Comparisons Between Ionic Diffusion and Electromigration in Moving Boundary System and Isotachophoresis Formed by Strong Electrolytes at Steady State: II. Computer-aided Analyses of Experimental Data 1 Li Renzhi1, Xu Hongbin2, Cao Chengxi3** (1Yijishan Hospital, 2Dept. of Biochemistry, Wannan Medical College, Anhui Wuhu 241001 3Dept. of Chemistry, University of Science and Technology of China, Hefei 230026) Abstract In this paper, with the computer-aided analyses and the mathematical expressions defined in the accompanying report, the authors compared the fluxes of ionic diffusion and electromigration in moving boundary system(MBS) and isotachophoresis(ITP) formed by strong electrolytes at steady state.Firstly, the results show that the ratios of the flux of ionic diffusion divided by that of ionic electromigration are, in general, less than 10 percent in a moving boundary of the steady state of MBS(or ITP) formed by different strong electrolytes, and the ratios in a stationary/or concentration boundary at steady state are all less than 1 percent. Thus, the flux of ionic diffusion is small and can be approximately omitted in contrast to that of ionic electromigration in the steady state of MBS(or ITP). Hence this results, coupled with the experiments and theoretical derivations by McInnes-Longsworth-Cowperthwaite about 60 years ago, show the validity of the assumption -- the omission of ionic diffusion in MBS or ITP. Secondly, the ratios(in the range from 0.01% to 1%) for a stationary boundary are much smaller than those(generally in the range from 1% to 10%) for a moving boundary, this results imply that the gradual widening of a stationary boundary is caused not only by ionic diffusion, but also by other reason like the un-existence of "self-regulating effect" in a stationary boundary . Thirdly, the results possess significance for biomedicine, such drug release by diffusion or electric field. Key words Boundary,Ionic diffusion,Electromigration,Electrophoresis,isotachophoresis 1 INTRODUCTION In the accompanying report[1], with the aids of Einstein-Nernst's equation describing the link of ionic mobility and diffusion, a series of mathematical expressions were defined and formulated for the comparisons between the fluxes of ionic diffusion in a boundary and of electromigration in a phase. As discussed in the accompanying paper, the assumption of omitting the flux of ionic diffusion as compared with that of ionic electromigration had been widely used in the fields of moving boundary system(MBS) and isotachophoresis(ITP) and of moving chemical reaction boundary(MCRB) and isoelectric focusing (IEF) again and again until recent time, however there are still little data to prove the rationality of the assumption, thus the assumption is just an assumption rather than a conclusion or fact. In addition, there exist the apparent un-appreciating opinion in MBS and ITP, which says the gradual widening of a stationary boundary, or concentration boundary, is caused only by ionic diffusion[2], this un-appreciating opinion should be corrected. Therefore, the main purposes of this paper are to compare the fluxes of ionic diffusion and electromigration in a MBS(or ITP) formed by strong electrolytes at steady state with the computer-aided analyses and the expressions formulated in ref. 1, and to find some data to show the validity of the assumption of omitting the flux of ionic diffusion as compared with that of ionic electromigration, additionally to correct the un-appreciating opinion in MBS and ITP mentioned above. 2 EXPERIMENTAL DATA AND TREATMENTS The experiments and related data of MBS and ITP are originated from refs. 3, 4, 5 and 6, for more details, one can see example 1-4 here or refs. 3,4, 5 and 6. Due to the existence of enough high ionic strength in those examples, thus the ionic absolute mobility should be corrected by using the empirical formula of Eq. 1, which is valid for both large ions with low electric charge intensity and small ions with high electric charge intensity as have been verified by Friedl, Reijenga and Kenndler[7,8] and the authors[9,10]
| (h=0.5, if z=1; h=0.78, if z2. used for I0.1mol/l) | (1)
where, m0,i is the absolute mobility of ion i, mact,i the actual mobility of ion i, z the number of ionic charge(s) and I the ionic strength given as
|  | (2)
In the next step, we will use Eqs. 15a-22f and Table 1 in ref. 1, together with Eqs. 1 and 2 here, to calculate the ratios of the flux of ionic diffusion divided by that of ionic electromigration in a MBS(or an ITP) formed by strong electrolytes at steady state. In order to calculate the values of Eqs. 14a-21g of ref. 1 for different boundaries formed by strong electrolytes (see those Examples in Section 3), a computer(486/DX-80, 8 Mbyte RAM, 540Mbyte hard disk, Copam Electronics Co., Taiwan, CHINA) is used and a program is written in QBASIC(Version 4.5, Microsoft, Redmond, WA, USA), additionally, the calculated data with the program written in QBASIC are compared with those with the Microsoft EXCEL 5.0. It was found that those data computed with the two methods are all coincidence with each others 3 RESULTS Example 1 The MBS formed by 0.1N KCl and LiCl initially is studied by Longsworth[3,4]. In the MBS at steady state, there exist two boundaries, one being a moving boundary formed by 0.1 N KCl and 0.065 N LiCl and another being stationary boundary formed by 0.065 N and 0.1 N LiCl(see Fig. 1A-1B). For more details, one can see ref. 3 and 4).
