample itself. The linearized dependence of the surface transmittivity on the progress of the photoreaction yields an approximate expression for S in terms of the surface concentration of photoreactant, a(0,u) (cf. eq 34):
S ( v ) i= Sf
+ (Si - Sf)a(O,v) = Sf+ (Si- Sf)g(u)
(101)
Recall the definition of the surface fluence, v(t) (eq 38). Let us define the dimensionless incident fluence, uo(t), in an analogous fashion; then
With use of the definition (eq 100) of the surface transmission coefficient, S ( v ) , the differential equation for v is then
or S ( U )= dv/dvo = Sf -b
(Si
- Sf)g(v) = Sf- M g ( v )
(104)
(24) Spencer, H. E.; Schmidt, M. W. J. Phys. Chem. 1970,74,3472-3475.
4928
The Journal of Physical Chemistry, Vol. 92, No. 17, 1988
X
4.00
-4.00
/
,,
I
*
v
-2.00
0.00
/
2.00
4.00
X
Figure 2. Function q ( x ) and its asymptotes, y = e“ and y = x . The function q satisfies the equation q ( x ) + In q ( x ) = x .
where AS = Sf - Si. We rewrite this, as usual, as an integral equation for vo: vo = JVSf - S g ( Y ) or
Sheats et al.
Y
”0°
I / p = x,, this quantity is undefined at the point x = x,, so a Taylor series expansion of the function g(x) about this point is not possible. The integral w(x) is
and thus that (1
-
P-W
Therefore, the quantum yield for the reaction upon continued exposure becomes, for v Iv,
(149) This reaches its maximum value at x = x,: w(xm) = 1/(1 + P )
(1 50)
The integral H(x) can be evaluated in closed form for x, I x:
where the expression for Q(p,v,) is given in eq 161. The quantum yield is observed to decrease as the applied dose increases after complete bleaching has been achieved. In the limiting case where only the reactant absorbs ({ = 0), the bleaching dose v, is v, = x,
P +1+P
so that the quantum yield is then constant: Q(Vmrp)/Qo
We define the function J(x):
Then G ( x ; w ) = J(x)
- J(v) + (1 + P h
(153)
We can find a closed form expression for J(x) when x, Ix:
Therefore, we can calculate the fluence v, needed to completely bleach the sample to a depth corresponding to a particular value of p:
4 h V m ) = xm
(155)
so the equation determining the relation between v, and x, is 0 = G(xm;p,vm) = J(Xm) - J ( v m )
+ (1 + P)CL (156)
or
J(ud -Jbm)
= (1
Then, from eq 151 and 154, we have
+ PIP
(157)
= 1+P
'
1
(166)
This raises the interesting possibility of the exploitation of photoreactions whose efficiencies increase (up to a certain point) upon exposure to radiation. This expression for the quantum yield should not be confused with the definition of the initial quantum yield Qo (eq 52); the latter is the quantum yield of the reaction of an infinitely thin sample exposed to an arbitrarily small dose of light. The expression of eq 166 is independent of p and holds for any value 0; in this case, however, the dose of of p, even in the limit p light driving the reaction is not arbitrarily small; it is fixed by p: it is exactly that dose required to completely bleach the sample to the depth corresponding to the particular value of p , In this case, one can assume that the functions Ax), w ( x ) ,and H(x) have reached their asymptotic limits corresponding to large values of the argument x (for example, w ( x ) = 1/(1 + p ) ) . On the other hand, in the calculation of the initial quantum yield Qo, one assumes that the functions g(x), w ( x ) , and H(x) should be evaluated at the opposite limit where x is small (for example, w(x) = x). These are two distinct limiting cases.
