V 2, (2.3) dang nua hinh iron (hinh 2.7.a) Dua vao duong Nyquisl ta xac dinh dugc cac thong so can ban he dien hoa :
2/ Dien cue xop, hat hmh ciu (hinh 3.2b), se ggi la mo hinh cdu
3.1.5.3 Dieu kien tai sdc toa do x=0 (R=0)
Tai gdc toa do dieu kien ciia dien irudng É = 0 nhu irinh bay 6 iren . Xet trudng hgp kh6ng cd đng van lai dien lich mach ngoai (no Mass Transfer Currenl), khi đ hien nhien dieu kien lai x=0 (R=0) nhu sau :
E*=0 an- op = 0 (3.56)
ox dx'
Voi cac didu kien (3.50), (3.51) va (3.56) ta co dii dieu kien bien dé giai he phuong trinh dao ham rieng phi tuyen 3.43 -> 3.45 hay he 3.46 -> 3.48.
Cac dieu kien bien 3.50, 3.51,^.56 hoan toan kliong phu thuoc vao kich
thudrc hat ban dkn trong mo hmh. Dac diem nay cho phep giai bai toan khong
bi han ché ve kfch thuoc ciia vat lieụ
3_,1_J>.4_ Dieu kien tai thoi diem ban đu t=0
Cac bien ciia he phuong trinh n'(x,t), p'(x,t), É(x,t) phu thuoc vao thdi gian. Nhu vay cáu hinh tai thcri diem t - 0 se quyet dinh tinh diing dan ve mat vat ly ciia nghiem toan hoc thu dugc. Néu theo nguyen ly mo phong tu dau
nhu da trinh bay o muc 3.1.1 chuong 3 , cáu hinh chac chan diing la cau hinh
CLia ban dan khi each ly hoan toan trong clian khong no , Po, E =0 . Nhu vay cáu hmh tai thoi diem t: 0 se la :
n*(x,t:-0).-0 ; p*(x,t:^0)'--0 ; E*(x,f--0) = 0 (3.57)
3.1.6 Phuong phdp sajUii >.e P-^^^^S ^'^^ '^-oo ham rieng phi tuyen
Gia sir ta co he gom n phuong trinh moc nhau, dao ham rieng, phi tuyén dang parabol nhu sau :
a 6 1 '•' dx-1=0 k.i
5c,(x,t)Y5cJx,t)
^ 5x dK
^ 5c,(x,t). ^
+ỵ..-^Ki^.(c.(>^-t).x,t) Vdi dieu kien bien B. ^c,(x,.t) '
ax . c , ( x , . t ) . t )
dx
= 0 (3.58
k,i = 0,l,...,n-l ; b la bien trai va phai . Tai day x - Ichong gian va t - thoi gian la cac bien doc lap ; c,(x,l) la cac bien phu thuoc ; k - so thu tu ciia phuong irinh; R^ la ham phi tuyén bai ky cua lai ca cac bien c,.
Dang t6ng quat ciia he phuong trinh tren nhu sau : a c , ( x , t ) di = G ^ 5 ' c , ( x , t ) 5c,(x,t) c , ( x , t ) , x , t (3.59) dx' dx
V6i Gj. la ham so phi tuyen nao do .