Fig. 1. The initial boundary formed by KCl and LiCl (A) and the moving and stationary boundaries at steady-state(B). The symbols, + and -, imply the anode and cathode, respectively. The stationary boundary bg in (B) is a concentration boundary, the follows are the same as the boundary. For other symbols, see the text. |  | Fig. 2. The initial boundary formed by KCl and KIO3(A) and the moving and stationary boundaries at steady state(B). For other symbols, see the text.
Table 1a. The ratios of the flux of ionic diffusion divided by that of ionic electromigration in the moving boundary formed by 0.1 N KCl(phase a) and 0.065 N LiCl(phase b) at steady state[3,4].
Ion(phase) | m0 (m2/Vs) | mact (m2/Vs) | c(N) | r(0.76)a) | r(0.85)a) | r(0.91)a) | r(0.94)a) | r(0.96)a) | r(0.98)a)
K+ (a) | 7.62×10-8 b) | 6.51×10-8 | 0.1 | 0.160 | 0.143 | 0.128 | 0.113 | 0.101 | 0.092
Li+(b) | 4.01×10-8 b) | 3.53×10-8 | 0.065 | 0.087 | 0.078 | 0.069 | 0.061 | 0.055 | 0.050
Cl-(a) | --- | --- | --- | 0.056 | 0.050 | 0.045 | 0.040 | 0.035 | 0.032
Cl-(b) | --- | --- | --- | 0.047 | 0.042 | 0.037 | 0.033 | 0.030 | 0.027
a) The values of r(w) are corespondent to w = 0.76, or 0.85, 0.91, 0.94, ··· and 0.98 in Table 1 in ref. 1. b) The data of m0 are quoted from ref. 6.
Table 1b.The ratios of the flux of ionic diffusion divided by that of ionic electromigration in the stationary boundary formed by 0.065 N LiCl(phase b) and 0.1 N LiCl(phase g) at steady state [3,4].
Ion(phase) | m0 (m2/Vs) | mact (m2/Vs) | c(N) | r(0.76)a) | r(0.85)a) | r(0.91)a) | r(0.94)a) | r(0.96)a) | r(0.98)a)
K+ (b) | 4.01×10-8 b) | 3.53×10-8 | 0.1 | 0.006 | 0.005 | 0.005 | 0.004 | 0.004 | 0.004
Li+(g) | 4.01×10-8 b) | 3.42×10-8 | 0.065 | 0.006 | 0.005 | 0.005 | 0.004 | 0.004 | 0.003
Cl-(b) | --- | --- | --- | 0.003 | 0.003 | 0.003 | 0.002 | 0.002 | 0.002
Cl-(g) | --- | --- | --- | 0.002 | 0.002 | 0.002 | 0.001 | 0.001 | 0.001
a) The values of r(w) are corespondent to w = 0.76, or 0.85, 0.91, 0.94, ··· and 0.98 in Table 1 in ref. 1. b) The data of m0 are quoted from ref. 6.