-
Discussion In the foregoing sections we have presented a mathematical formalism for computing the concentration and optical transmittance profile of photoactive solid media. A comparison to experimental data was made previously for a first-order rea~ti0n.l~ In this section we will briefly discuss some of the ways in which this analysis may be used. We also discuss prior work on this subject and its relation to our own work. The solutions to eq 1 and 2 for a first-order rate law have been reported by several authors. The solutions for the case { = 0 were apparently discovered by Wirtinger no later than 1922, as reported by Wegscheider.16 Extension to the case of nonzero {was made
4934
The Journal of Physical Chemistry, Vol. 92, No. 17, 1988
Sheats et al.
by Kirkwood in 1947; this work was cited by HerrickI7J8in 1966 for both stirred and quiescent systems. In Appendix I, we compare as a personal communication from Professor Kirkwood. The { our formalism with the corresponding equations that are appro= 0 solutions have also been reported by Kessler,19Mauser,20and priate in perfectly stirred photochemical reactions. Simmons.21 Simultaneously with publication of our earlier comActinometry. Chemical a ~ t i n o m e t r ycan ~ ~ be simplified and m ~ n i c a t i o n , 'Babu ~ and Barouch reported the exact solution of generalized extensively by use of the present formalism. For a eq 1 and 2 for a first-order rate law.23 Their method of solution given solid-state reaction (that has been previously calibrated), is equivalent to that of Kirkwood if the transformation of eq 70 actinometry consists simply of obtaining a transmittance versus is made. In all of these reports, the assumption that all species time curve and analyzing it according to the present equations; are uniformly distributed prior to exposure to light is made to the variable Y consists of the product of time, incident intensity, and parameters that are all constants of a given material (cf. eq simplify the mathematics and render expressions that are im38). The analysis will be simplest if the reaction is first order mediately integrable. The approach of HerrickL8is interesting and there is no product or medium absorbance ({ = 0), but these because it includes the possibility that several independent are not necessary restrictions. The large variety of photoreactions first-order photoreactions may be occurring simultaneously. In this set of papers, each reported solution to the first-order described by eq 1 should provide a convenient and accurate problem is essentially the integral equation (73). As we have chemical actinometer in every part of the ultraviolet and visible discussed above, the integrand of eq 73 is singular at the origin; spectrum. An example that is useful in the region of 300-450 this presents some numerical problems that our method can avoid. nm is given in ref 13. Brauer and Schmidt have reported some actinometric systems Also, the representation of the scaled reactant concentration and that can be analyzed by transmittance at a wavelength other than light intensity of eq 60 and 74 is more convenient, since this yields explicit expressions for the dependent variables in terms of the the irradiation w a ~ e l e n g t h . ~The ~ ? present ~~ equations can be used independent ones. In addition, our method is not limited to the for their chemical systems also, with analysis at the wavelength first-order rate law, as is that of the previous authors. of irradiation. It can be shown38that the precision of the latter Photochemical Kinetics. Many of the characteristics considered method can be made better than for the former. here do not arise in stirred fluid solutions. The study of photoThe greatest advantage of the present method lies in its genchemical kinetics is quite similar to that of other types of chemerality and versatility; the actinometric photoreaction can be chosen to suit the needs of the individual case. As an example of a unique istry. In solids, however, there are special complications. Not application, one could accurately determine the intensity and only must one deal with the nonlinear coupling of eq 1 and 2, but the conventional chemical analysis used to obtain product and diameter of a tightly focused Gaussian laser beam. This problem kinetic information may be difficult; in the case of a cross-linked is not readily amenable to conventional solutions when, for expolymer, it cannot be done at all. Thus it is useful to have a means ample, the beam is contained in a m i c r o ~ c o p eit; ~continues ~ to by which reaction mechanisms and kinetic parameters may be be a subject of concern even without this complication.40 