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A modified diffusion-based model for radiolytically evolved gases during ion implantation in polymers
Nuclear Instruments and Methods in Physics Research 836 (1989) 38-42 North-Holland, Amsterdam
38
A MODIFIED DIFFUSION-BASED MODEL FOR RADIOLYTICALLY DURING ION IMPLANTATION IN POLYMERS Ramji
PATHAK,
V.J. MENON,
Yan de Graaff Laboratory,
U.K.
Physics Department,
CHATURVEDI
EVOLVED
GASES
and A.K NIGAM
Banaras Hindu University,
Varanasi-221
005, India
Received 30 November 1987 and in revised form 22 August 1988
A refined theoretical model, based on 3-~mensiona1 diffusion equation, has been proposed to explain our earlier experimental resufts of dynamic partial pressure variation of CO, evolved during 250 keV D+ ion implantation in Mylar. This model gives a better fit with the experimental points than the percolation model in the pre-maximum as well as in the postmaximum region of the curve. The exnerimental H, evolution curve obtained bv Davenas et al. in 500 keV Ar+ implantation of PMMA, has also been explained by our mode1 in a much better way than by the percolation model.
1. Introduction
In our previous communication [l] we made an attempt to measure continuously the dynamic partial pressure (i.e. variation of pp with time) of the evolved CO, during 2.50 keV D+ ion implantation of Mylar and developed a diffusion based theory to explain the rising portion of the pp curve. In that paper, the theoretical predictions of our “diffusion based model” against the “percolation tube model” of Davenas et al. [2], were found to be more satisfactory in the premaximum region only as compared to the percolation model fit [2], which was, however, somewhat satisfactory in the postmaximum region. The aim of the present paper is to refine our diffusion based model further to explain the post-maximum region also both of our’s as well as Davenas’s et al. experimental curves [2]. The refinement of our model has been achieved by the following three steps: (i) By considering the 3-dimensional diffusion equation taking the sample of finite size, instead of one-dimensional diffusion equation with the tacit assumption of regarding the other two Cartesian directions to be infinite, as done in our earlier paper [l]. (ii) By considering the diffusion in accordance with the actual experimental conditions of the sample. Due to the small range (I, = 5 pm) of the impinged ions as compared to total thickness i.e. 160 pm [L3 in fig. l(a)] of the sample, most of polymer thickness (L,) remains unaffected by the ion beam. Hence, most of the radiolytically evolved gases although diffuse in the +Z direction, the gradient of the gas concentration will get saturated at a smaller value on account of the virgin polymer’s continuity almost up to infinity [as shown in fig. l(b)]. This will inhibit fast diffusion of evolved 0168-583X/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
gases in the +Z direction as compared to that in the - Z direction where a sharp gas concentration gradient gets established (fig. l(b)) as the evolved gases tend to exit towards vacuum at the vacuum-polymer interface. It is clear from fig. 1 that the lateral diffusion of the evolved gases in the radial direction with respect to the beam axis, takes place through the virgin sample which is very close to the surface (= 5 pm). As the Fickian diffusion is a zig-zag process, the diffused molecules have a fair chance to come to the interface and to
VACUUM
(VACUUM-
POLYMER
1
e
Oepth
in
polymer
-
Fig. 1. (a) Schematic view of the ion beam affected and unaffected (virgin) thickness of the sample with the diffusion direction of evolved gases shown by arrows. The curved arrows show diffusion in the lateral direction. (b) A schematic view of saturated concentration profile of the radiolytically evolved gases in the beam affected and virgin zones. A sharp gradient is maintained in the - Z direction due to the continuous exit of diffusing molecules into vacuum. (See text.)
R. Pathak et al. / Radiobtically
escape into the vacuum. This phenomenon is depicted by curved arrows in fig. l(a). Hence, effectively all these molecules which diffuse in the lateral direction also, come to the vacuum-polymer interface maintaining a sharp concentration gradient. Thus in the mathematical formulation one can safely assume that diffusion is taking place predominantly in the - Z direction and in lateral directions only. (iii) By deriving and determining numerically the more convenient integral representation of the evolution rate Q(t) and pressure p(t) instead of estimating their complicated integrals by the steepest descent technique as done before. However, in this case, some error functions were approximated to be equal to unity in the integrals to reduce the computer time. This worked reasonably well as far as (fit to the data upto) ‘not too large’ time was concerned. Other correlated studies (e.g. study of the delayed emission [3] of hydrogen from ion bombardment, (anti) correlation studies (41 between the number of free radicals and H, evolution etc.) of the evolved species from polymer surfaces during ion bombardment have been made by several authors [3-81.
