Langmuir 1994,10, 790-796
790
Gas-Sensing Mechanism of Phthalocyanine Langmuir-Blodgett Films Hong-Ying Wang and Jerome B.Lando' Department of Macromolecular Science, Case Western Reserve University, Cleveland, Ohio 44106 Received June 7,1993. I n Final Form: December 3,199P Gas-sensing properties of phthalocyanine [(CsH&SiOSiPcOGePcOH] Langmuir-Blodgett (L-B) films for halogen gases were investigated. The L-Bfilmswere deposited on siliconchips with a built-in microheater and a temperature-sensing diode for rapid control and monitoring of the sensing film temperature. Sensitivitiesto C12, Br2, and 12 at various temperatures were studied. The temperature at peak sensitivity for Cl2 is 130 "C, for Brp is 60 "C,and for 12 is 20 "C. Sensors with different film thickness were fabricated and tested in Clp gas at room temperature. The resulta show that the conductivity change arises from the both surfaceand bulkeffects. An analytical model is proposed, consideringthe gas adsorption and desorption on the film surface as rate processes and Fickian diffusion through the film, to describe the behavior of the gas sensor. The model fib the experimental data.
Introduction It has long been known that the conductivities of phthalocyankes (PCS)are very sensitive to the presence of certain electrophilic gases such as NO2 and C12,1t2which leads to an interest in their use as microelectronic gas sensors. The gas sensing is realized through the charge transfer interaction in which the gas molecule to be sensed acta as a planar r-electron acceptor forming a redox couple, and the positive charge produced is delocalized over the phthalocyanine ring causing the increase of the conductivity. Thus phthalocyanines are known as p-type semiconductors. Thin films of phthalocyanines from vacuum sublimation and Langmuir-Blodgett (L-B) deposition have been studied extensivelyfor gas sensing.'-l1 It has been reported for vacuum sublimation films that there are two stages of conductivity changes, an initial fast change followed by a slow drift to a steady-state value, when the films were exposed to the incoming gases.3-7 It was suggested that the fast process is due to the surface adsorption of gas molecules and the slow process is due to the bulk diffusion of gas molecules into the film.sS An alternative explanation was given by Archer et al.? who proposed a heterogeneous surface site model in which the fast initial change is due to NO2 adsorption on easily accessible adsorption sites from which oxygen is easily displaced and the slower change is caused by the replacement of 0 2 by NO2 molecules on Abstractpublishedin Advance ACSAbstracts, February 1,1994. (1)Bott, B.; Jones, T. A. S e w . Actuators 1984,5,43-53. (2) Jones, T. A.; Bott, B. Sens. Actuators 1986,9, 27-37. J.;Tamizi,M.; Wright, (3) Archer,P.B.M.;Chadwick,A.V.;Miasik,J. J. D. Sens. Actuators 1989, 16,379-392. (4)Pizzini, S.; Timo, G. L.; Beghi, M.; Butta, N.; Mari, C. M.; Faltenmaier, J. Sene. Actuators 1989,17,481-491. (5) Dogo, 5.;Germain, J.-P.;Maleysson, C.; Pauly, A. Thin Solid Films 1992,219,251-256. (6) Gentry, S. J.; Waleh, P. T. R o c . 2nd Int. Meet. Chem. Sens. 1986, 209-212. (7) Sadaoka, Y.; Matsuguchi, M.; Sakai, Y.;Mori, Y . J. Mater. Sci. 1992,27,5215-5220. (8)Baker, S.; Roberta, G. G.;Petty, M. C. Proc. Inst., Electr. Eng. 1988,130,260-263. (9) Wohltjen, H.; Barger, W. R.;Snow,A.W.; Janis, N. L. lEEE Trans. Electron Devices 1985, 7,1170-1174. (10) Wang, H.Y.; KO,W. H.; Babel, D. A.;Kenney, M. E.; Lando, J. B. Sens Actuators 1990, B l , 138-141. (11) Lu, A. D.; Jiang, D. P.; Li, Y.J.; Liu, W. N.; Pang, X. M.; Fan, Y.;Chen, W. Q.; Li, T. J.; Dong, X. J.; Zhu, Z. Q. Thin SolidFilms 1992, 210/211, 60f2-609. 9
strong adsorption sites. For L-B films without exposure to gases, a linear dependence of conductance on f i i thickness is obtained, which suggests that the conduction process is primarily a bulk phenomenon rather than a surface e f f e ~ t . ~InJ ~our previous work on phthalocyanine L-B film Cl2 gas sensors, the effect of film temperature on sensitivity, conductivity rise and decay times, as well as pulsed operation have been studied.1° The response and recovery curves of the sensor ais0 showed two stages of conductance change. Since the sensor measurement electrodes are only in contact with the bottom face of the sensing film, the permeation of the gas into the bulk of the L-B film appears to be necessary for maximum response. It should be noted that the filmsare stable at temperatures well above 200 "C. In this paper we present our work on the gas-sensing mechanism, which shows that the conductivity change of the phthalocyanine L-B film arises from the adsorption of the gas molecules on the film surface and subsequent diffusion into the film.
