Co/β-Mo2C-decorated Carbon Paper Anode for High-performance Microbial Fuel Cells
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摘要: 采用热处理辅助的化学法制备了Co/β-Mo2C双金属碳化物,将其作为微生物燃料电池(MFCs)阳极催化剂. 采用扫描电子显微镜(SEM),X-射线衍射(XRD)和X射线光电子能谱(XPS)等手段研究了形貌和组成,采用电化学交流阻抗(EIS)法、循环伏安(CV)法和单室空气阴极微生物燃料电池(MFC)研究了其电化学性能. 结果表明:Co/β-Mo2C具有良好的电催化性能以及出色的微生物相容性. 含有3 mg/cm2 Co/β-Mo2C的阳极MFC最大输出功率密度和库伦效率分别为483.3 mW/cm2和3.7%,分别是未修饰的碳纸阳极MFC的1.30倍和1.68倍. 该研究可为开发高性能的微生物燃料电池阳极催化材料提供新思路.Abstract: Bimetallic carbide Co/β-Mo2C was prepared with the thermal treatment-assisted chemical method and used as an anode catalyst for Microbial Fuel Cells (MFCs). The morphology and content of Co/β-Mo2C was studied with the scanning electron microscope (SEM), X-ray diffraction (XRD), and X-ray photoelectron spectroscopy (XPS). Furthermore, the cyclic voltammetry (CV), the electrochemical impedance spectroscopy (EIS) and a single-chamber air-cathode Microbial Fuel Cell(MFC) were used to study the electrochemical performance of Co/β-Mo2C. The results showed that due to the prominent electrochemical activity and excellent biocompatibility, the MFC device equipped with 3 mg/cm2 Co/β-Mo2C-decorated carbon paper anode delivered a power density of 483.3 mW/cm2, and a coulomb efficiency of 3.7%, which are 1.30 and 1.68 times those of the pristine carbon paper anode MFCs respectively. This work provides a new path for the explorer of high-performance anodic catalysts for MFCs application.
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Keywords:
- microbial fuel cells /
- anode /
- cobalt-doped molybdenum carbide /
- composite
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链回归点、非游荡点集、极限点集和回归点集是动力系统中重要的定义,在动力系统的发展中有着重要的作用,与系统的混沌、链传递密不可分。