2. Theory
-D
7%
= kn,(O)
eek’
for t > 0,
(1)
where D is the diffusion coefficient of the evolved gas, n,(O) represents the number of bonds of appropriate type per unit volume in the virgin sample and k is the time constant for their decay. In order to solve (l), the free retarded Green’s function [II] g(xt, x’r) for the problem, is employed, which is given by g=B(X)(rrp)-3’2
exp(-p2),
x = t - I’,
/.t=4L)X
and
p=-
x-x’
(3) 6’ With the help of this Green’s function, g, the solution of eq. (1) which vanishes at t < 0, turns out to be n(x,
t) =kn,(O)
eVkr
‘dX ekX /0
3 @Pi+) - erfb,-1
XI-I J=l
2
(4)
where
Here it is assumed that the beam is incident on an area = I, I,, a square which is not significantly different from the real beam of circular cross section. This assumption simplifies the mathematical formulation considerably. The beam penetrates an average distance 1s into the solid target. By Ficks first law, the rate of diffusive flow of the gas across a surface (x1 = constant) normal to the j th coordinate axes, is given by dxj dx, aELi
-D/j”
r, ,
where i, m and j are permutations of 1, 2, 3 and the integral with respect to dx,dx,, runs over the given surface of area L,L,. The net gas evolution rate Q(t) is obtained by adding contributions of the form (6) arising from the five faces xj = 0, x1 = *Lr/2 and xz = f L,/Z. Although the algebraic value of R X(L,, t) is nonzero, we must drop this term from Q because diffusion in the large positive Z direction is prevented for the reasons discussed above. Recalling eq. (4) for n, the expression for the net evolution rate becomes Q(t)=A
eMkr ‘dXeikXH(h)/fi, I0
(7)
where A = (D/4n)1/2kn,(0)1,12 is a constant and H(X) is a sum of five terms containing error functions and exponentials. The full expression for H(h) is algebraically complicated and numerically inconvenient because the evaluation of the integral (7) requires a large amount of computer time. Hence in the present paper we replace H(h) by a simple ansatz valid for X not too large compared to Lz/D: H(h)
=
(1 -e -&/A)
+ C(e-~:+/~
_ e-&/A),
(8)
where we have used a square geometry (L, = L,, f, = i2) and defined a If=-
(2)
39
where 0 is the unit function and
Rj(xi, t) =
To explain theoretically the post-maximum as well as the pre-maximum regions of the curve, the above mentioned modifications were made. The polymer sample was regarded as a rectangular slab of sides L,, L,, L, parallel to the axes of a Cartesian coordinate system whose origin is supposed to be at centre of the face where the beam is incident normally at the time t = 0. Despite the constant beam flux, it is assumed that the rate of scissioning of bonds of a particular type, will vary with time as G(t) = G(0) eekr, where G(0) is the value of G (scission) in the virgin polymer sample and k is the time constant for the decay [9]. The diffusion of the evolved gases will take place in the -Z and lateral directions for the reasons discussed above. The space-time dependent concentration n(r, t) of the given gaseous species, will obey the 3-dimensional inhomogeneous Fick’s diffusion equation [9-111. g
evolved gases in polymers
L,fI, __
i 4fi
i
-13
* 03-=
5
R. Pathak et al. / Radiolytically evolued gases in polymers
40
The first bracket in eq. (8) represents the emission of gas across the xy plane (at t = 0) and the second bracket gives the cont~bution from the remaining faces with an effective relative strength C (assumed constant) and with parameters ut, having only an effective significance not rigidly related to the actual dimensions L,, L, because the molecules tend to move towards the interface along curved zig-zag paths. If p(r) is the partial pressure of radiolytically evolved species at time t then [4] Ap(t)
=p(t>
=
-P(O)
$j’dt’
ep"Q(t'),
0
where /3 = &./Y is the pumping speed parameter and p(O) is the partial pressure at t = 0. Substituting for Q(t) from eq. (7) and intergrating partially we get e-W-r’)
Ap(r)
_ e-B(r-r’)
H(
1’)
= $r;dt’
i
P-k
fl’
which is amenable to Simpson quadrature in the variable s’=/i’. In the development of this theory, we have introduced seven parameters, viz. A/V, 8, k, a:_; a:,, 2 and C. The parameter A/V carries information a1 about the number of bonds n,(O) per unit volume of the virgin sample; it is fixed by normalizing the theoretical curve eq. (11) to the experimental value at the peak. Next, /3 is essentially the pumping speed for the system, and can be estimated experimentally. The remaining parameters are usually unknown and are to be fitted from the experimental data. Here k is the time constant for the rate of bond scission. (1 - e--O:-/‘) is the contribution to the gas evolution rate across the xy plane (at z = 0). Also, (e-af+/r - e-“f-/‘) is the contribution to gas emission from the remaining faces. Finally, the coefficient C measures the relative strength of the above mentioned contributions. It is very interesting to note that the rate Q(t) depends directly on k [cf. eq. (7)] and the standard formula for partial pressure depends crucially on fi [cf. eq. (lo)] but the final formula in our model contains p and k ‘symmetrically’ [cf. eq. (ll)]. This predicts that the final value of Ap(t) remains unchanged if /I and k inverted.
