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 :
3.1 Mo hinh thiet lap can bing ban din-dien giai (SEI)
3JJ Mo hinh tiep xuc bdn đn-dien gidi cho he nano-xop
Mo hinh ciia tiep xiic dien cue ban din vdi chat dien giai chiia cap oxy
Ox/Red
Dedu Cbiidkat^l
l l - D
fhcB eve dill
Lini kni
Dcd^n Ban d^n loai-n f)ien giai
f «-c.B-^ E p Cv b - y , _ - " ^ 4 * ^ ' ^ ^ « ^ ^ * * E v »
Hinh 3.1: al Mo hinh hí dien hod (PEC Cell) : hi Mo hinh moi chieu SEI
hoa khijr (cap Redox) da tumg duoc xay dimg [12.22,55], dua tren ly thuyei
M6 hinh he dien hoa dien cue ban dSn (PEC Cell) hinh 3.1.a va muc nang lugng ciia tiep xiic ban din-dien giai chua cap oxy hoa khir AVA""- khi can bing tren hinh 3.1.b. Cac th6ng sd cua dien cue ban din nhu ndng do hat lai n, p , dien trudng E dugc coi la ham ciia bien x theo mot chieu khong gian (hinh 3.Lb) vdi bien tiep xiic tai x=L .
M6 hinh tiep xiic ban din-dien giai nay tuong tu nhu chuyen tiep Schottky ciia ban din-kim loai ma d đ miic Fermi ciia kim loai thay bing . miic Redox ciia dung dich. Tinh chit Idp gin be mat ciia chat ban din trong chuyen tiep SEI dugc m6 la tuong tu nhu uong chuyen liep Schottkỵ Do đ trong ly thuyét dien hoa ciia dien cue ban din ta gap nhirng dac trung gidng nhu ly thuyét Schottk}' [22].
Theo ly thuyet ve cac qua trinh dien tu gin be mat chit ban din [71] : nféu ton tai mot thé be mat Vg (hinh 3.Lb), thi phan bd dien the d trang thai chuin dung (Steady Slate) iriing vdi phan bd nong do hat laị Phan bd nay thu dugc tijr nghiem cua phuong trinh Poisson (thudng d dang khong giai dugc). "Phuong an gin diing cho ta quan he Mott-Schottky giua dien dung Cg^ va the be mat Vg nhu cdng thiic (2.5). Trudng hgp long quit chi cd Idi giii sd cua phan bd dien the cung nhu nong do hat tai trong vimg dien lich khdng gian.
Trudng hgp ban din khdng d trang thai chuin dimg, phan bd dien the va nong do hat tai phai xac dinh tu he phuong trinh lien tuc theo thdi gian va phuong trinh Poisson cho ca hai loai hat taị Khi đ dong hgc (kinetic) cua dieu kien bien anh huong manh den hanh vi thiet lap can bing cua ban din. Su khac nhau ve ban chat vat ly giira chuyen tiep kim loai-ban din va SEI se din dén dong hgc cua dieu kien bien khac nhau [55].
Ddi vdi dien cue nano-xop, m6 hinh th6ng thudng cua tiep xiic ban dan- dien giai (hinh 3.1b) khdng the ap dung dugc. Tren hinh 3.2a , dien cue xop gom nhOng hat ban din kich thude nho nhiing trong dung dich dien giaị Khi đ dung dich tham vao cac 16 xop tao ra liep xiic dien vdi lung hai ban dan.
AVVVVV
rrcoj ;
Diea giai
Hat ban dao
Red
Hinh 3.2: al Mo hinh dien cue nano-xop ; bl Mo hinh hat bdn đnldien gidi
De nghien curu he dien cue nano-x6'p phai thiet lap m6 hinh tiep xiic ban din-dien giai mod. Mo hinh phai dap iing dugc 2 van de dac thu ve hinh thai hgc (Morphology): xop ; va kich thude nho (Nano). Do dac thii ciia dien cue dang nano-xop, ve nguyen tie, ton tai SEI tren be mat cua titag hat ban din cua mang dien cue (hinh 3.2a).
