Comparisons Between Ionic Diffusion and Electromigration in Moving Boundary System and Isotachophoresis Forme By Strong Electrolytes at Steady State:I.Theory1 Cao Chengxi1，Li Renzhi2，Xu Hongbin3 (1Dept. of Chemistry, University of Science and Technology of China,230026 Hefei CHINA 2Yijishan Hospital, 3Dept. of Biochemistry, Wannan Medical College, Anhui Wuhu 241001 CHINA) Abstract In this paper, a series of mathematical expressions for the comparisons between the fluxes of ionic diffusion and electromigration in moving boundary system(MBS) and isotachophoresis(ITP) formed by strong electrolytes at steady state are formulated with the aids of Einstein-Nernst's equation describing the relationships between ionic diffusional coefficient and mobility. Those expressions possess following apparent academic significances. Firstly, the expressions supply a theoretical method for the comparisons, at least approximately quantitative ones, between the fluxes of ionic diffusion and electro-migration in MBS and ITP, possibly including MCRB and IEF. Secondly, with the expressions, we can calculate ratio between the fluxes of ionic diffusion and electromigration in a MBS(or an ITP) formed by strong electrolytes at steady state, and show evident data to demonstrate that the flux of ionic diffusion can be omitted in contrast with that of ionic electromigration in MBS and ITP formed by strong electrolytes at steady-state. Thirdly, with the data calculated with the expressions, we can correct the un-appreciating opinion presenting in MBS and ITP, which believes that the gradual widening of a stationary boundary, viz., concentration boundary, in a MBS(or an ITP) is caused by ionic diffusion. Key words Boundary, Ionic diffusion, Electromigration, electrophoresis, isotachophoresis 1 INTRODUCTION The moving boundary system(MBS) is a key important boundary, the theory of MBS is of great importance in the determination of ionic transference number and mobility, in isotachophoresis (ITP), in computer simulation of electrophoresis and in the formation of natural pH gradient for isoelectric focusing(IEF), as will be mentioned below. The researches of MBS began in Kohrausch’ years(1890s)[1,2]. During 1890-1930, the relative theory of MBS and moving boundary method(MBM) were developed and a lot of data of ionic transference numbers and mobilities were monitored with MBM. The MBS theory and MBM, together with a number of data of transference numbers and mobilities of ions, were reviewed by MacInnes and Longsworth in the paper entitled as 'Transference Numbers by the Method of Moving Boundaries'[3]. It is assumed, in this paper, that 'the ionic diffusion that takes place at the boundary has no influence on ionic motion’(see page 176 and 219 in ref. 3). The assumption of omitting ionic diffusion was, partly but not completely, demonstrated by the theoretical derivations(see Eq. 4 in page 225 in ref. 3) given by MacInnes and Longsworth, because the derivations are relied upon another assumption(see page 220 in ref. 3) that says diffusion influences the thickness of the boundary but not its rate of motion. Obviously, there existed little data to show the validity of the assumption, and the comparisons between the ionic diffusion and electromigration in a boundary system were not performed. During 1930-50, the systemic theory of MBS was developed and proved by Longsworth[4-8], Dole[9], Alberty[10,11], Nichol[12] and Svennson[13]. In those studies, evidently, the assumption of omitting ionic diffusion was used by them again and again, but there were still little data to demonstrate the validity of the assumption mentioned above. Later, the ITP, viz., displacement electrophoresis[1,2,14-16], one of the most important electrophoretic techniques, was developed from MBM(for more details, one can see the historical reviews in page 1-9 in ref. 1); some computer simulations of ITP were, according to the theory of MBS, performed by numerous scientists[1,2,16-23]; and the formation of natural pH gradient for isoelectric focusing(IEF)[24,25] was done, which was based upon the mechanism of ITP, viz., MBM. Apparently, the assumption of omitting ionic diffusion was also used by some researchers repeatedly(see the example in ref. 24), whereas there were still little data to prove the validity of the assumption. Recently, the concept and theory of moving chemical reaction boundary(MCRB) were advanced by the authors[26-28] from the ideas of electromigration reaction by Deml and Rigole[29,30] and of stationary neutralization boundary by Pospichal et al.[31,32]. The theory of MCRB has proved by some experiments quantitatively[27,33-37]. In addition, the relationships between MCRB and IEF[38-40] were excellently shown by the authors. In those studies by Deml and Rigole[29,30], Pospichal et al.