3. Results and discussion In order to test the validity of our model, numerical calculations of Ap(t) at various points have been made and compared with our experimental results on CO, evolution in Mylar during D+ implantation [l] and
those of Davenas’s et al. on H, evolution with Ar+ ion implantation [2]. 3.1. Comparison with our experimental
in PMMA
results
Values of Ap( t) were read off at N = 52 points i.e. 1, =o, 12,‘‘., t, from our experimental curves [l] which is a proportionately reduced partial pressure trace of the 40-48 amu repeated scan showing pp variation of CO, with time while the sample was bombarded with 250 keV D+ ions. The numerical fitting procedure adopted by us was as follows. As mentioned above, the coefficient A/V appearing in eq. (11) can be fixed by normalizing Ap( t) to its correct value at the peak position. Let the symbol {IX> = (/3, k, a$_, Q:+, a:_, C} denotes the remaining set of parameters and consider the unweighted squared deviation to be minimized, viz. x2=
E { AP exp(l,) i=l
- Aptheo(ti)}2.
02)
Since x2 is a highly non-linear function of {a} we first subject it to a coarse scanning program over a wide range of {o) so as to locate suitable initial values for the parameters. Next starting from these initial values of the set (cu}, we varied them in small steps until x2 becomes smallest. The final values of {(u} so obtained were essentially unique to within an error not exceeding 10% typically. However, we must emphasize that tthe roles of p and k are interchangeable in our fit, and it is not possible to give the statistical error matrix for the parameters because our a2 is unweighted. Knowing the values of parameter a$_ and range t, it is possible to estimate the diffusion coefficient from D = 1:/4a:_.
(13)
Fig. 2 shows the experimental partial pressure points (x) with theoretical curves A and B drawn on the basis of the percolation model for fi = 0.10 and 0.07 respectively, and curve C drawn on the basis of our modified diffusion-based model for p = 0.07. The scales in the pre-maximum and post-maximum regions are different. It can be seen that with this modification our post-maximum region fits very well with the experimental one, at little cost to the pre-maximum region. The pre-maximum of the theoretical curve is quits close to the experimental curve. The apparent discrepancy in the pre-maximum region is due to different time scales only. However, there is a little discrepancy in the tail regions of the theoretical and experimental curves, which is mainly due to the replacement of some error functions by unity in arriving at the ansatz [equation (8)], which in fact begins to break down when time f becomes very large. The theoretical curves A and B obtained on the basis of percolation model with two p-values (0.01 and 0.07) lie on either side of the experi-
41
R. Parhak et al. / Radiolyrically evolved gases in polymers Time 12*0°
10
,
,
(sec.1 -
20,
,
30
,
,
,
40 ,
50
,
,
10'0 -
100
50
200
150
Time
250
300
350
(sec.) -
where Fig. 2. Our experimental points (crosses) [l] and the theoretical curves based on our modified diffusion model ( -) p = 0.07 s -I, k = 0.145 s-‘, ai_ = 2.0 s, a:+ = 32.0 s, a:_ = 100.0 s, and C = 1.3 are taken, leading to the value D = 3 x lo-’ cm’/s for the diffusion coefficient of CO, in Mylar. Dotted lines A and B show the curve based on the percolation model of Davenas et al. [l]. Note the difference in the time scales of the pre-maximum and post-maximum regions.
mental
curve
in the post-maximum
fig. 2. But both
of them
are at quite
region,
as shown
a distance
from
in
quite
experimental curve in the pre-maximum region. One may say by looking at the post-maximum region of the curves, that one should try with some intermediate p-value and obtain the theoretical curve which may be
close
to the
experimental
one
in the
post-maxi-
But in the pre-maximum region the resultant curve will fall in between A and B which themselves are at quite a distance from the experimental one. This suggests that the percolation model is not very suitable for this. However, our modified diffusion-based mum
the
region.
t
0
25
50
75
100 Time
125
150
(sec.)