Néu gia thiel cac hat ban din chi noi vdi nhau tai mot diem cd dien tich rit nho so vdi dien lich con lai cua be mat hat đ, ta cd the gia thuyét cac hat nhu nhirng hinh ciu dugc bao phu loan bg bdi chat dien giai va din dien qua day din ly tudng nd'i giira tam cua cac hinh ciu đ (hinh 3.2b). Khi đ tinh chat cua hat ban din la cac bien theo 3 chieu khong gian. Do tinh d6'i xiing cua hinh ciu nen thuc té ta cd the chuyen bai loan ve dang chi phu thu6c vao khoang each den tam ciia hinh ciu R (mot bien khong gian).
Dieu kien bien : Néu thiet lap dugc dieu kien bien khdng thaj' d6i theo
kich thude hat thi mo hinh cd the sir dung de nghien cuu tinh chat cua dien
cue nano-xop theo kfch thude hat ban din (hieu ung kfch thude hat).
Van de kich thuoc : Ddi vdi mo hinh dien cue phang (hinh 3.1b), khi
kfch thude ban din Idn ban nhieu kfch thude vimg dien tfch khdng gian, phan bd a trang thai can bing Iheo ly thuyet Schottky (kh6ng phu thugc vao kfch thude hat). Khi kfch ihudc ban din cd the so sanh dugc vdi kfch thude vung
dien tfch khdng gian (\'ai dp dai Debye), phan bd dien the va nong do hat lai la ham phu thugc vao kich thuoc hai [27] .
Van de hinh hoc cua hat : Mo hinh hat ban din h'mh cau co duong
kinh v6 ciing Idn chinh la trudng hgp m6 hinh phang. Nhung doi đi vdi kich
thude hiru han (dac biet la co kich thude \'iing dien tich khong gian), van de
hinh hgc cua mo hinh se xuat hien.
Van de cdu hinh tai thdi diem khdi đu (t=0) : Ciu hinh can bing la
nghiem tai trang thai chuan dimg cua phuong trinh m6 la qua trinh theo thdi gian. Do đ dieu kien tai thdi diem t = 0 cua phuong trinh mo ta qua trinh phai cd do tin cay caọ Qua trinh thiet lap can bing cua SEI trong luan an dugc mo
phong theo nguyen ly tu diu (Up Inito).
CB
VB
Baa d^D loai-o Ckin Uwttg Dien giii
Á/Â
^A
^ ^ /
a/ Bdn đn, Dien gidi trong chdn khong bl Trao doi dien tich khi tiep xuc
Ban dan loai-n
•
Dien giai
r
Urp irunR Lop d i ^ lich =
hoi dien lich khun^ p v i J
Oe d^n B a n d a n luai^n , i>ien ^iui
c/ Thiet lap cac lap dien tfch d/ Cdu hinh vung ndng luong cdn hang
Hinh 3.3 : Md hinh md phong cdn hdng ban đn-dien gidi theo nguyen ly tir đu (Up Inito)
Hinh 3.3 mo ta nguyen ly Up Inito : Gia thuyet lai thdi diem dau tien (t=0) , chit ban din loai n va dien giai chiia cap Redox AVẤ each ly nhau hoan loan trong chan khong. Khi đ, trong ban din dien tfch se trung hoa, cau triic vimg nang lugng Ê, E\ la phing va muc Fermi Ep theo thang nang lugng tuyet đi cao hon muc Redỏ: Ê^^jô (hinh 3.3a). Khi cho ban din tiep xiic vdi cap Redox, se xay ra qua trinh trao đi dien tich giira cac hat tai (dien tir va 16 trong) vdi trang thai oxy hoa (Ox) va khir (Red) cua cap oxy boa khir (cap Redox) (hinh 3.3b). Hau qua la trang thai Irung hoa dien lich trong ban din bi pha vd tai viing gin be mat tiep xiic, lao ra Idp dien tich khdng gian (hinh 3.3c). Ddi dien vdi Idp dien tich kh6ng gian cua ban din, trong dung dich dien giai hinh thanh Idp dien tfch gom cac ion trai diu gin be mat tiép xiic, tao ra Idp dien tfch kep Helmholtz (hinh 3.3c). Su hinh thanh cac lop dien tich tren be mat tiep xiic tao ra dien trudng (rao the) ngan can qua trinh trao doi dien lich giiia ban din va cap Redox. Khi rao the be mat du Idn, su trao doi dien tfch se dimg (thuc ra la chuin dimg) va can bing dugc thiet lap. Khi đ muc Fermi cua ban din se can bing vdi miic Redox trong dung dich (hinh 3.3d). Bit cii su pha va can bing nao xay ra cung se din den qua trinh trao doi dien tich neu tren de thiet lap lai can bing. .