Experimental Measurements The material used in this research was the axially substituted amphiphilicphthalocyanine dimer [(C&I~~)aSiOSiPcOGePcOHl (where Pc is the phthalocyanato dianion), Figure 1,which was specially designed and prepared for L-Bfilm deposition.'* The fabrication and characterization of the silicon-based gas sensor shown in Figure 2 and ita coating with the phthalocyanine by the L-Bf i i technique have been reported.12J4J6The sensor is a 3-mm-by-2-mm-by-0.3-mmdevicethat has an interdigitatedgold electrode in ohmic contact with the sensing film on top of it, which measures the conductivity change. The electrodes consist of three pairs of fingers, each 20 pm wide with a gap of 20 pm between the fingers and a finger overlap distance of 320 pm. A heater and a diode were integrated in the sensor substrate to control and monitor the temperature of the film up to 300 "C. The gas-sensing measurement system is shown in Figure 3. The conductance of the phthalocyanine film was monitored by measuring the voltage drop acrose the resistor box using an electrometer connected to an IBM personal computer. The resistor box is connected in series with a 1.5-Vbattery and the sensor. Previous work10 on the study of film temperature effecta (12) KO,W. H.; Fu,C. W.; Wang, H. Y.;Babel, D. A.;Kenney, M. E.; Lando, J. B. Sene. Matenale 1990,2,39-65. (13) Shutt, J. D.;Batzel, D. A.;Sudiwala,R. V.; Rickert,S. E.;Kenney, M. E. Langmuir 1988,4, 124W1247. (14) Wu, Q. H.; KO, W. H. Sens. Actuators 1990, B l , 183-187. (15) Fu, C. W. Ph.D. Thesis, Case Western Reserve University, Cleveland, OH, 1988.
0743-7463/94/2410-0790$04.50/0Q 1994 American Chemical Society
Langmuir, Vol. 10, No.3, 1994 791
Gas-Sensing Mechanism of Phthalocyanine 0 SICCs H13)3
I
1
1.2
-
0.9
-
0.6
-
0
c
E. 8 9 0
0
0
0.3 -
.
0 0
0.3
0.6
0.9
1.2
1.s
1.8
0
0.3
0.6
0.9
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1.5
1.8
I
OH
Figure 1. Structure of [ ( C ~ H ~ ~ S S ~ O S ~ P C O G ~ P C O H I .
C
5
I
i
3 mm
top vlrw Time ( 8 ) (x
lo3)
Figure 4. Response of [(CaH3~SiOSiPcOGePcOHl(29 layers) to 111ppm chlorine in nitrogen at various temperatures.
Dlaphrqn
I
L sectlonal view
Figure 2. Structure of silicon-based gas sensor. Heating element 8
-
0
20
40
60
80
1m120
140
160
I
Temperature (degrees) Figure 5. Normalized conductance change vs temperature for C12, Br2, and I2 gases.
Figure 3. Gas testing measurement system. on the sensor performance is shown in Figure 4. It was observed that the conductance increased sharply with time at fiist when the Cl2 gas contacted the sensor and then increased slowly. The response increasedwith temperature up to 130OC,then decreased again. Figure 5 shows the plot of the normalized conductance change versus temperature, in which data were taken 14 min after the sensor was exposed to Cl2. It can be seen from Figure 5 that the temperature at peak sensitivity for Cl2 is 130 'C, for Bra is 60 OC, and for 1 2 ie 20 OC. The peak temperature is directly related to the electron affiiity of the ionizinggas. When operated at high temperature (>130 "C), the sensor response had a slow downward drift when Cl2 was still present, indicating the bond
formation between the Cl2 and the Pc rings which c a d the ring to become chlorinated and lose its conductivity. The recovery also showed a rapid decrease followed by a slower decrease of conductance when the C12 gas was removed. At lower temperature ( 4 3 0 "C) the conductance did not return to the base line after C4 was switched off for 15 min. This again ehows that Cl2 interacted strongly with the Pc film forming a strong redox couple,andelevatedtemperatures (>13OoC) were required for complete desorption. The 1 2 sensor showed very good reversibility, as the sensor completely recovered to the baseline at room temperature. Sensors with 9,29,49, and 69 layers, around 103,331,669, and 787 A thick (monolayer thickness is 11.4A from the ellipsometry measurementlz), respectively, were tested in premixed 93 ppm chlorine in nitrogen at room temperature (18-20 O C ) . Before each test, the sensor was cleaned by heating it to 140 OC in Nn to drive off allthe impurities and residual gas molecules, as shown by the return of the device conductance to the original baseline.