175
200
225
250
-
obtained by Davenas et al. [2] on the basis of the percolation model, and (-.-) by us using our Fig. 3. Theoretical curves ( -) present model where B = 32.0 s, k = 0.155 s-t, a:_ = 2 00 s a:+ = 32.0 s, a:_ = 125.0 s and C = 1.15 are taken to explain the ) of Ha partial pressure in PM& measured by Davenas et al. [2], due to bombardment by PMMA by experimental curve (500 keV Ar+ beam.
42
R. Pathak et al. / Radiolyrically evolved gases in polymers
model calculations fit extremely well in the post-maximum region and are much better compared to the curve of percolation model in the pre-maximum region. It will not be out of place here to mention that the species evolved from the polymer were mainly coming from the polymer itself during bombardment, instead of those arriving from gas phase reaction in vacuum. This is because the prior estimation of the gas phase reaction and its products, was made without the polymer sample at the site during bombardment. It was found to be quite negligible. 3.2. Comparison with Davenas’s experimental
tion tubes due to to which the percolation tubes no longer remain clean. Consequently, a diffusion flow results instead of viscous flow. The study of the diffusion coefficients of different species obtained under different implantation conditions (e.g. energy, flux, type of ions etc.) is in progress. A possible limitation of our study is the treatment of D as a time-independent constant which was done for mathematical simplicity. This might not be strictly correct because of the fact that the polymer in the path of the beam gets more and more damaged as time progresses in the given experiment.
results
Our modified diffusion-based model is also tested by applying it to Davenas’s experimental curve [2] for the evolved H, gas from PMMA polymer target due to its implantation by 500 keV Ar+ ions. The experimental values of Ap(t) were read off at N = 22 points i.e. t, = 0, tz-thl according to Davenas’s data [2]. As done in the previous case, the theoretical curve using our diffusion-based model, was obtained. The theoretical curves based on our diffusion-based model and on the percolation model for H, evolution are shown in fig. 3, along with the experimental points (x) mentioned above. In the pre-maximum region the diffusion-based theoretical curve B is evidently very close to the experimental points while the percolation based theoretical curve rises a bit too sharply. In the post-maximum region also, the curve B matches better with the experimental points rather than the curve A.
4. Conclusion Hence one can see that the modified diffusion-based model fits better not only with our CO, evolved experimental curve but also gives better results with H, evolved experimental curve obtained by Davenas et al. As already explained in our earlier paper [l], the reasons for the diffusion mechanism as a mode of flow of the gases through the polymer seems to be justified as compared to their viscous flow through completely clean percolation tubes. It is because the ions passing through the target in reality break the polymer chains into small segments whose broken ends come inside the percola-
This work is financially supported by DAE project No. 37/6/84-G. One of us (V.J.M.) is grateful to the UGC for awarding him a Research Scientist (A) position. The financial support from CSIR (to R.P. in the form of JRF and (to U.K.C.) in the form of a pool Scientistship, is gratefully acknowledged.
References [l] U.K. Chaturvedi, V.J. Menon, Ramji Pathak and A.K. Nigam, Nucl. Instr. and Meth. B24/25 (1987) 353. [2] J. Davenas, X.L. Xu, C. Khodr, M. Treilleux and G. Steffan. Nucl. Instr. and Meth. B7/8 (1985) 513 and refs. therein. [3] W.L. Brown, G. Foti, L.J. Lanzerotti. J.E. Bower and R.E. Johnson et al. Nucl. Instr. and Meth. B19/20 (1987) 899. [4] B. Wasserman, Phys. Rev. B34 (1986) 1926. [5] MS. Dresselhaus et al., Proc. Mater. Res. Sot. Symp., Vol 27, Ion Implantation and Ion Beam Processing of Materials, eds. G.K. Hufles, O.W. Holland, CR Clayton and C.W. White (North-Holland, New York,, 1983) p. 413. [6] T.C. Smith, in Ion Implantation Equipment and Techniques, eds. H. Ryssel and H. Glawisehnig (Springer, Berlin, 1982). [7] T. Venkatesan, T. Wolf, D. Allara, B.J. Wilkens and G.N. Taylor, Appl. Phys. Lett. 43 (1983) 934. [8] T. Venkatesan, D. Edelson and W.L. Brown, Nucl. Instr. and Meth. Bl (1984) 286. [9] C. Yamaguchi, Nucl. Instr. and Meth. 154 (1978) 465. [lo] R. Glang, R.A. Holmwood and J.A. Kurtz, in Handbook of Thin Film Technology, eds. L.I. Maissel and R. Glang (McGraw-Hill, New York, 1970) p. 247 and refs. therein. [ll] P.H. Morse and H. Feshback, Method of Theoretical Physics (McGraw-Hill, New York, 1953)) pp. 173-857.