[31,32] and the authors[26-28,33-41], the assumption of omitting ionic diffusion was also used once more. From above introduction, it is evident that there is still little experimental data to reveal the rationality of the assumption. Thus, the assumption is just an assumption, rather than a fact or conclusion. Owing to this, some scientists doubted the validity of the assumption. Therefore, in the paper and the accompanying report, we try to manifest the validity and rationality of the assumption. In addition, there exists an incorrect opinion in MBS and ITP[42], which believe that the gradual widening of a stationary boundary, or a concentration boundary, in MBS is only caused by ionic diffusion. This un-appreciating opinion should be corrected in this and the accompanying papers. In this paper, we firstly define and deduce some mathematical expressions for the comparisons the fluxes of ionic diffusion and electromigration in a MBS(or ITP) formed by strong electrolytes at steady state, the analyses of experimental data with the expressions will be given in the accompanying report[43]. 2 NOTATIONS J : the flux of an ion(equiv./s). The superscripts, a, b and ab, indicate the ionic flux in phase a, b, and boundary ab, respectively, the subscripts, diff and elec imply the ionic fluxes caused by diffusion and electric field, respectively, and the subscripts, i, i+1 and co, mean the fluxes of ion i, i+1 and the corresponding ion, respectively. c : the equivalent concentration(equiv./l ). the superscripts, a, b and ab, indicate the concentrations of ion in phase a, b, and boundary ab, respectively, and the subscripts, i, i+1 and co, mean the concentrations of ion i, i+1 and the corresponding ion, respectively. m : the mobility(m/Vs). r : the ratio between the fluxes of ionic diffusion in a boundary and of ionic electromigration in a phase. R : the gas constant(= 8.31 J/mol K). F : Faraday constant(= 96500 C/mol). D : the diffusional coefficient(m2/s). d ab : the width of a boundary(m). k : Bolzmann's constant(=1.38×10-23 J/K). T : the absolute temperature(K ). e : the electric charge of an electron(= 1.60×10-19 C). z : the number of electric charge of an ion. : the concentration gradient(equiv./m). : the gradient of electric potential(V/m). E : the electric field strength(V/m). i : the electric current density(A/m2). v : the velocity of a boundary/or a phase(m/s). the superscript, ab, indicates the boundary ab. n : the number(= 4, 5, · · ·, 9. see Table 1). w : the percentage of a given concentration change in a boundary(see Table 1). x : the x-axis coordinates. 3 THEORY Fig. 1a shows a moving boundary formed by ion i in phase a and ion i+1 in phase b, at steady state, Fig. 1b indicates the concentration distributions of ion i, i+1 and the corresponding ion in boundary ab.
| Fig. 1. The diagram of moving boundary(A) and the distribution of concentration of ion in the boundary (B). The symbols,+ and -, indicate the anode and cathode, respectively. For other symbols, see the text.
The total flux of an ion, , can be, according to the Nernst-Planck's formula, expressed as[16,18], if there are no convection, inter-ionic interactions, electro-osmostic flow(EOF) and temperature gradients
|  | (1)
where, the first term in the right side of Eq. 1 is the flux of ionic diffusion, and the second term is the flux of ionic electromigration. In a MBS(or an ITP) at steady state, a phase can be considered as a uniform solution of electrolyte(s), and the concentration gradient of an ion, such as ion i, mainly exists in boundary ab. Thus, the flux of ionic electromigration in a phase, such as ion i in phase a, can be expressed as and the flux of ionic diffusion in boundary ab may be given as
|  | (2)
|  | (3)
Since, our purpose is to compare the fluxes of ionic diffusion and electromigration in MBS(or ITP) at steady state, thus one may define the ratio of the flux of diffusion in a boundary divided by that of electromigration in a phase for ion i as
|  | (4)
By using the Einstein-Nernst's equation which describes the relationship between the diffusion coefficient and mobility of an ion[44]
|  | (5)
one can obtain from Eq. 4
|  | (6)
As have been computed and simulated by numerous scientists[3,16-18,23], the most of concentration changes of an ion in a boundary at steady state can be, approximately, considered as a linear concentration gradient. Thus, the concentration gradients of different ions in boundary ab may be, approximately, expressed as(Note, the width of a boundary should be defined as those in Table 1)
| (for ion i) | (7)
| (for ion i+1) | (8)
| (for corresponding ion) | (9a)
|  | (9b)
and by using the electroneutrality condition[45] for Eq. 9b, Eq. 9a can be expressed as
|  | (9c)
In order to define the width of a boundary, Eq. 10 was derived by McInnes and Longsworth[3]
|  | (10)
Table 1. The different widths of a boundary in MBS(or ITP) formed by strong electrolytes at steady-state*
|  |  | w | n | width of a buondary | Eq.
e2.0 ~ e-2.0 | 12.0-88.0% | 88.0-12.0% | 76% | 4 |  | (11a)
e2.5 ~ e-2.5 | 7.60-92.4% | 92.4-7.60% | 85% | 5 |  | (11b)
e3.0 ~ e-3.0 | 4.70-95.3% | 95.3-4.70% | 91% | 6 |  | (11c)
e3.5 ~ e-3.5 | 2.9-97.1% | 97.1-1.80% | 94% | 7 |  | (11d)
e4.0 ~ e-4.0 | 1.80-98.2% | 98.2-1.80% | 96% | 8 |  | (11e)
e4.5 ~ e-4.5 | 1.1-98.9% | 98.9-1.10% | 98% | 9 |  | (11f)
* We try to use the equation of boundary width(see Eq. 12 here) defined by Thormann-Mosher[16,17], but it seems to me there are print errors in the equation.