Wang and Lando
792 Langmuir, Vol. 10, No.3, 1994 1.40 I
AG = AGmU - B, ex(-
wmw
1
t)
$)- B2 ex(-
-:
0.84
0.56
;Ix
69 laycn
1.12
I '
co
h(C'-Cs)
kdcs
t
I
0.28
1
___)
0
1
Figure 8. Model of gaa eensing. 0.00
1
0
0
0
"
4
"
6
0
0
rate of desorption = kdc,
Time (seconds)
Figure 6. Response to 93 ppm Clz at room temperature with f i i thickneseof 29,49,and 69 layers,solidl i n e s are the theoretical fittings.
-r
-1
Fick's first law of diffusion:
ac J =D ax
where COis the concentration of the incoming gas, C* is the concentration in the surface region of the film by mass transfer, and C, is the concentration on the film surface. k,, ka, and kd are the reaction rate constanta of the mass transfer, adsorption, and desorption, respectively. J is the diffusion flux and D the diffusion coefficient. When a steady state is established at the surface, the rate of mass transfer is equal to the rate difference between the adsorption and desorption and equal to the rate of diffusion at the surface into the film:
In the film,Fick's second law of diffusion applies Concentration (ppm) Figure 7. Room-temperature conductance change vs Clz concentration for sensor with 29 layers.
The sensor waa cooled back to room temperature and exposed to a ClrNz gaa mixture for 10 min (loo0cma/min) and then waa taken out of the testing chamber in air and heated to 140 OC again for recovery. The results are shown in Figure 6. The sensor with nine layers did not show any substantial conductance change within the experimental error both at room temperature and 130 OC. It was observed that the sensor response increased with the f i i thickness within the thickness range from 29 to 69 layera and a two-stage conductance change occurred, which indicated that the sensor conductance changes arise from both surface and bulk effects. The room-temperature conductance changes for the sensor with 29 layers on exposure to various concentrations of ClTNZ mixture are shown in Figure 7. The relationship is linear.
E=,-" at ax2
(Fick's second law)
(2)
At the electrode, no further diffusion can occur: at x = I ,
ac
(3)
(ZL=l= O
When t = 0, the gas concentration in the film equals zero: C(x,O) = 0
(4)
Equations 1and 3 become the boundary conditions of the diffusion equation (2) and eq 4 is the initial condition of eq 2. The solution (see Appendix) is: OD
Theoretical Modeling Considering both the surface and the bulk effect, we assume that the gas molecules reach the surface region of t h e film by mass transfer which is a rate process. Then the gas molecules adsorb on the film surface and diffuse into it, which causes an increase in the film conductivity. The gas adsorption and desorption on the film surface are also rate processes, and the gas diffusion in the film obeys Fick's laws. As shown in Figure 8, x = 0 is the f i surface exposed to the incoming gas, the x = 1 is the bottom of the film in contact with the sensor substrate. 1 is the film thickness. Thus assuming (Figure 8)
mass transfer rate from gas phase = k,(Co - C*) rate of adsorption = k,(C*
- C,)
C(x,t) = a - E A , , e x p ( - y p t ) cos[A,(x
- 113
(6)
n=l
where
A, =
4 asin(A,l) (7) 2A,1 + sin(2AJ)
A,, is a characteristic value of eq 2 determined from the boundary conditione (1)and (3)by eq 6. Figure 9 indicates that An is the abscissa of the point of intersection between the curve y = cot(An1) and the line y = -Anb.