38
A MODIFIED DIFFUSION-BASED MODEL FOR RADIOLYTICALLY DURING ION IMPLANTATION IN POLYMERS Ramji
PATHAK,
V.J. MENON,
Yan de Graaff Laboratory,
U.K.
Physics Department,
CHATURVEDI
EVOLVED
GASES
and A.K NIGAM
Banaras Hindu University,
Varanasi-221
005, India
Received 30 November 1987 and in revised form 22 August 1988
A refined theoretical model, based on 3-~mensiona1 diffusion equation, has been proposed to explain our earlier experimental resufts of dynamic partial pressure variation of CO, evolved during 250 keV D+ ion implantation in Mylar. This model gives a better fit with the experimental points than the percolation model in the pre-maximum as well as in the postmaximum region of the curve. The exnerimental H, evolution curve obtained bv Davenas et al. in 500 keV Ar+ implantation of PMMA, has also been explained by our mode1 in a much better way than by the percolation model.
1. Introduction
In our previous communication [l] we made an attempt to measure continuously the dynamic partial pressure (i.e. variation of pp with time) of the evolved CO, during 2.50 keV D+ ion implantation of Mylar and developed a diffusion based theory to explain the rising portion of the pp curve. In that paper, the theoretical predictions of our “diffusion based model” against the “percolation tube model” of Davenas et al. [2], were found to be more satisfactory in the premaximum region only as compared to the percolation model fit [2], which was, however, somewhat satisfactory in the postmaximum region. The aim of the present paper is to refine our diffusion based model further to explain the post-maximum region also both of our’s as well as Davenas’s et al. experimental curves [2]. The refinement of our model has been achieved by the following three steps: (i) By considering the 3-dimensional diffusion equation taking the sample of finite size, instead of one-dimensional diffusion equation with the tacit assumption of regarding the other two Cartesian directions to be infinite, as done in our earlier paper [l]. (ii) By considering the diffusion in accordance with the actual experimental conditions of the sample. Due to the small range (I, = 5 pm) of the impinged ions as compared to total thickness i.e. 160 pm [L3 in fig. l(a)] of the sample, most of polymer thickness (L,) remains unaffected by the ion beam. Hence, most of the radiolytically evolved gases although diffuse in the +Z direction, the gradient of the gas concentration will get saturated at a smaller value on account of the virgin polymer’s continuity almost up to infinity [as shown in fig. l(b)]. This will inhibit fast diffusion of evolved 0168-583X/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
gases in the +Z direction as compared to that in the - Z direction where a sharp gas concentration gradient gets established (fig. l(b)) as the evolved gases tend to exit towards vacuum at the vacuum-polymer interface. It is clear from fig. 1 that the lateral diffusion of the evolved gases in the radial direction with respect to the beam axis, takes place through the virgin sample which is very close to the surface (= 5 pm). As the Fickian diffusion is a zig-zag process, the diffused molecules have a fair chance to come to the interface and to
VACUUM
(VACUUM-
POLYMER
1
e
Oepth
in
polymer
-
Fig. 1. (a) Schematic view of the ion beam affected and unaffected (virgin) thickness of the sample with the diffusion direction of evolved gases shown by arrows. The curved arrows show diffusion in the lateral direction. (b) A schematic view of saturated concentration profile of the radiolytically evolved gases in the beam affected and virgin zones. A sharp gradient is maintained in the - Z direction due to the continuous exit of diffusing molecules into vacuum. (See text.)
R. Pathak et al. / Radiobtically
escape into the vacuum. This phenomenon is depicted by curved arrows in fig. l(a). Hence, effectively all these molecules which diffuse in the lateral direction also, come to the vacuum-polymer interface maintaining a sharp concentration gradient. Thus in the mathematical formulation one can safely assume that diffusion is taking place predominantly in the - Z direction and in lateral directions only. (iii) By deriving and determining numerically the more convenient integral representation of the evolution rate Q(t) and pressure p(t) instead of estimating their complicated integrals by the steepest descent technique as done before. However, in this case, some error functions were approximated to be equal to unity in the integrals to reduce the computer time. This worked reasonably well as far as (fit to the data upto) ‘not too large’ time was concerned. Other correlated studies (e.g. study of the delayed emission [3] of hydrogen from ion bombardment, (anti) correlation studies (41 between the number of free radicals and H, evolution etc.) of the evolved species from polymer surfaces during ion bombardment have been made by several authors [3-81.