M6 phong thiet lap can bing lir diu nhu tren dam bao ciu hinh cua ban din d trang thai i=0 la hoan loan chfnh xac.
Cau hinh can bing cua SEI la nghiem cua he phuong trinh m6 la qua trinh thiet lap can bing lai trang thai chuan dimg. dudi dang bg sd. tuong iing vdi cac tham sd cua vat lieu va idp tiep xiic nhu : miic vung, ndng do tap
(donor), miic Redox, nong do ion ....vv va kich thuoc hat . Hanh vi cua chat
ban din khi bi kfch thich cung nhu qua trinh thiet lap lai can bing dugc mo phong bat dau tu cau hinh can bina neu tren.
3J.2 phuong trinh mo td qud trinh thiet lap cdn bdng
Trong vat ly ban din, qua trinh thiet lap can bing dugc m6 ta bdi phuong trinh lien tuc (Equations of Continuity) cho hai loai hat tai nhu sau :
^ = D,V^n + ^ , n V £ + ^ ^ V n . E - R , + G , . (3.I)
J = D,V^p-^,pV.E-fi,Vp.E-Rp+G^ (3.2)
trong đ n,Dg,|i^,R^,G^ la ndng do, he sd khuech tan, do linh dong, tdc do tai hgp, kfch thfch (neu cd) cua dien lir. p,Dj^,|â,Rp,Gp la ndng do, he sd khuech tan, do linh dong, tdc do tai hgp, kfch thfch (neu cd) cua 16 trong. E la dien trudng (noi va ngoai) dugc m6 ta bing phuong trinh Poisson :
V.E = - ^ (3.3) vdi p la nong do dien tfch du (excess charge);
£0, e la hing sd dien mdi cua chan khong va chit ban din .
Cac phuong trinh tren bieu dien dudi dang tong quit con cac bien sdn,p,Ela cac bien vector cua he phuong trinh theo khdng gian.
Tai thdi diem diu tien t=0 ciu hinh (cac bien n, p, E) cua ban din each
ly vdi chit dien giai (hinh 3.3a) se la no , Po , miic Fermi E% va dien trudng
E=0. Cac thdng sd nay la hien nhien, da dugc nghien cuu va kieim chiing. Khi cho tiep xiic vdi pha thii hai (dien giii chiia cap Redox), xay ra qua trinh trao d6i dien tfch tren bien tiep xiic, din tdi su thay doi ciia n, p, E theo khdng gian. Giaị he phuong trinh tren vdi dieu kien bien m6 ta qua trinh trao doi dien tfch cho dén khi dieu kien chuin dimg suit hien, cho la cau hinh can bing ciia tiep xiic SEI tuong iing vdi phan bd n, p, Ẹ
Neu dieu kien bien thay d6i (thay d6i miic Redox, kich thfch anh sang...w) trang thai chuin dung mdi se dugc thiet lap tuong iing \'di phan bd
n, p, E mdị Nhu vay nghiem cua he phuong trinh tren dudi cac lac dong khac nhau se cho ta nhimg thdng tin ve tinh chat vat ly tuong iing.
Thanh phin tdc do lai hgp R^ p hay kich thfch G^p se dugc bieu dien qua cac bien cua phuong trinh tuy theo gia dinh ve co che lai hgp hay kfch thfch.
Thuc hien bien doi he he phuong trinh tren đi vdi hai loai m6 hinh :