With the present approach, thin (=l Mm) films of a photobleachable determined by direct, in situ optical analysis. A specific example of such an application is the photobleaching polymer, readily prepared by spin-casting,' can be placed in the of 9,lO-substituted anthracenes in methacrylate polymers.29 We focus of the beam and bleached to determine the intensity; from have found that the rate law f ( a ) derived from solution phase a knowledge of the total beam power the diameter follows. The studies2*gives a close but imperfect fit to the data, indicating the experimental apparatus required is just a photodetector, and the presence of additional mechanistic features in the polymer system. computation (including a numerical integration over the Gaussian Another example is given by Gupta and Goyal in a review of beam profile) is elementary. A second example concerns the photochemical experiments on visual pigment extracts.30 difficulty of intensity measurements in the poorly accessible ilMicrophotolithography'~'1~13 is an important field in which lumination plane of a lithographic projection camera;41the present solid-state photochemistry plays a central role. The industrial method is ideally suited to this app1i~ation.l~ Mechanistic Studies. In industrial and other practical appliapplication constrains concentration and film thickness to specific cations of photochemistry, knowledge of concentration or bleach ranges, and the measurement of photokinetic parameters, conprofiles is often important in the context of some process, and centration profiles, and bleaching characteristics must be made optimization of such profiles with respect to various specifications on materials whose optical density does not in general allow any is needed. Photolithography'.'' is a characteristic example. In simplifying assumptions. We have shown e l ~ e w h e r e that ' ~ the addition to the conventional process of exposure of a material to use of the present formalism to determine the photochemical change its development rate in some solvent, photobleachable efficiency of a commercial resist is superior to the conventional chemicals are being introduced for use as image enhancer^.^^^^^^^ method of extrapolating data to zero exposure. Another method that has been applied to the study of solid-state The nonlinear dependence of transmittance on incident dose photoreactions is that of h ~ l o g r a p h y . ~ 'The . ~ ~irradiating laser provides a means of sharpening an image that is degraded by beam is split into two beams (typically of equal intensity), which diffraction. Several different versions of this scheme can be are recombined on the sample at some angle, creating a sinusoidal implemented, depending on whether the exposure of the bleachable interference pattern. The scattering of a probe beam is then used dye occurs separately from the resist or not. In all cases, to detect the growth of the interference pattern. This method is particularly useful for cases where the absorption of the actinic (33) Gauglitz, G.; Hubig, S. J . Photochem. 1985, 30, 121-125. beam is small, since as a zero-background technique it is quite (34) Mauser, H. Z. Nuturforsch. 1975, 30, 157-160. (35) Hatchard, C. G.; Parker, C. A. Proc. R. SOC.London, A 1956, 235, sensitive for such cases, while absorption measurement is difficult. 5 18-536. For cases where the absorption is strong or where the crossed-beam (36) Brauer, H.-D.; Schmidt, R. Photochem. Photobiol. 1983, 37, laser irradiation is either impossible or inconvenient, the present 587-591. technique is advantageous because of its experimental and com(37) Brauer, H.-D.; Schmidt, R.; Gauglitz, G.; Hubig, S. Photochem. Photobiol. 1983, 37, 595-598. putational simplicity. (38) Sheats, J. R.; Diamond, J. J., to be submitted for publication. Although our interest is primarily in solids, the analysis is (39) Wu, E.-% Jacobson, K.; Papahadjopoulos, D. Biochemistry 1977, 16, applicable to solutions provided that mass transport does not 1916-19Al __-significantly alter the concentrations during the reaction. Various (40) Kimura, S.; Munakata, C. Opt. Lett. 1987, 12, 552-554. (41) Shaw, J. M.; Hatzakis, M. J . VUC.Sci. Technol. 1981,19, 1343-1347. approximate methods of dealing with the nonlinear coupling between eq 1 and 2 have been discussed in the l i t e r a t ~ r e ~ ~ * ~ ~ (42) J ~ West, P. R.; Griffing, B. F. SPIE Proc. 1983, 394, 39-48.
__
(29) Sheats, J. R., manuscript in preparation. (30) Gupta, B. D.; Goyal, I. C. J . Photochem. 1985, 30, 173-206. (31) Burland, D. M. Arc. Chem. Res. 1983, 16, 218-224. (32) Deeg, F. W.; Pinsl, J.: Brauchle, C.; Voitlander, J. J . Chem. Phys. 1983, 79, 1229-1234.