Langmuir, Val. 10, No. 3, 1994 793
Gas-Sensing Mechanism of Phthalocyanine "
y
Table 1. Fitting Parameters of the Sensor Response at Different Film Thickness (Room Temperature)
= COt(hl)
AG- (st-') 1.51 X 10-' 7.23 X 10-' 1.47 X 1V
11 (8)
72 (6)
350 550 795
112 125
140
given specific magnitudes of D and 1, one time constant is not enough to describe the process. It can be seen from Figure 9 that the solutions of eq 6 for X, are 1r/2 < A11 < A, 3 ~ / 2< X z l < 27r, etc. Lettering X,l = un which is a particular value for a certain n term, the response constants are
Figure 9. Curve y = cot(X,l) and the line y = -A&.
It is shown from the solution (5) that when time t approaches infinity, the gas concentration in the film reaches a maximum value which is: a = (kaCo)/(ka + kd). This means that the diffusion of the gas molecules into the film results in the accumulation of the gas molecules in the film. When the elapsed time is sufficient, the film is oxidized by all the gas molecules, and the concentration in the film reaches a maximum controlled by the concentration adsorbed at the surface. This condition is implied in the boundary condition (1)in which when t - 0 ,
thickness (A) 331 (29 layers) 559 (49 layers) 787 (69 layers)
ac
J=D-=O ax
r2
So when 1 is large T will be long, and when D is large T will be small. The diffusion coefficient, D = DOexp(-EJkT), will increase with temperature, which causes the time constants to decrease. When n increases, vn An1 will increase, which causes both Tn and Bn (eq 11)to decrease rapidly. Therefore the terms with large n in eq 10 will vanish rapidly with time. In other words all those large n terms with small 7% will be effective only when time is very short, and at longer times, the n = 1term with the largest T will dominate. So with moderate film thickness, temperature, and diffusion coefficient, we use only the first two terms in eq 10to describe the sensor performance: AG = AG-
At this time, the rate of adsorption is equal to the rate of desorption, and the whole system reaches a steady state. We further assume that the conductance change in the film is
")
- B, exp( - L) 71 - B, exp( - 7 2
It is predicted that if the sensing film is very thin or the sensor is operated at high temperature (or D is large), one time constant is enough to describe the sensor response, since the sensing film is saturated rapidly by the gas molecules to reach the maximum response value. Discussion
where B is a scaling constant and Go is the conductance of the film before doping. We have
1. Thickness Effect. As mentioned above, the maximum conductance change is
AGsin' X,l
AG = Bal- 4 a0 n=l
= Bal = -
exp(-X:Dt)
2Xn1 + sin(2Xn1)
(9) An
The conductance change AG also can be written as
where
Therefore the conductancechange is a maximum value minus a summation of multiple exponential terms. The maximum conductance change is AG- = @a1= [k$(ka + kd)lBCol, where /? should be related to the film temperature. The response time constant is T~ = 1/Xn2D and thus is determined by A,, and D. Since the gas molecules accumulate in the fiim and the concentration gradient across the film changes during the gas doping,
Thus it varies linearly with film thickness. The response time constants also increase with film thickness. Figure 6 shows the fitting of eq 13 to the experimentaldata, which indicates that the model of surface adsorption/desorption and bulk diffusion is a suitable description of the observed data. The fitting parameters for AG,, 71, and 7 2 are shown in Table 1. A plot of the maximum conductance change versus film thickness is shown in Figure 10, which shows the linear relationship. Figure 11shows the normalized conductivity change (AG/AG-) vs time for films of 29 and 69 layers, plotted from the fitting parameters in Table 1. These data indicate that the thicker the film, the longer the time needed to reach AG,. It needs to be mentioned that in Figure 10 the linear fitting line has a negative y axis intercept, which indicates that there exists a minimum film thickness. Below this film thickness, there is no measurable conductance (the minimum measurable conductance for our experiment setup is 10-10 a-1). A possible explanation for this is that because of the geometry of the sensor measurement electrode, an extra contact resistance is introduced into the measurement circuit. The gold electrode has a
794
Langmuir, Vol. 10, No. 3, 1994
Wang and Lando Thinner film causes poor contaCtS around the comers
I .60 .e
9
1.28
-
-
L
0.96
-
e-sx
0.64
-
0.32
-
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E
4
Thicker film gives better contacts
s
'
0.00 20
I 30
M
40
60
70
2.20
Film Thickness (layers)
AG = A G , , , ~ - B) ex(-
Y
z
h
Figure 10. Maximum conductance change vs film thickness.
5
6
+)- B,
ex(-
+)
-
1.76
130 'C
c
E
1.2
29 layers
-
1.0
0.4
1.32
c8
0.88
i
0.44
I
-
0.8-
e
-
0.6
&
3
-
0.00 0
150
300
450
600
750
900
0.2 -
Time (seconds) 0.0
' 0
1600
8M)
2400
I 3200
Time (seconds)
Figure 11. Normalized conductance change vs time for films of 29 and 69 layers.