Cac bien de)c4ap x, t dugc roi rac hoa (discretized): N diem niit khong gian Xj;
j = 0,1,...,N-1 ; Tai thoi diem t, vdi 1 = 0,1,... Cac gia tri cua nghiem Cy(x,.) se chi dugc tinh tai cac diem niit luoi cua x va t nhu neu tren va ky hieu la c^{],\) •
Cong thiic vi sai ciia dao ham theo thcri gian t cua c,; bieu dien nhu sau :
ac,(x,t) ,c,(j,i+i)-c,(j,i)
dt t w - t : (3.60) Ta dat: Cj (j) = TC^ (j, 1) + (1 - T)CJ (j, 1 +1) ; T - tham so thay doi dugc 0 < x < 1 Khi do cong thiic vi sai 3 diem cia dao ham theo x tai diem nut j nhu sau :
5'c,(x,t) 2 dx 8 6 0 + 0 . 0 + 0 , dc.(x,X) c,(j + l ) - c , ( j - l ) + 0 ( 5 , - 8 J + 0 ( 5 , - 5 . ) (3.61) (3.62) ax 5 _ + 6 ,
vdi 5_ = X - Xj_| va S, = x^,, - Xj ; ham 0(x) la ham chi bach xáp xị Cong thiic vi sai 3 diem tien tai bien trai B^ „„ (diem j=0) ciia dao ham theo x
a:,(x,t)
dx 1 . I|c,(aị,).llfc.(ụi)-^c.aị.) •a6,(S,+6,))(3.63)
5,
voi 5, = x , -Xo ; h^=x^-x, va
Cdng thiic vi sai 3 diem liii tai bien phai B,„^, (j=N-l) ciia dao ham theo x : 5c.(x.t) 1 N - l . 2f + l 1 ^ (N_i,l + l ) - ( l + f ) c , ( N - 2 , i + l) + — - c , ( N - 3 , l + l) f + 1 * + ' + 0 ( 5 , . , ( 5 , . , + 6 , . , ) ) (3-64) Vdi 5 , . , = x , . , - x , _ 3 ; 5 , . , = x , . , - x , . 3 va f = 5 , _ , / 5 , . ,
Thay (3.60) -(3.64) vao he phuong trinh (3.59) ta cd he nxN phuong trinh :
F,J(ẹ(m,l + l),c,(m,l))=0 (3.65) Trong he phuong trinh (3.69) la cd nxN ^in sd c.(m, I + 1) tai cac diem luoi
thdi gian "mdi" ; Cdn cac gia tri tai ludi thdi gian "cu" Ci(m,l) dugc coi la cac thong soda biet. Cac chi sd cua (3.69) nhu sau ; j=OJ,..,N-l ; m=0,l,2 đi vdị j=0 ; m=j-l, j , j+1 đi vdi 1< j < N-2 ; m= N-3, N-2, N-l đi vdi j=N-l (he phuong trinh cd dang dudng cheo).
F^^(...) la cac ham phi tuyen . He phuong trinh dai so phi tuyen nhu vay cd the giai dugc b ^ g phuong phap sd su dung quy trinh lap Newton-Raphson [85,41] nhu sau:
Néu khai trien ham sd xung quanh diem nut nao đ (ky hieu IQ ) b^ng khai trien chudi Taylor ta se cd bieu dien :
+ Z{Alc!![c,(J-U + l)-Ci(j-l,l + l)|J + B^;i[c,(j,l + l)-c,(j,l + l)|„;
1=0 Dl^j[c,(j + l,l + l)-c,(j + l,l + l ) | J (3.66) Dl^j[c,(j + l,l + l)-c,(j + l,l + l ) | J (3.66) '(J) _ F ( J ) voi: A (j) _ ap, 5c, (j-1,1 + 1) Bi^: = dh 5c, (j, 1 + 1) D!^: = 9F„ 5c, (J+ 1,1 + 1)
Chu y rang : A\^] la ma tran nxn dinh nghia đi vdi j=l to N-l
Bj,^; la ma tran nxn dinh nghia đi vdi j=0 to N-l
D['] la ma tran nxn dinh nghia đi vdi j=0 to N-2
Cdn" thiic vi sai duoc sir dung tai cac bien nen can hai ma tran niia cho bien
X,.,= 5F,
5c, (2,1 + 1) đi j=0 va ^k.i
5F,
Ta cd the viet phuong trinh (3.66) dudi dang ma tran J.dc = - F
Vdi dinh nghia cac ma tr^n nhu sau:
(3.67) J = J = g(0)j-j(0)^(0) A ( 1 ) B 0 ) D ( 1 ) ^j^O)B(j)D(j)^ Y(N-l)^(N-l)g(N-l) ; d c = 'dc'°' dc<" dc"' _dc'^-"_ ;F|o = lo pO)| lo p(J)l lo p(N-l)| lo_ (3.68)
Quy trinh Nev^on - Raphson dugc ap dung cho he phuong trinh: Tinh
ma tr|n J va FIQ tai diem "doan" CIQ = c^°^ , giai phuong trinh (3.67) vdi dc .