学者们对链回归点、非游荡点集、极限点集和回归点集的拓扑结构和动力学性质进行了研究[1-11]。如:证明了在群作用下的逆极限空间中移位映射的G-回归点集等于自映射在其G-回归点集形成的逆极限空间[1];在正上密度回归点集稠密的条件下,证明了可迁系统等价于E系统[2];证明了在群作用下的逆极限空间中移位映射的G-链回归点集等于自映射在其G-链回归点集形成的逆极限空间[3];证明了转移映射的强非游荡点集等于自映射f在其强非游荡点集形成的逆极限空间[4];研究了右高类帐篷映射链回归点集和强链回归点集的关系[5];指出映射f的每个负轨道的a-极限集是G上某个点的w-极限集[6];讨论了强一致收敛下序列映射非游荡点的保持性[7];在广义树上研究了连续自映射f回归点集的拓扑结构[8];证明了回归点集对映射f强不变[9];证明了x是一致回复点当且仅当x是Birkhoff回复点[10];在具备一定条件下的可交换的C-系统中,证明了任意传递点是等度连续点[11]。
链回归点一定是G-链回归点,非游荡点一定是G-非游荡点,极限点一定是G-极限点,回归点一定是G-回归点集,但反之不成立。本文尝试将链回归点、非游荡点集、极限点集和回归点集的动力学性质进行推广,在映射f是G-等度连续的条件下,在度量G-空间中研究了G-链回归点、G-非游荡点、G-极限点和G-回归点之间的动力学关系,拟充实度量G-空间中G-链回归点、G-非游荡点、G-极限点和G-回归点的理论。
1. 预备知识
定义1[1] 设X是度量空间,G是拓扑群。若映射φ: G×X→X满足
(1) ∀x∈X,有φ(e, x)=x,其中e为G的单位元;
(2) ∀x∈X和g1, g2∈G, 有
φ(g1,φ(g2,x))=φ(g1g2,x), 则称(X, G, φ)是度量G-空间,简称X是度量G-空间。为了书写方便,通常将φ(g, x)简写为gx。
若X是紧致度量空间,则称X是紧致度量G-空间。
定义2[1] 设(X, d)是度量空间,如果f: X→X是一一映射且f和f-1都是连续的,则称f是同胚映射。
定义3[1] 设(X, d)是度量G-空间,f: X→X连续。如果∀x, y∈X,∃{ni}⊂N+,∃{gi}⊂G,使得limi→∞gifni(x)=y,则称y是x的G-极限点,用wG(x, f)表示。记WG(f)≡⋃x∈XwG(x,f),称WG(f)为f的G-极限点集。
定义4[1] 设(X, d)是度量G-空间,f: X→X连续,x∈X。如果对任意包含x的开集U,∃n∈N+,∃g∈G,使得gfn(x)∈U,则称x是f的G-回归点。f的G-回归点集用RG(f)表示。
定义5[3] 设(X, d)是度量G-空间,f: X→X连续,x∈X。如果对任意包含x的开集U,∃n∈N+,∃g∈G,使得gfn(U)∩U≠Ø,则称x是f的G-非游荡点。f的G-非游荡点集用ΩG(f)表示。
定义6[12] 设(X, d)是度量G-空间,f: X→X连续。如果∀g∈G,∀x∈X,有f(gx)=gf(x),则称f是等价映射。
定义7[12] 设(X, d)是度量G-空间,f: X→X连续。如果∀g∈G,∀x∈X,∃h∈G,有f(gx)=hf(x),则称f是伪等价映射。
定义8[13] 设(X, d)是度量G-空间,f: X→X连续。如果∀ε>0,∃δ>0,当d(x, y) < δ时,∀n∈N+,∃gn, pn∈G,有d(fn(gnx), fn(pny)) < ε,则称f是G-等度连续。
备注1 G-等度连续点的概念见文献[13], 交换群的概念见文献[14]。
定义9[15] 设(X, d)是度量G-空间,f: X→X连续,ε>0。如果∀i(0≤i < n),∃gi∈G,使得d(gif(xi), xi+1) < ε,则称{xi}i=0n是f作用下的(G, ε)链。
定义10[3] 设(X, d)是度量G-空间,f: X→X连续。如果∀ε>0,存在f作用下的(G, ε)链{xi}i=0n,其中x0=xn=x,则称x是f的G-链回归点。f的G-链回归点集用CRG(f)表示。
引理1[1] 设(X, d)是紧致度量G-空间,G是紧致,则∀ε>0,∃0 < δ < ε,当d(u, v) < δ时,∀g∈G,有d(gu, gv) < ε。
引理2[3] 设(X, d)是度量G-空间,G是紧致的,f: X→X同胚等价,则f(WG(f))=WG(f)。
引理3[3] 设(X, d)是紧致度量G-空间,G是紧致的,f: X→X同胚等价,则f(CRG(f))=CRG(f)。
2. 主要定理
定理1 设(X, d)是紧致度量G-空间,G是可交换的紧致群,f: X→X伪等价。若f是G-等度连续的,则RG(f)=WG(f)=ΩG(f)。
证明 由G-回归点、G-极限点和G-非游荡点的定义易知,RG(f)⊂WG(f)⊂ΩG(f)。下证:ΩG(f)⊂RG(f)。由引理1知,∀ε>0,∃0 < ε0 < ε,当d(z1, z2) < ε0时,∀s∈G,有