Figure 13. Theoretical fitting curves of sensor response to 111 ppm Cl2 in N2 at film temperature of 90, 110, and 130 O C (29 layers).
where the rate constant k's and Bare all related to the film temperature. P can be considered to be a description of the extrinsic conduction of the film with temperature (eq 8). The film was doped by the incoming electron acceptor gas which forms an acceptor level in the Pc energy gap. The electrical conduction in a doped semiconductor is related to the temperature as follows:16 In the lowest energy state of the acceptor atom, the extra hole of the acceptor is localized at the acceptor site. Thus at very low temperature, the acceptor holes are tied to the acceptor
sites and this effect is called carrier freeze-out. With increase of the temperature, more and more holes are ionized and enter the valence band as free holes. Thus the conductivity of the semiconductor increases with temperature. At high temperatures, all those shallow acceptor levels are ionized and there is little change in free carrier density with temperature; i.e. the conduction is saturated with temperature. Therefore it is expected that P will increase first with temperature and then saturate at a constant Pmmwith temperature. Equation 15 tells us that the maximum conductance change varies with temperature as a result of the variation of the equilibrium between the adsorption and desorption of the gas on the film surface with temperature. It can be seen from eq 15 that k J ( k a+ kd) I 1. When k, >> kd and P = @ m u , AGmm reaches a peak value. This has been demonstrated experimentially as shown in Figure 5. At higher temperature, kd is greater in relation to ka, and thus the conductance change will decrease. For the electron acceptor gas with higher electron affinity, it is more difficult for the induced positive holes to escape the Coulomb field of the negatively charged adsorbed species to give a large conductivity enhancement.17 Thus it is expected that Cl2 will give a peak response at higher temperature (i.e. P- occurs at higher temperature) than Br2 and 12, as demonstrated experimentally in Figure 5. The response time constant (eq 121, which is inversely proportional to the diffusion coefficient D and vn2, will decreasewith increase of temperature, as mentioned above. Figure 13 shows the fitting of eq 13 to the experimental data (Figure 4). The fitting parameters for AG-, 71 and 72, are shown in Table 2.
(16) Singh, J. Physics of Semiconductors and TheirHeterostructures; McGraw-Hill, Inc.; N e w York, 1993; Chapter 8.
(17)Van Ewyk, R. L.; Chndwick, A. V.; Wright, J. D. J. Chem. SOC., Faraday Trans. 1 1980,76,2194-2206.
thickness of 3000 A on the Si substrate with 5000 A of Si02. A Cr layer of 300 A was deposited under the gold layer to improve the adhesion between Au and Si for the in situ heater and the temperature monitor. As shown in Figure 12, the gap between the two electrode fingers is 20 pm, the electrode step is 3300 A, and the monolayer thickness is only 11.4 A. If the film thickness is small, the contact between the electrode and the film around the corners of the electrodes will be poor, which introduces an extra contact resistance. With increase of the film thickness, the contact will be improved and the contact resistance decreases. This also explains why the sensor with a film thickness of nine layers did not have any measurable response. 2. Temperature Effect. The maximum conductance change is related to the film temperature as follows AGmm a ka P ka
+ kd
Gas-Sensing Mechanism of Phthalocyanine
Langmuir, Vol. 10, No. 3, 1994 795
Table 2. Fitting Parameters of the Sensor Response at Different Film Temperature (29 layers) temperature ("C) 90 100 110 120 130
AG8.11 x 1.05 X 1.45 X 1.59X 1.64X
lo-' 10-6
10-6 10-6 10-6
71 ( 8 )
72 (8)
320 275 195 140 68
80 65 53 35
Note that at 130 "C, the sensor response can be fitted with only one time constant, as predicted by the model. 3. Concentration Effect. Since the rate constant k's and 0 are independent of the gas concentration, the maximum conductance change (eq 14) is linearly proportional to the chlorine gas concentration, as demonstrated experimentally in Figure 7.8J8 It has been reported that instead of the linear relationship, the sensor response versus the gas concentration follows an empirical power law relationship: AG a 0, where a is an arbitrary numbel.4 which in our case is 1. This may be due to the fact that some doping gases not only contribute to the conduction carriers but also affect the carrier mobilities. This results in a power factor a deviation from the linear relationship. In this case, our model can be simply modified to describe the results. As shown in eq 1, if the mass transfer rate from the gas phase = k,(Co C*) becomes k,(Coa - C*), then nothing else in the solution changes except that the maximum conductance change becomes AG,,
ring stack is much faster than between the stacks, the observed result that maximum conductivity increases with film thickness (increases the probability of interstack transport) makes sense. However, the response time will increase with thickness since the undoped Pc dimer is a good insulator. Thus diffusion of the dopants to the electrodes is necessary.