By using Eqs. 15a-18f in ref. 1, we compute the ratios of the fluxes of diffusion divided by those of electromigration of potassium, lithium and chloride ions in the moving boundary at steady state, the results are listed in Table 1a. With the aid of Eq. 19a-22f in ref. 1, we calculate the ratios for the ions of lithium and chloride ions in the stationary (or concentration) boundary, the analytical data of the calculation are collected in Table 1b. Example 2 Fig. 2A shows the MBS formed by 0.1 N KCl and KIO3 initially[3,4]. After the steady-state is achieved, there are also a moving boundary formed by 0.1 N KCl and 0.06 N KIO3 and a stationary(or concen-tration) boundary formed by 0.06 and 0.1 N KIO3 as given in Fig. 2B. Similarly, the results of ratios for the moving boundary are shown in Table 2a, and those for the stationary(or concentration) boundary are given in Table 2b.
Table 2a. The ratios between the fluxes of ionic diffusion and electromigration in the moving boundary formed by 0.1 N KCl(phase a) and 0.06 N KIO3(phase b) at steady state[3,4].
Ion(phase) | m0 (m2/Vs) | mact (m2/Vs) | c(N) | r(0.76)a) | r(0.85)a) | r(0.91)a) | r(0.94)a) | r(0.96)a) | r(0.98)a)
Cl- (a) | 7.91×10-8 b) | 6.75×10-8 | 0.1 | 0.151 | 0.135 | 0.121 | 0.107 | 0.095 | 0.087
IO3-(b) | 4.25×10-8 b) | 3.76×10-8 | 0.006 | 0.084 | 0.075 | 0.067 | 0.059 | 0.053 | 0.048
K+(a) | --- | --- | --- | 0.060 | 0.054 | 0.048 | 0.043 | 0.038 | 0.035
K+(b) | --- | --- | --- | 0.056 | 0.050 | 0.045 | 0.040 | 0.035 | 0.032
a) The values of r(w) are corespondent to w = 0.76, or 0.85, 0.91, 0.94, ··· and 0.98 in Table 1 in ref. 1. b) The data of m0 are quoted from ref. 6.
Table 2b. The ratios between the fluxes of ionic diffusion and electromigration in the stationary boundary formed by 0.06 N KIO3(phase b) and 0.1 N KIO3 (phase g) at steady state[2,4].
Ion(phase) | m0 (m2/Vs) | mact (m2/Vs) | c(N) | r(0.76)a) | r(0.85)a) | r(0.91)a) | r(0.94)a) | r(0.96)a) | r(0.98)a)
IO3-(b) | 4.25×10-8 b) | 3.75×10-8 | 0.06 | 0.007 | 0.006 | 0.005 | 0.005 | 0.004 | 0.004
IO3-(g) | 4.25×10-8 b) | 3.63×10-8 | 0.1 | 0.006 | 0.006 | 0.005 | 0.004 | 0.004 | 0.004
Cl-(b) | --- | --- | --- | 0.004 | 0.004 | 0.004 | 0.003 | 0.003 | 0.003
Cl-(g) | --- | --- | --- | 0.003 | 0.002 | 0.002 | 0.002 | 0.002 | 0.001
a) The values of r(w) are corespondent to w = 0.76, or 0.85, 0.91, 0.94, ··· and 0.98 in Table 1 in ref. 1. b) The data of m0 are quoted from ref. 6.
Example 3 See Figs. 3A-B. The MBS formed by 0.0025 N KCl and NaCl initially was used to study the computer simulations of ITP[5]. As have been simulated by Radi and Schumacher[5], at steady state, the moving boundary is formed by 0.0025 N KCl and 0.002 N NaCl and the stationary is formed by 0.002 and 0.0025 N NaCl. The results for both the moving and the stationary boundaries, which are respectively computed with Eqs. 15a-18f and Eqs. 19a-22f in ref. 1, are collected in Table 3a and 3b, respectively.
Fig. 3. The initial boundary formed by 0.0025 N KCl and NaCl(A), and the moving/stationary boundaries at steady-state(B). |  | Fig. 4. The initial boundary formed by 0.0025 N KBrO3 and KIO3(A), and the moving/stationary boundaries at steady-state(B).