(43) Sheats, J. R. Appl. Phys. Lett. 1984, 44, 1016-1018. (44) Vollenbroek, F. A,; Nijssen, W. P. M.; Kroon, H . J. J.; Yilmaz, B. Microcircuit Eng. 1985, 3, 245-252. (45) Sheats, J. R.; Hargreaves, J. S. Proc. ACS Diu. Polym. Muter. 1986, 55, 616-618. (46) Bartlett, K.; Hillis, G.; Chen, M.: Trutna, R.; Watts, M . SPIE Proc. 1983, 394, 49-56.
The Journal of Physical Chemistry, Vol. 92, No. 17, 1988 4935
Photochemistry in Strongly Absorbing Media knowledge of the bleaching profiles is essential for process development, and this has in the past required numerical solution of the differential equations.42 The analytical equations derived here will provide greater intuitive feel for such processes and provide for easier and more accurate optimization. We have previously reported some preliminary analysis of this type.13 Another topic of current interest to which our results are relevant is that of photoablation by high-energy pulsed For example, if the beam of an A r F laser (193 nm) is directed onto virtually any organic polymer, for sufficiently high fluences the polymer will be removed due to breakdown into small molecular fragments. The thickness removed per pulse is typically quite small because the penetration depth of the light is small. This phenomenon has potential relevance to surgery and drilling of microscopic high aspect ratio holes, among other possible uses.53-55 Previous ana lyse^^^,^^ have not considered the depth dependence of the photochemistry. Keyes et al.53applied eq 12 but did not consider the effect of absorbing products or inactive medium, both of which must be important for typical polymers at 193 nm.
Conclusions We have presented solutions of the kinetic equations for photochemical reactions in strongly absorbing media with negligible mass transport, which allow for arbitrary amounts of absorbance of photoproducts and inert media, arbitrary reaction rate laws and temporal variations of incident irradiance, and the concentration dependence of surface reflectance. Solutions are obtained in terms of a parameter that must be determined as the root of a nonlinear algebraic equation and that usually involves the numerical computation of a simple integral. These numerical efforts are elementary and can in fact be carried out with a programmable calculator. Application to a variety of problems in photochemistry are described. Acknowledgment. J.J.D. thanks Professor Shepley Ross, Jr., of the Department of Mathematics of Bates College for fruitful discussions regarding this article and for introducing us to the pleasure of TEX. Appendix I: Comparison with Reactions in Perfectly Stirred Media In perfectly stirred photochemical reactions, the concentrations of chemical species are uniform inside the reaction vessel, and the average scaled concentration of reactant a(v) depends only on the dose. Here it represents the spatially averaged reactant concentration: n ( v ) = -l:'a(z',v) 1 dz' ZI
Once again this can be recast in integral form as
so that
ab) = g(4) as was the case for diffusion-free reactions (eq 60). The Beer-Lambert equation is
The scaled light intensity b now decays exponentially into the sample:
6 = e-r;
r = rr+ (ri- r,)a(v) 6 = (1 - e - r ) / r
(1.7) (1.8)
Equations 1.3 and 1.8 can be combined to yield a differential equation for (5: (1.9) or, in integral form
One can solve this equation and so determine the reaction concentration. However, one may use the integral transformation (eq 70) to obtain the integral equation for ;b itself