2. Theory
-D
7%
= kn,(O)
eek’
for t > 0,
(1)
where D is the diffusion coefficient of the evolved gas, n,(O) represents the number of bonds of appropriate type per unit volume in the virgin sample and k is the time constant for their decay. In order to solve (l), the free retarded Green’s function [II] g(xt, x’r) for the problem, is employed, which is given by g=B(X)(rrp)-3’2
exp(-p2),
x = t - I’,
/.t=4L)X
and
p=-
x-x’
(3) 6’ With the help of this Green’s function, g, the solution of eq. (1) which vanishes at t < 0, turns out to be n(x,
t) =kn,(O)
eVkr
‘dX ekX /0
3 @Pi+) - erfb,-1
XI-I J=l
2
(4)
where
Here it is assumed that the beam is incident on an area = I, I,, a square which is not significantly different from the real beam of circular cross section. This assumption simplifies the mathematical formulation considerably. The beam penetrates an average distance 1s into the solid target. By Ficks first law, the rate of diffusive flow of the gas across a surface (x1 = constant) normal to the j th coordinate axes, is given by dxj dx, aELi
-D/j”
r, ,
where i, m and j are permutations of 1, 2, 3 and the integral with respect to dx,dx,, runs over the given surface of area L,L,. The net gas evolution rate Q(t) is obtained by adding contributions of the form (6) arising from the five faces xj = 0, x1 = *Lr/2 and xz = f L,/Z. Although the algebraic value of R X(L,, t) is nonzero, we must drop this term from Q because diffusion in the large positive Z direction is prevented for the reasons discussed above. Recalling eq. (4) for n, the expression for the net evolution rate becomes Q(t)=A
eMkr ‘dXeikXH(h)/fi, I0
(7)
where A = (D/4n)1/2kn,(0)1,12 is a constant and H(X) is a sum of five terms containing error functions and exponentials. The full expression for H(h) is algebraically complicated and numerically inconvenient because the evaluation of the integral (7) requires a large amount of computer time. Hence in the present paper we replace H(h) by a simple ansatz valid for X not too large compared to Lz/D: H(h)
=
(1 -e -&/A)
+ C(e-~:+/~
_ e-&/A),
(8)
where we have used a square geometry (L, = L,, f, = i2) and defined a If=-
(2)
39
where 0 is the unit function and
Rj(xi, t) =
To explain theoretically the post-maximum as well as the pre-maximum regions of the curve, the above mentioned modifications were made. The polymer sample was regarded as a rectangular slab of sides L,, L,, L, parallel to the axes of a Cartesian coordinate system whose origin is supposed to be at centre of the face where the beam is incident normally at the time t = 0. Despite the constant beam flux, it is assumed that the rate of scissioning of bonds of a particular type, will vary with time as G(t) = G(0) eekr, where G(0) is the value of G (scission) in the virgin polymer sample and k is the time constant for the decay [9]. The diffusion of the evolved gases will take place in the -Z and lateral directions for the reasons discussed above. The space-time dependent concentration n(r, t) of the given gaseous species, will obey the 3-dimensional inhomogeneous Fick’s diffusion equation [9-111. g
evolved gases in polymers
L,fI, __
i 4fi
i
-13
* 03-=
5
R. Pathak et al. / Radiolytically evolued gases in polymers
40
The first bracket in eq. (8) represents the emission of gas across the xy plane (at t = 0) and the second bracket gives the cont~bution from the remaining faces with an effective relative strength C (assumed constant) and with parameters ut, having only an effective significance not rigidly related to the actual dimensions L,, L, because the molecules tend to move towards the interface along curved zig-zag paths. If p(r) is the partial pressure of radiolytically evolved species at time t then [4] Ap(t)
=p(t>
=
-P(O)
$j’dt’
ep"Q(t'),
0
where /3 = &./Y is the pumping speed parameter and p(O) is the partial pressure at t = 0. Substituting for Q(t) from eq. (7) and intergrating partially we get e-W-r’)
Ap(r)
_ e-B(r-r’)
H(
1’)
= $r;dt’
i
P-k
fl’
which is amenable to Simpson quadrature in the variable s’=/i’. In the development of this theory, we have introduced seven parameters, viz. A/V, 8, k, a:_; a:,, 2 and C. The parameter A/V carries information a1 about the number of bonds n,(O) per unit volume of the virgin sample; it is fixed by normalizing the theoretical curve eq. (11) to the experimental value at the peak. Next, /3 is essentially the pumping speed for the system, and can be estimated experimentally. The remaining parameters are usually unknown and are to be fitted from the experimental data. Here k is the time constant for the rate of bond scission. (1 - e--O:-/‘) is the contribution to the gas evolution rate across the xy plane (at z = 0). Also, (e-af+/r - e-“f-/‘) is the contribution to gas emission from the remaining faces. Finally, the coefficient C measures the relative strength of the above mentioned contributions. It is very interesting to note that the rate Q(t) depends directly on k [cf. eq. (7)] and the standard formula for partial pressure depends crucially on fi [cf. eq. (lo)] but the final formula in our model contains p and k ‘symmetrically’ [cf. eq. (ll)]. This predicts that the final value of Ap(t) remains unchanged if /I and k inverted.