Table 3a. The ratios between the fluxes of ionic diffusion and electromigration in the moving boundary(or ITP) formed by 0.0025 N KCl(phase a) and 0.002 N NaCl(phase b) at steady state[5]
Ion(phase) | m0 (m2/Vs) | mact (m2/Vs) | c(N) | r(0.76)a) | r(0.85)a) | r(0.91)a) | r(0.94)a) | r(0.96)a) | r(0.98)a)
K+(a) | 7.62×10-8 b) | 7.43×10-8 | 0.0025 | 0.088 | 0.079 | 0.071 | 0.062 | 0.056 | 0.051
Na+(b) | 5.18×10-8 b) | 5.07×10-8 | 0.0020 | 0.060 | 0.054 | 0.048 | 0.041 | 0.038 | 0.035
Cl-(a) | --- | --- | --- | 0.018 | 0.016 | 0.014 | 0.012 | 0.011 | 0.010
Cl-(b) | --- | --- | --- | 0.015 | 0.014 | 0.012 | 0.011 | 0.010 | 0.009
a) The values of r(w) are corespondent to w = 0.76, or 0.85, 0.91, 0.94, ··· and 0.98 in Table 1 in ref. 1. b) The data of m0 are quoted from ref. 5.
Table 3b. The ratios between the fluxes of ionic diffusion and electromigration in the stationary boundary (or ITP) formed by 0.002 N NaCl(phase b) and 0.0025 N NaCl(phase g) at steady state[5]
Ion(phase) | m0 (m2/Vs) | mact (m2/Vs) | c(N) | r(0.76)a) | r(0.85)a) | r(0.91)a) | r(0.94)a) | r(0.96)a) | r(0.98)a)
Na+(b) | 5.18×10-8 b) | 5.07×10-8 | 0.0020 | .0007 | .0006 | .0006 | .0005 | .0005 | .0004
Na+(g) | 5.18×10-8 b) | 5.05×10-8 | 0.0025 | .0007 | .0006 | .0006 | .0005 | .0004 | .0004
Cl-(b) | --- | --- | --- | .0001 | .0001 | .0001 | .0001 | .0001 | .0001
Cl-(g) | --- | --- | --- | .0002 | .0002 | .0001 | .0001 | .0001 | .0001
a) The values of r(w) are corespondent to w = 0.76, or 0.85, 0.91, 0.94, ··· and 0.98 in Table 1 in ref. 1. b) The data of m0 are quoted from ref. 5.
Example 4 The MBS created by 0.0025 N KBrO3 and KIO3 initially, as studied by Radi and Schumacher[5], will become a moving and a stationary boundary. As done by us in Example 1-3, the data of the calculations for the moving and stationary boundaries are, respectively, given in Table 4a and 4b.
Table 4a. The ratios between the fluxes of ionic diffusion and electromigration in the moving boundary(or ITP) formed by 0.0025 N KBrO3(phase a) and 0.0021 N KIO3(phase b) at steady state[5]
Ion(phase) | m0 (m2/Vs) | mact (m2/Vs) | c(N) | r(0.76)a) | r(0.85)a) | r(0.91)a) | r(0.94)a) | r(0.96)a) | r(0.98)a)
BrO3+(a) | 5.76×10-8 b) | 5.62×10-8 | 0.0025 | 0.067 | 0.060 | 0.053 | 0.048 | 0.043 | 0.039
IO3-(b) | 4.25×10-8 b) | 4.15×10-8 | 0.0021 | 0.050 | 0.045 | 0.040 | 0.035 | 0.031 | 0.029
K+(a) | --- | --- | --- | 0.011 | 0.010 | 0.009 | 0.008 | 0.007 | 0.006
K+(b) | --- | --- | --- | 0.009 | 0.009 | 0.008 | 0.007 | 0.006 | 0.005
a) The values of r(w) are corespondent to w = 0.76, or 0.85, 0.91, 0.94, ··· and 0.98 in Table 1 in ref. 1. b) The data of m0 are quoted from ref. 5.
Table 4b. The ratios between the fluxes of ionic diffusion and electromigration in the stationary boundary (or ITP) formed by 0.0021 N KIO3(phase b) and 0.0025 N KIO3(phase g) at steady state[5].