Acknowledgment. The authors acknowledge visiting professor T. C. Tan from National University of Singapore for helpful discussions during the initial construction of the sensor model and Mr. Tao Li from the Department of Macromolecular Science for his help in solving the diffusion equation. Financial support by the Edison Polymer Innovation Corporation (EPIC) & NED0 is gratefully acknowledged. Appendix From the boundary condition
one can get
and
= Oal= -
The value of a equals 1 or is an arbitrary number, depending on the type of gas molecules, the range of gas concentrations, and the sensing film and its structure and morphology.
Conclusions A model describing the gas sensing of Pc LB film has been proposed which fits the experimental results very well. The model considers the gas adsorption and desorption on the film surface, and the gas diffusion through the film. The solution obtained from the model for the sensor response has multiple time constants, depending on the sensing film thickness, temperature, and diffusion coefficient, as demonstrated experimentally. The model indicates that the thicker the film, the larger the response but the slower the response. Thus there must be a film thickness compromise between the gas sensitivity and response time. Also the model predicts that there is a conductance peak with temperature, which is a result of the variation of the equilibrium between the gas adsorption and desorption on the film surface with temperature. The higher the temperature and the larger the diffusion coefficient, the faster the sensor response, although above the maximum steady-state adsorption, AG,, decreases. The model decribes the sensor response with gas concentration as a power law relationship in which the power factor can be 1 or an arbitrary number. The film's electrical response to gas exposure should depend on the ordered nature of the film. If one assumes that charge carrier transport along the phthalocyanine (18) Jiang, D. P.; Lu, A. D.; Li,Y. J.; Pang, X. M.; Hua, Y. L. Thin Solid hlma 1991,199,173-174.
Therefore
where
Write
The problem becomes C, = DC,,
(0< x
< 1)
C(x,O) = 0 C(0,t)+ bC,(O,t) = a C,(l,t) = 0 To change the nonhomogeneous boundary condition to the homogeneous boundary condition, let C(x,t) = u(x,t) + a . Then ut = Dux,
(0 < x
u(x,O), = - a
< 1)
(1) (2)
796 Langmuir, Vol. 10, No. 3, 1994 u(0,t)
Wang and Lando
+ bu,(O,t) = 0
(3)
u,(l,t) = 0
We have
ut = DuEE
(4)
(0< x
< 1)
U(g',O) = - a
Seperating variables u(x,t) = X ( x ) T ( t )
u(l,t)- bu,(l,t) = 0
(5)
U((0,t) = 0
Substituting (5) in (l),we have Thus
cos Xg' + B sin Xg']
u(€,t)= e-'?A
where X is a positive constant. It follows that
From the boundary conditions
+ h2X = 0 T' + D X ~ = To
u [ ( ~ , t=) e-'zDt[- AX sin xg'
X"
+ BX cos X~'I~=,,= o eB=O
The solution is
+ B sin k c ]
u(x,t) = T ( t ) X ( x )= e-'*Dt[A cos XX
From the boundary conditions u(0,t)
+ bu,(O,t) = A + BbX = 0
u,(l,t) = e-'aDt[-AX sin XX
=+
+ BX cos
A = -BbX
We have proven that [cos X,xl series under this condition is orthogonal. Therefore
=0 * cot(X1) = -bX
therefore
An
can be obtained from the initial condition:
OD
e-&lDt[A, cos(X,x)
u(x,t) =
+ B , sin(X,x)]
fl=l
We need to prove the orthogonality of the Fourier series under the boundary condition cot(h,l) = -bXn $cos X,x cos X,x dx = 0
n#m Thus the final solution is
But C(x,t) = a
J'sin ~ , xsin ~ , x dx #
o
n
ax
at'
a2u - a2U ax2 ag'2
0
zm
Thus [cos X,x] series is orthogonal, but [sin X,xl is not. To solve this problem, let 6 = 1 - x , therefore -au= - - au
+ u(x,t) =
where cot(X,l) = - bX,,
4a sin(X,I)
A, = 2X,1
+ sin(2X,1)