3. Results and discussion In order to test the validity of our model, numerical calculations of Ap(t) at various points have been made and compared with our experimental results on CO, evolution in Mylar during D+ implantation [l] and
those of Davenas’s et al. on H, evolution with Ar+ ion implantation [2]. 3.1. Comparison with our experimental
in PMMA
results
Values of Ap( t) were read off at N = 52 points i.e. 1, =o, 12,‘‘., t, from our experimental curves [l] which is a proportionately reduced partial pressure trace of the 40-48 amu repeated scan showing pp variation of CO, with time while the sample was bombarded with 250 keV D+ ions. The numerical fitting procedure adopted by us was as follows. As mentioned above, the coefficient A/V appearing in eq. (11) can be fixed by normalizing Ap( t) to its correct value at the peak position. Let the symbol {IX> = (/3, k, a$_, Q:+, a:_, C} denotes the remaining set of parameters and consider the unweighted squared deviation to be minimized, viz. x2=
E { AP exp(l,) i=l
- Aptheo(ti)}2.
02)
Since x2 is a highly non-linear function of {a} we first subject it to a coarse scanning program over a wide range of {o) so as to locate suitable initial values for the parameters. Next starting from these initial values of the set (cu}, we varied them in small steps until x2 becomes smallest. The final values of {(u} so obtained were essentially unique to within an error not exceeding 10% typically. However, we must emphasize that tthe roles of p and k are interchangeable in our fit, and it is not possible to give the statistical error matrix for the parameters because our a2 is unweighted. Knowing the values of parameter a$_ and range t, it is possible to estimate the diffusion coefficient from D = 1:/4a:_.
(13)
Fig. 2 shows the experimental partial pressure points (x) with theoretical curves A and B drawn on the basis of the percolation model for fi = 0.10 and 0.07 respectively, and curve C drawn on the basis of our modified diffusion-based model for p = 0.07. The scales in the pre-maximum and post-maximum regions are different. It can be seen that with this modification our post-maximum region fits very well with the experimental one, at little cost to the pre-maximum region. The pre-maximum of the theoretical curve is quits close to the experimental curve. The apparent discrepancy in the pre-maximum region is due to different time scales only. However, there is a little discrepancy in the tail regions of the theoretical and experimental curves, which is mainly due to the replacement of some error functions by unity in arriving at the ansatz [equation (8)], which in fact begins to break down when time f becomes very large. The theoretical curves A and B obtained on the basis of percolation model with two p-values (0.01 and 0.07) lie on either side of the experi-
41
R. Parhak et al. / Radiolyrically evolved gases in polymers Time 12*0°
10
,
,
(sec.1 -
20,
,
30
,
,
,
40 ,
50
,
,
10'0 -
100
50
200
150
Time
250
300
350
(sec.) -
where Fig. 2. Our experimental points (crosses) [l] and the theoretical curves based on our modified diffusion model ( -) p = 0.07 s -I, k = 0.145 s-‘, ai_ = 2.0 s, a:+ = 32.0 s, a:_ = 100.0 s, and C = 1.3 are taken, leading to the value D = 3 x lo-’ cm’/s for the diffusion coefficient of CO, in Mylar. Dotted lines A and B show the curve based on the percolation model of Davenas et al. [l]. Note the difference in the time scales of the pre-maximum and post-maximum regions.
mental
curve
in the post-maximum
fig. 2. But both
of them
are at quite
region,
as shown
a distance
from
in
quite
experimental curve in the pre-maximum region. One may say by looking at the post-maximum region of the curves, that one should try with some intermediate p-value and obtain the theoretical curve which may be
close
to the
experimental
one
in the
post-maxi-
But in the pre-maximum region the resultant curve will fall in between A and B which themselves are at quite a distance from the experimental one. This suggests that the percolation model is not very suitable for this. However, our modified diffusion-based mum
the
region.
t
0
25
50
75
100 Time
125
150
(sec.)