yielding 4,a, and 6 from eq 1.5-1.8. The integrand of this equation is nonsingular for finite, real values of the argument s. This integral can be evaluated by standard methods.34 The amount of reactant consumed, AC,, is
AC, = C,O(Zl - zo)(l - (5) = C,O(Zl - zo)[l - g($)]
(1.12)
and the amount of light absorbed, AE, is
(1.13)
- 20
where the reaction vessel is in the light beam for a distance of zI - zo; we will use a similar notation for other quantities (for example, 4). The rate now depends only on the average reactant concentration, so the rate equation is (cf. eq 36) -aulav = f ( a ) b (1.2) This can be spatially averaged, yielding a rate equation depending only on average quantities:
With use of eq 1.9, this can be written as
From the definition of { (eq 39), this becomes
~
(47) Srinivasan, R. J. Vac. Sci. Technol., B 1983, I , 923-926. (48) Srinivasan, R.; Leigh, W. J. J . A m . Chem. SOC. 1982, 104, 6784-6785. (49) Geis, M. W.; Randall, J. N.; Deutsch, T. F.; Efremow, N. N.; Donnelly, J. P.; Woodhouse, J. D. J . Vac. Sci. Technol., B 1983, 1, 1178-1181. (50) Andrew, J. E.; Dyer, P. E.; Forster, D.; Key, P. H. Appl. Phys. Lett. 1983, 43, 717-719. (51) Koren, G.; Yeh, J. T. C. Appl. Phys. Lett. 1984, 44, 1112-1114. (52) Latta, M.; Moore, R.; Rice, S.;Jain, K. J . Appl. Phys. 1984, 56, 586-588. (53) Keyes, T.; Clarke, R. H.; h e r , J. M. J . Phys. Chem. 1985, 89, 4194-41 96. (54) Srinivasan, R. Science 1986, 234, 559-565. (55) Singleton, D. L.; Paraskevopoulous, G.; Taylor, R. S.; Higginson, L. A. J. IEEE J. Quantum Electron. 1987, QE-23, 1772-1782. (56) Garrison, B. J.; Srinivasan, R. Appl. Phys. Lett. 1984,44, 849-851. (57) Jellinek, H. H. G.; Srinivasan, R. J . Phys. Chem. 1984, 88, 3048-305 1.
Since 4 has been determined, we can rewrite this in familiar form: n o
T-
where w ( x ) has been previously defined (eq 67). The quantum yield Q ( v ) for the photoreaction is then (cf. eq 90) (1.17) with Qo = kF(C;)/ri2 as before. By comparison with eq 78, we note that this is simply the expression Q ( v ) / Q o = (1
+ iW'[?4~)1
(1.18)
4936 The Journal of Physical Chemistry, Vol. 92, No. 17, 1988
Sheats et al.
The expression for the quantum yield in diffusion-free media (eq 90) may be rewritten by using the mean value theorem:
And the scaled transmission coefficient b(p,v) is determined by $ as well (cf. eq 43 and 61):
H ( v ) - H(4) Q(W) -- (1 + n
b ( w ) = a $ / a v = exp(-rXp) - l’g[9(d9v)l dp’]
Q”
v-9
The partial differential equation for 9 is no longer integrable (cf. eq 72):
Since H ( x ) is a continuous and differentiable function of x in the interval [$,VI, there is some point $* such that H ( v ) - H ( $ ) = S ” H ’ ( x ) dx = H’($*)(v - 4)
(1.20)
0
The function $(p,v) is itself a continuous function of p, so we can write the quantum yield as (1.22) Q ( p ? v ) / Q o = (1 + OH’[$(g*,v)I where 0 Ip* 5 p. It is interesting to note that these last two equations (eq 1.18 and 1.22) are identical in form. Of course, the dynamics (specifically, the equations that determine the argument of H’(x) in each case) are quite different. We may also calculate the contrast y. From eq 94, we have
aa ar Y = v a l l v(rf = - - ri)a v = v(rf- ri)f(a)6
(1.23)
This illustrates the mathematical similarities between photochemistry in diffusion-free media and perfectly stirred media. Given the more complicated form of eq 1.10, we observe that the kinetic equations for solid-state photochemical reactions are no more inherently difficult to solve than the corresponding equations in perfectly stirred solution.