| (here d = 9.190 N= 1; d =4.394 N=10) | (12)
According to Eq. 10, we have different widths of a boundary(see Eqs. 11a-f listed in Table 1). Eq. 11a is defined by McInnes and Longsworth[3]. Eqs. 11b-f are defined by the author. Apparently, Eqs. 11b-f are equivalent to Eq. 12 defined by Thormann and Mosher[16,17] as the region where the mole-fraction changes its value from N percent to 100-N percent, since Eqs. 11b-f define the widths of 85 to 98 percent of the concentration change in a boundary and Eq. 12 gives the widths of 80 to 98 percent of the changes in a boundary. Note here, the widths of a boundary(see Eqs. 11a-f), together with Eq. 12, are defined for a MBS(or ITP) formed with strong electrolytes, hence it is unclear whether the definitions, viz., Eqs. 11a-f, have validity for a MBS(or ITP) formed with weak electrolytes. Owing to the gradient of electric potential in a phase is uniform and constant, if the steady state of a MBS(or ITP) is achieved. Hence, we have
| ; | (13a, 13b)
In the steady state of a MBS(or ITP), the boundary velocity is constant and can be expressed as[2]
|  | (14a-b)
Thus, after the insertion of Eqs. 13a, 7 and 14a into Eq. 6, we obtain the ratio of flux of diffusion divided by that of electromigration for ion i in MBS(or ITP) at steady state
|  | (15a-f)
where, if n = 4, w =0.76; n = 5, w = 0.85; ···(see Table 1). Similarly, with aid of Eqs. 13b and 14b, one obtains the ratio for ion i+1
|  | (16a-f)
where, if n = 4, w =0.76; n = 5, w = 0.85; ···(see Table 1). And with the aid of Eq. 9c, one also has the ratio(see Eqs. 17a-f) for the corresponding ion in boundary ab and phase a, and the ratio(see Eqs. 18a-f) for the ion in boundary ab and phase b
|  | (17a-f)
|  | (18a-f)
where, if n = 4, w =0.76; n = 5, w = 0.85; ···(see Table 1). Eqs. 15a-18f are used to compute the ratios of the flux of ionic diffusion divided by that of ionic electromigration in a moving boundary(or ITP). However, for a stationary, usually concentration, boundary(see Fig. 2), we should use Eqs. 19a-22f to calculate the ratios
|  | (19a-f)
|  | (20a-f)
|  | (21a-f)
|  | (22a-f)
where, if n = 4, w =0.76; n = 5, w = 0.85; ···(see Table 1). Those mathematical expressions of Eqn. 15a-22f, joined with Table 1, are for the comparisons of the ionic diffusion and electromigration in MBS and ITP formed with strong electrolytes at steady-state. 4 DISCUSSIONS AND CONCLUSIONS Both ionic electromigration and diffusion are important physical phenomena in the fields of boundary chemistry and electrophoresis. To authors' knowledge, the relationships between ionic electromigration and diffusion, except for Einstein-Nernst's equation(see Eqn. 5 here), have not been given yet and the comparisons between them in the fields of boundary and electrophoresis have not been performed up to now. The theoretical links between ionic electromigration and diffusion in MBS or ITP were shown and a series of mathematical expressions(see Eqn. 15a-22f and Table 1) were derived by using the Einstein-Nernst's equation that describes the affinity between ionic mobility and diffusional coefficient. Those expressions at least give the approximate quantitative comparisons for the fluxes of ionic electromigration and diffusion in MBS and ITP formed by strong electrolytes at steady-state. Those expressions possess following significances. Firstly, as discussed just above, the expressions supply a theoretical method for the comparisons, at least approximately quantitative ones, between the fluxes of ionic diffusion and electromigration in MBS and ITP, possibly including MCRB and IEF[26-28,34-41]. Secondly, the assumption of omitting ionic diffusion in contrast to ionic electromigration was widely used in MBS, ITP, MCRB and IEF as discussed in the Introduction here, while there are little data to verify the rationality and validity of the assumption, so the assumption is just an assumption. It is, in academy, necessary to give documentations to prove this assumption. With the expressions of Eqs. 15a-22f, we can at firstly calculate the ratio of flux of ionic diffusion divided by that electromigration in a MBS(or an ITP) formed by strong electrolytes at steady state, and show evident data to demonstrate that the flux of ionic diffusion can be omitted in contrast with that of ionic electromigration in MBS and ITP, as will be discussed in the accompanying report[43], and it is found by us the conclusion relied on the expressions is coincident with that based on other principle (not be shown here). 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[45] Hjerten S, Liao J-L, Yao K-Q, theoretical and experimental study of high performance electrophoretic mobilization of isoelectric focused protein zones, J.Chromatogr.,1987,387:127. 1 This project was supported mainly by the National Natural Scientific Foundation of China(No.29775014) and by the Health Committee of China(No.98-2-334), and partly by the Education Committee of Anhui Province(No.97JL154) and Wannan Medical College. 2 Correspondence to Cheng-Xi Cao(36 years old, Associated Professor in Toxicolagy & Toxicological Analysis), Tel:0551-3648377, Fax:0551-3631760, e-mail:cxcao@mail.ustc,edu.cn Received 3 Feb.1999，Revised 14 May.1999