d(sz1,sz2)<ε。 (1) ∃0 < ε1 < ε0,当d(z1, z2) < ε1时,∀s∈G,有
d(sz1,sz2)<ε02。 (2) 设f是G-等度连续的,则对ε1>0,∃0 < ε2 < ε1,当d(z1, z2) < ε2时,∀n≥0,∃gn, kn∈G,使得
d(fn(gnz1),fn(knz2))<ε12。 (3) 设x∈ΩG(f), 则∃m>1,∃g∈G,∃y∈X,使得
d(x,y)<ε2, (4) d(gfm(y),x)<ε2。 (5) 由式(3)和式(4)知
d(fm(gmx),fm(kmy))<ε12。 由于f是伪等价映射,故∃pm, tm∈G,使得
d(pmfm(x),tmfm(y))<ε12。 (6) 由式(2)和式(5)知
d(tmgfm(y),tmx)<ε02。 (7) 由式(2)和式(6)知
d(gpmfm(x),gtmfm(y))<ε02。 由于G是可交换的,故
d(gpmfm(x),tmgfm(y))<ε02。 (8) 由式(7)和式(8)知
d(gpmfm(x),tmx)<d(gpmfm(x),tmgfm(y))+d(tmgfm(y),tmx)<ε0。 由式(1)知
d((tm)−1gpmfm(x),x)<ε, 则x∈RG(f),从而有ΩG(f)⊂RG(f),故RG(f)=WG(f)=ΩG(f)。证毕。
定理2 设(X, d)是紧致度量G-空间,G是紧致的,f: X→X同胚等价。若f是G-等度连续的,则WG(f)=CRG(f)=∞⋂n=1fn(X)。
证明 由引理2知,∀n≥1,有
fn(WG(f))=WG(f)。 又fn(WG(f))⊂fn(X),则∀n≥1,有WG(f)⊂fn(X)。因此
WG(f)⊂∞⋂n=1fn(X)。 由引理1知,∀ε>0,∃0 < ε0 < ε,当d(z1, z2) < ε0时,∀s∈G,有
d(sz1,sz2)<ε。 (9) 设f是G-等度连续的,则对上述ε0>0,∃0 < δ < ε0,当d(z1, z2) < δ时,∀n≥0,∃gn, kn∈G,使得
d(fn(gnz1),fn(knz2))<ε0。 (10) 设y∈∞⋂n=1fn(X),则∀n≥1,∃xn∈X,使得
y=fn(xn)。 (11) 根据X的紧致性,存在子列{xni}i=0∞满足xni→x(i→∞)。因此,对上述δ>0,∃m>0,当i>m时,有
d(xni,x)<δ。 由式(10)知,当i>m时,有
d(fni(gnix),fni(knixni))<ε0。 由式(11)和f是等价映射知
d(fni(gnix),kniy)<ε0。 由式(9)和f是等价映射知,当i>m时,有
d((kni)−1gnifni(x),y)<ε。 从而有y∈WG(f),则∞⋂n=1fn(X)⊂WG(f),故
WG(f)=∞⋂n=1fn(X)。 下证CRG(f)=WG(f)。由G-极限点和G-链回归点的定义知,WG(f)⊂CRG(f)。由引理3知,∀n≥1,有
fn(CRG(f))=CRG(f)。 又fn(CRG(f))⊂fn(X),则∀n≥1,有
CRG(f)⊂fn(X)。 从而有
CRG(f)⊂∞⋂n=1fn(X), 则CRG(f)⊂WG(f),故CRG(f)=WG(f)。因此WG(f)=CRG(f)=∞⋂n=1fn(X)。证毕。
定理3 设(X, d)是紧致度量G-空间,G是可交换的紧致群,f: X→X伪等价,则WG(f)中的所有点都是G-等度连续点当且仅当f是G-等度连续的。
证明 (⇒)假设WG(f)中的所有点都是G-等度连续点。∀x∈X, 则wG(x, f)≠Ø。取y∈wG(x, f),则y是G-等度连续点。由引理1知,∀ε>0,∃0 < ε0 < ε,当d(z1, z2) < ε0时,∀s∈G,有
d(sz1,sz2)<ε。 (12) 由y是G-等度连续点知,对ε0>0,∃0 < δ0 < ε0,当d(z, y) < δ0时,∀n≥0,∃gn, kn∈G,使得
d(fn(gnz),fn(knz))<ε02。 (13) 由引理1和y是G-等度连续点知,对上述δ0>0,∃0 < δ1 < δ0,当d(z, y) < δ1时,∀n≥0,∃tn, hn∈G,使得
d(fn(tnz),fn(hny))<δ02。 (14) 当d(z1, z2) < δ1时,∀s∈G,有