Neu dc va FIQ nho hon gia tri sai so dinh irudc nao đ thi phuong trinh coi
nhu da giai xong. Cdn ngugc lai ta se dat CIQ = c^^°^ ""^ = dc + c*^^ ""^ va lap lai
qua trinh giai nhu tren cho den khi nghiem hoi tu hoac d^'u hieu phan ky (divergence) hay dao đng (oscillation) dugc phat hien. Trong trudng hgp nay
ta se chgn diem "doan" ban dau ciia in sd ej,(j,I-hl) la diem noi suy tuyen
tfnh ciia c^. (j, 1 -1) va ê (j, 1) den ludi thdi gian tiep theo t,+, .
Gia tri ciia tham sd thay doi x trong khai trien (3.60), theo kinh nghiem [41], thudng xáp xi 0,5 se cho do hoi tu ciia quy trinh Newton-Raphson la tdị
3.7.7 . Luu do quy trinh mo phong qud trinh thiet lap cdn bdng SEI Luru do 3.1 Cac thong D e , , . b . K Cac thong Tinh ra : r\^ so ban d^n: n„. Eg, c^n ^
...v\' va Kich thude hat. so dien giai : Ê,,^^ . Ê ,
1 ^ ^ p ^ •'^n ' ^ - l ^ D e b y e • •"• -Nc ^ O X ' ,Nv ^recl • l^c.v ..V\' ,T, i Thai diem t* = 0 : n*(x*,t*) = p*(x*,t*) = 0 ; E*(x*,t*) = 0 I
Tang thdi gian t* den gia tri liep theo : t* + At
1
Giai he phuong trinh (3.43) - (3.45) hoac (3.46) - (3.48)
vdi dieu kien bien (3.50), (3.51) va (3.56).
I
Nghiem la phan bd tai thdi diem t* = I* + At 'n*(x*,i*),p*(x*,t*),E*(x*,t*)
i
Kiem tra dieu kien chuan dung — • Khong dat
T"
i
Dal
I
Kei qua la cau hinh d trang thai can bang :n*(x*), p*(x*),Ê(x
3. 2 phuong phap sd tinh dien dung dien hoa dien cue ban d^n nano-xop
3.2.7 Cdc lop dien tich cua tiep xiic bdn đn - dien gidi
Trong mo hinh dien cue ban din nano-xop (ph^n 3.1.1), do tinh chat xop cua dien cue, cac be mat tiep xiic trong (Internal Surfaces) ve nguyen t^c deu đng gdp thanh ph^n vao dien dung dien hoa ciia dien cuẹ
Xem xet tiep xuc cua mot hat ban dSn vdi dung dich dien giai va vai tro cua cac Idp dien tich kep tren be mat tiep xiic đi vdi dien dung dien hoạ Mo hinh cac lop dien tich ciia
tiep xiic ban din-dien giai [5] nhu tren hinh 3.6. Lop dien tich khong gian tai be mat tiep xiic ciia ban din q^*^ se đng gdp thanh phin dien dung Cg^ vao dien dung dien hoa ciia hat ban din.