Appendix II: Solution to the General First-Order Equations The dimensionless equations describing a first-order reaction are
a
-- In a(p,v) = b(p,v); a(p,O) = 1
av
a
ab
In b(p,v) =
{(p)
+ a(p,v);
b(0,v) = 1
(11.1) (11.2)
where ~ ( t =) k s t Z ( z o , t ’ )dt’
(11.7)
We rearrange this to a form that will yield a convenient integral approximation (the form of this equation was suggested by examining the integral equation for $ with the aid of the calculus of variations): --a9 - [1
(1.21)
l # ) < $ * I V
+ 4411 = - [ [ ( P I $ + 1 - e-+]
a $ / a P = -[5%)$
where
--
(11.6)
aP
+ {(p)]$ = 1 - $ - 8
We note that the initial optical density (cf. eq 11.4 and 11.5):
(11.8)
rican be found from { ( p )
so we may transform the differential equation for 6:
a
-eri(4+(p,v) aP
= -eri(p)[l - &,v)
-
(11.10)
or in integral form
$(w) =
S:
eri(eo)-ri(r)$(po,v) - e-ri(r)
eri(p’)[
1 - ,+(p’,v)
- e-+(w‘s4 I dw’ (11.1 1)
We will demonstrate that this integral equation may be solved to arbitrary precision over a suitable interval by representing the exact solution $ as the limit of a monotonically convergent sequence of upper or lower bounds to the exact solution. We will assume that the quantity 1 {is positive. The extension of the argument to negative values (corresponding to photoantibleaching reactions) will be apparent. We assume that the exact solution $(p,v) has already been found in some interval [O,pO]and we wish to extend the solution to the region ( p o , p l ] . We assume that all approximate solutions satisfy the physical boundary conditions, $(p,O) = 0, $(O,v) = v, and &v) 2 0. Crude bounds to the exact solution in this region are obtained by noting that derivatives of the initial and final optical densities bound -8 In $ / d p :
+
(11.3)
(11.12)
10
p(z)
=
ri - rf = (a, - h,)
C,O(Z’)dz’
(11.4)
Integrating this equation in a positive direction yields
With no loss of generality, we assume that the initial distribution of reactant C:(z) does not vanish (except possibly on a set of measure zero). This ensures that both p(z) and r f ( p ) are continuous functions. Regions containing no photoreactant are inert, and the transmission coefficient of such a layer is time independent. It should be noted that no other assumptions have been made concerning the initial distributions of material Cy(z) or the time dependence of the incident light. Also, the auxiliary function { ( p ) (no longer simply a parameter) has its sign fixed by the constant a, - Xa,. Once again, the scaled reactant concentration a(p,v) is determined by the local fluence $(p,v) and the solution to the integral equation fixed by the reaction rate law (eq 46): a(w,v) = g [ @ ( p , v ) ]= e-+(’.”)
or @(po,v)er*(‘@-’I(p)I @ ( p , v ) I$(po,v)erl(w)-rd”);
po
I p (11.15)
We may systematically improve the bounds to the exact solution by employing the integfal equation for $ (eq 11.1 1). We define a sequence of iterates & ( p , v ) : = er,(m)-r,(p)$(po,v) - e-r,(d
$i+i(k,v)
epi(p’)[
1 - $i(w’,v) - e-5Ar‘34] dwLI
(11.16) We define the error in an iteration: & ( p , v ) = &,u)
- $(w,v).