d(sz1,sz2)<δ02。 (15) 由y∈wG(x, f)知,∃g′∈G,∃m>0,使得
d(g′fm(x),y)<δ12。 (16) 由式(14)和式(16)知,∀n≥0,有
d(fn(tng′fm(x)),fn(hny))<δ02。 由于f是伪等价映射,故∃t′n, h′m∈G,使得
d(t′nfn+m(x),h′nfn(y))<δ02。 (17) 又f是一致连续的,故∀0≤i≤m,∃0 < δ2 < δ1,当d(z1, z2) < δ2时,有
d(fm(z1),fm(z2))<δ12。 (18) 设d(z, x) < δ2,由式(18)知,∀0≤i≤m,有
d(fi(z),fi(x))<δ12。 (19) 再结合式(15)知
d(g′fm(z),g′fm(x))<δ02。 (20) 结合式(16)和式(20)知
d(g′fm(z),y)<d(g′fm(z),g′fm(x))+d(g′fm(x),y)<δ0。 由式(13)知
d(fn(gng′fm(z)),fn(kny))<ε02。 由于f是伪等价映射,故∃g′n, k′n∈G,使得
d(g′nfn+m(z),k′nfn(y))<ε02。 由式(12)知
d(h′ng′nfn+m(z),h′nk′nfn(y))<ε2。 (21) 由式(12)和式(17)知
d(k′nt′nfn+m(x),k′nh′nfn(y))<ε2。 由于G是可交换的,故
d(k′nt′nfn+m(x),h′nk′nfn(y))<ε2。 (22) 由式(21)和式(22)知,当n>0时,有
d(k′nt′nfn+m(x),h′ng′nfn+m(z))<d(k′nt′nfn+m(x),h′nk′nfn(y))+d(h′nk′nfn(y),h′ng′nfn+m(z))<ε。 (23) 当0 < i≤m时,取li=si=e。当i≥m+1时,取li=k′i-mt′i-m, si=h′i-mg′i-m。由式(19)和式(23)知,∀i≥0,有
d(lifi(x),sifi(z))<ε, 则x是G-等度连续点,因此X中的所有点都是G-等度连续点。
假设映射f不是G-等度连续的,则∃δ3>0,∀n∈N+,∃kn≥0,∃x′n, y′n∈X且满足d(x′n, y′n) < 1/n,对∀s, t∈G,有
d(fkn(sx′n),fkn(ty′n))⩾δ3。 (24) 由于X是紧致的,因此,存在列{x′ni}和{y′ni},使得x′ni→x′, y′ni→y′。因为d(x′ni, y′ni) < 1/ni,所以x′=y′。
由于x′是f的G-等度连续点,则对δ3/2>0,∃0 < δ4 < δ3/2,∀n∈N+,∃gn, kn∈G,当d(x′, z) < δ4时,有
d(fn(gnz),fn(knx′))<δ32。 (25) 取m>0,满足d(x′m, x′) < δ4, d(y′m, x′) < δ4,d(x′m, y′m) < 1/m。由式(25), 可得
d(fn(gnx′m),fn(knx′))<δ32, d(fn(gny′m),fn(knx′))<δ32, 则∀n∈N+, 有
d(fn(gnx′m),fn(gny′m))<d(fn(gnx′m),fn(knx′))+d(fn(knx′),fn(gny′m))<δ3, 与式(24)矛盾。因此f是G-等度连续的。
(⇐)由G-等度连续和G-等度连续点的定义立刻可以得到。证毕。
3. 总结
本文首先引入G-链回归点、G-非游荡点集、G-极限点集和G-回归点集的定义,然后在G-等度连续的条件下研究了这些点集之间的关系。结论如下:(1)RG(f)=WG(f)=ΩG(f); (2)WG(f)=CRG(f)=∞⋂n=1fn(X); (3)f是G-等度连续的当且仅当WG(f)中的所有点都是G-等度连续点。这些结论推广了度量空间中链回归点、非游荡点集、极限点集和回归点集的结果,并为其在实际中的应用提供了理论支撑。
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表 1 Co/β-Mo2C的EDS结果
Table 1 The EDS results of Co/β-Mo2C
元素 质量分数/% 原子数百分比/% C 63.30 87.41 O 6.51 6.75 Co 5.73 1.61 Mo 24.46 4.23 -
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