Cac ion trong dung dich (sat be mat tiep xiic) ngugc dau vdi dien tich du tai vLing dien lich khong
I i J- .- L L d p dien lich khong gian
Ban dai^
kep Helmholtz
0 , <& L<^ khuech tan
Cation dung dich
••CB
c
e D
Anion bi hap phu
(y^ = Phan tur dung moi
Hinh 3.6: Cdc Idp dien tich cua tiep xiic bdn đn - dien gidi
gian ciia ban din, tao ra lop dien tich kep Helmholtz . Lop nay đng gdp thanh phin dien dung Helmholtz C^ vao dien dung dien hoạ
Trong dung dich, vung gan be mat liep xiic, ion lai lop dien lich do chenh lech nong do ion giira vung gan be mat dien cue so vdi vung nam trong the khdi CLia dung dich (bulk solution). Ldp nay dugc ggi la Idp dien tfch khuech tan (ldp GCS : theo ly thuyet ciia Gouy-Chapman-Slem). Ldp GCS se
Trong ly thuyet kinh dien ve dien dung dien hoa : dien dung dien lich khong gian Cg^ đng vai tro chii dao tai vung the phan cue duong (gan diing Mott-Schottky). Nhimg đi vdi he dien cue nano-xop, gia thuyet nhu vay khong the ap dung dugc. Diem mdi trong phuong phap linh dien dung dien hoa ciia luan an nay la linh den sir tham gia ciia ca 3 thanh phin dien dung ciia tiep xiic ban din - dien giai (SEI) vao dien dung dien hoạ Nhu vay phep do thuc nghiem tdng trd (Impedance) cd the lien hanh d tin sd thap, nhim tranh anh hudng ciia thanh phin de din nhu da lung dugc de cap [23] .
3.2.2 Cong thuc tinh dien dung cua cdc ldp dien tich tren SEI
Dien dung vi phan ciia ldp dien tich Q ndi chung duoc dinh nghia dQ
C =
dcp (3.69)
Q la long dien tich va cp la sut the tren ldp dien lich.
Su dung dinh nghia tren đi vdi Idp dien lich khong gian ciia ban din dQsc
C =
d(P: (3.70)
(ps la the be mat (Voh) va Q^ la idng dien lich khong gian (Coulomb) ta cd :
Qsc= jp-d? (3.71)
Vung dien tich khdng gian
la tich phan ciia mat do dien lich du p theo khong gian dr .
Theo ky hieu va cong thuc bien doi loan hgc nhu d muc 3.1.5 ciia luan an :
Ddi vdi md hinh phdng :
Qsc=noLoc..|^(ế-l)-(e"-l)jdx'
' 0 "o J (3.72)
Doi vdi mo hinh cdu :
1 R' R' L 0 V ^ . / P ^ ( e ^ ' - l ) - ( e " ' - l ) d R - (3.73) n„
Ddi vdi mo hinh phdng (r C = 5Q Sc ^•^O-'^Debve J ^ ( e " - l ) - ( e - - l ) d x t?(p. kT/e dip es, i P^(é--l)-(e"--l)lx- n^ L Debye 9(p' (3.74)
Ddi vdi mo hinh cdu
^LW^.\^ C = 5 Q s c ^ g - ^ p L p e b y e C = 5 Q s c ^ g - ^ p L p e b y e 5(ps I Pl(eP--i)_(e"--i)dR* nri kT/e r,- ^t, w Pl(eP'_i)-(e"'_l)liR-d(p* Debye
Cac ky hieu thong sd : L Debye ^kTsSo 1 2e e n 0 J ccp p = In - ^ n = in — Po ' 1^0 (3.75) (p = kT ' V = x = L Debve Debye R" = R L Debvi
; L - do day (ban kinh) ciia mang (hat) ban din . Cong ihde loan hgc de linh dien dung ciia ldp dien tich Helmholtz va ldp khuech tan (ldp GCS) (khong bi anh hudng bdi kich thude hat ban din) da dugc trinh ba\' trong cac giao trinh ve dien hoa dien cue ban din [5.6] :
C H -
£„.£,)
(3.76)
^ * ^ 0 - ^ s o l GCS