The Journal of Physical Chemistry, Vol. 92, No. 17, 1988 4937
Photochemistry in Strongly Absorbing Media Subtraction of eq 11.1 1 from eq 11.16 yields an integral equation for the error: 6$i+l(w,u) = e-ri(r)~~e’i(i)[~,(p’,U) + e-&i(p‘,v)- +(p’,v) ,-4(r’+’)] dp‘
or the lower bound of eq 11.15. Then the sequence of iterates, (&,converges monotonically to the exact solution: lim IS&(p,v)l Ilim ISi( I(So( !im (wl - wOli I(60( lim M’ IO I-m
j-m
I-m
j-m
(11.25)
=
e - r i ( @ ) & ~ ( @ j ( w /-, ve-4(dqv)[ ) 1 - e-@h’+’)]] dw‘ (11.17) Now we show that the sequence of iterates (eq 11.16) preserves the sign of the error. The functionflx) = x e-‘ is monotonically increasing for positive values of the argument x. Thus 0 Ix1 < x2if and only if 1 I f l x l ) 4 I 0 andfl4J > f ( 4 ) , and the integrand of eq 11.17 is positive. Therefore the error in the subsequent iteration is also positive. The argument for negative errors follows the same line, with the related conclusion: if S& is negative in the interval ( w o , p l ] ,then is also negative in the interval. In addition, I 0, and no unphysical solutions have been if I 0, then introduced by the iteration. Thus convergence of the sequence 11.16 is monotonic after some cycle i if the error at some iteration has a definite sign:
+
4,
4,
The monotonicity of the convergence follows from eq 11.23. Therefore, we have determined the exact solution in the interval [ O , p l ] . By using both the upper and lower initial bounds to the exact solution, we can simultaneously bound the exact solution from above and below. This same strategy can be repeated over subsequent intervals, so that upper and lower bounds to the exact solution can be propagated to larger values of the argument p.
Appendix 111: Summary of Mathematical Equations Here we list the principal definitions and equations used in the development of our analytical solution to the coupled partial differential equations. The basic coupled partial differential equations describing the rate of reaction and absorbance of light are
(11.18)
az(zJ) - [curCr(z,t)+ apCp(z,t)+ cu,C,(z,t)]l(z,t) (2)
Next, we bound the error upon iteration. We assume that we have an upper bound to the error in the ith iteration. Then we can determine an upper bound to the error in the next iteration. Suppose that 0 I6&(w,u) ISi for PO II.L Iwl:
The dimensionless dependent variables are a(p,v), the scaled reactant concentration, and b(p,u), the scaled light intensity:
0 IS& I6,
b(w,v) = I(z,t)/lo(t) (16) where the initial concentration of reactant C; is assumed to be constant, as are the initial concentrations of all other species. These are functions of the dimensionless independent variables p, the change in the optical density at a depth below the surface z - zo, and u, the scaled surface fluence:
{@i(w,u)
> 0) * { @ j + l ( P , u ) > 0);
PO IP
(11.19)
Then 0 I1 - e-% 0 2 -e+[l - e-@i]
6, I S& - e-411 - e-*,(]
az
a ( ~ - v=) Cr(z,t)/C,O
w ( z )=
c:(ar
- l a p ) ( z- zo) =
(ri.- ri)(z - zo)
(15)
(37)
Now we integrate over a positive direction to preserve the sense of the inequalities: The basic equations involve a parameter {related to the change in optical density:
or, from eq 11.17 and 11.18 (11.20)
(39)
This latter integral may be bounded from above by using the mean value theorem. Since ri is a continuous function of p (this is the reason we assumed that the reactant concentration does not vanish), there is some value 1.1* such that
The coupled partial differential equations describing a and b are then aa -(42) a u = f ( a ) b ; a(w,O) = 1
0I
J:eri(”’)
Iag-ri(fl)
dw’= eri(r’)(p- M O ) ;
po Iw* Ip
(11.21)
Since the initial optical density increases with w, we have {w* Ip ) j (erdfi’) Icrib))
--ab = ( { + a)b; b(0,u) = 1
aw
(43)
where the dimensionless rate-law function is
so that
e-ri(r)L:
eri(r’)dw’ Ip - po Ipl - wo
This gives us the upper bound to
(0 IS 6 j IS i ) * (0 I&+I
(11.22)
These may be solved by representing a and b in terms of the auxiliary function 4, which describes the dimensionless local fluence:
@J~+~:
ISj+l ISi(K1 - Po))
(11.23)
The above argument (eq 11.17-11.23) holds, mutatis mutandis, for negative errors S& < 0 also. The conclusion is the same as eq 11.23, with the inequalities reversed. Finally, we prove the convergence of the sequence of iterations to the exact solution d(p,u). Suppose that pI - po is bounded: O-