175
200
225
250
-
obtained by Davenas et al. [2] on the basis of the percolation model, and (-.-) by us using our Fig. 3. Theoretical curves ( -) present model where B = 32.0 s, k = 0.155 s-t, a:_ = 2 00 s a:+ = 32.0 s, a:_ = 125.0 s and C = 1.15 are taken to explain the ) of Ha partial pressure in PM& measured by Davenas et al. [2], due to bombardment by PMMA by experimental curve (500 keV Ar+ beam.
42
R. Pathak et al. / Radiolyrically evolved gases in polymers
model calculations fit extremely well in the post-maximum region and are much better compared to the curve of percolation model in the pre-maximum region. It will not be out of place here to mention that the species evolved from the polymer were mainly coming from the polymer itself during bombardment, instead of those arriving from gas phase reaction in vacuum. This is because the prior estimation of the gas phase reaction and its products, was made without the polymer sample at the site during bombardment. It was found to be quite negligible. 3.2. Comparison with Davenas’s experimental
tion tubes due to to which the percolation tubes no longer remain clean. Consequently, a diffusion flow results instead of viscous flow. The study of the diffusion coefficients of different species obtained under different implantation conditions (e.g. energy, flux, type of ions etc.) is in progress. A possible limitation of our study is the treatment of D as a time-independent constant which was done for mathematical simplicity. This might not be strictly correct because of the fact that the polymer in the path of the beam gets more and more damaged as time progresses in the given experiment.
results
Our modified diffusion-based model is also tested by applying it to Davenas’s experimental curve [2] for the evolved H, gas from PMMA polymer target due to its implantation by 500 keV Ar+ ions. The experimental values of Ap(t) were read off at N = 22 points i.e. t, = 0, tz-thl according to Davenas’s data [2]. As done in the previous case, the theoretical curve using our diffusion-based model, was obtained. The theoretical curves based on our diffusion-based model and on the percolation model for H, evolution are shown in fig. 3, along with the experimental points (x) mentioned above. In the pre-maximum region the diffusion-based theoretical curve B is evidently very close to the experimental points while the percolation based theoretical curve rises a bit too sharply. In the post-maximum region also, the curve B matches better with the experimental points rather than the curve A.
4. Conclusion Hence one can see that the modified diffusion-based model fits better not only with our CO, evolved experimental curve but also gives better results with H, evolved experimental curve obtained by Davenas et al. As already explained in our earlier paper [l], the reasons for the diffusion mechanism as a mode of flow of the gases through the polymer seems to be justified as compared to their viscous flow through completely clean percolation tubes. It is because the ions passing through the target in reality break the polymer chains into small segments whose broken ends come inside the percola-
This work is financially supported by DAE project No. 37/6/84-G. One of us (V.J.M.) is grateful to the UGC for awarding him a Research Scientist (A) position. The financial support from CSIR (to R.P. in the form of JRF and (to U.K.C.) in the form of a pool Scientistship, is gratefully acknowledged.
References [l] U.K. Chaturvedi, V.J. Menon, Ramji Pathak and A.K. Nigam, Nucl. Instr. and Meth. B24/25 (1987) 353. [2] J. Davenas, X.L. Xu, C. Khodr, M. Treilleux and G. Steffan. Nucl. Instr. and Meth. B7/8 (1985) 513 and refs. therein. [3] W.L. Brown, G. Foti, L.J. Lanzerotti. J.E. Bower and R.E. Johnson et al. Nucl. Instr. and Meth. B19/20 (1987) 899. [4] B. Wasserman, Phys. Rev. B34 (1986) 1926. [5] MS. Dresselhaus et al., Proc. Mater. Res. Sot. Symp., Vol 27, Ion Implantation and Ion Beam Processing of Materials, eds. G.K. Hufles, O.W. Holland, CR Clayton and C.W. White (North-Holland, New York,, 1983) p. 413. [6] T.C. Smith, in Ion Implantation Equipment and Techniques, eds. H. Ryssel and H. Glawisehnig (Springer, Berlin, 1982). [7] T. Venkatesan, T. Wolf, D. Allara, B.J. Wilkens and G.N. Taylor, Appl. Phys. Lett. 43 (1983) 934. [8] T. Venkatesan, D. Edelson and W.L. Brown, Nucl. Instr. and Meth. Bl (1984) 286. [9] C. Yamaguchi, Nucl. Instr. and Meth. 154 (1978) 465. [lo] R. Glang, R.A. Holmwood and J.A. Kurtz, in Handbook of Thin Film Technology, eds. L.I. Maissel and R. Glang (McGraw-Hill, New York, 1970) p. 247 and refs. therein. [ll] P.H. Morse and H. Feshback, Method of Theoretical Physics (McGraw-Hill, New York, 1953)) pp. 173-857.