Ion(phase) | m0 (m2/Vs) | mact (m2/Vs) | c(N) | r(0.76)a) | r(0.85)a) | r(0.91)a) | r(0.94)a) | r(0.96)a) | r(0.98)a)
IO3-(b) | 4.25×10-8 b) | 4.16×10-8 | 0.0021 | .0004 | .0004 | .0004 | .0003 | .0003 | .0003
IO3-(g) | 4.25×10-8 b) | 4.15×10-8 | 0.0025 | .0004 | .0004 | .0004 | .0003 | .0003 | .0003
K+(b) | --- | --- | --- | .0001 | .0001 | .0001 | .0001 | .0001 | .0001
K+(g) | --- | --- | --- | .0001 | .0001 | .0001 | .0001 | .0001 | .0001
a) The values of r(w) are corespondent to w = 0.76, or 0.85, 0.91, 0.94, ··· and 0.98 in Table 1 in ref. 1. b) The data of m0 are quoted from ref. 5.
4 DISCUSSIONS It is evident from Table 1a-4a that, in those moving boundaries, the highest values of the ratios of the fluxes of ionic diffusion divided by those of ionic electromigration are those for the leading ions, the lowest values of the ratios are those for the corresponding ions, and the values of the ratios for the following ions moving just next the leading ions are between the highest and lowest values of the ratios. The similar results can also be found from Table 1b-4b. It is also clear from Table1a-4a that the ratios of the fluxes of ionic diffusion in the moving boundaries divided by those of ionic electromigration in phases are, generally, from 1 percent to 10 percent, this indicates, to some extent, that the flux of ionic diffusion in a moving boundary is small and can be approximately omitted as compared with that of ionic electromigration in a phase. Table 1b-4b show, very obviously, that the values of ratios of the fluxes of ionic diffusion divided by that of ionic electromigration in the stationary, viz., concentration, boundaries are all less than 1 percent, and in some cases less than 0.1 percent and even less than 0.01 percent. Those data show, very evidently, that the flux of ionic diffusion in a stationary boundary is very small and can be overlooked in contrast to that of ionic electromigration in a phase, completely. Compare the ratios in Table 1b-4b and those in Table 1a-4a respectively, the ratios for a concentration boundary are very low in contrast to those for a moving boundary. Thus, according to the opinion which indicates the gradual widening of a stationary boundary in an ITP is caused by ionic diffusion[2], one can conclude the width of a moving boundary should be larger than that of a stationary boundary due to the higher values of the ratios for a moving boundary in contrast to very low values of ratios for a stationary boundary as shown in Table 1a-4a and 1b-4b. In fact, the width of a moving boundary is much smaller than that of a stationary boundary, this fact is obviously against the conclusion -- a wide moving boundary but a thin stationary boundary existing in MBS and ITP -- dragged from the opinion[2]. Clearly, there must be other reason that is responsible to the gradual widening of a stationary boundary in MBS and ITP. The other reason is the "self-correct effect" or "Restoring effect" that exists in a moving boundary, while is not present in a stationary boundary or concentration boundary in MBS and ITP, as will be discussed below. The reasons why the ratio for a stationary boundary are such low are: (1) the difference of concentrations, , in a stationary boundary is low as compared with that in a moving boundary; (2) the width of a stationary boundary is, evidently, larger than that of a moving boundary, this has been proved by the experiments performed by Longsworth(see Fig. 1, 4, 5 and 8 in ref. 11); (3) the difference of ionic mobilities, , in a stationary boundary is, generally, smaller than those in a moving boundary. The reasons why the values of ratios for corresponding ions in both moving and stationary boundary are the lowest as compared with those for other two ions/or another ion in two phases are: the difference of concentrations of corresponding ions, , is the smallest as compared with those for other two ions /or another ion in two phases. As calculated by McInnes and Longsworth[3], the accuracy of the method is within 0.02 percent. Thus, the MBM used to determined the transference number and mobility of ion is a very precise method. The first reason why the method is of such high accuracy is mainly due to the 'Restoring effect' called by McInnes and Longsworth[3], or the "Self-correcting effect"[12], or the "Self-sharpening effect"[5], which is excellently proved by the experiments performed by McInnes and Cowperthwaite[3,4], and the mechanism of which has been described(for more details, one can refer refs. 3, 4 and 5). The second is, possibly, that the diffusion influences the thickness of boundary but not the ionic transference number, as has been assumed by McInnes-Longsworth[3]. The third, as shown in this paper, is the low flux of ionic diffusion in contrast to high flux of ionic electromigration in a MBS at steady state. The results in this paper also possess significance to biomedicine, such as drug release. Drug release is originally caused by diffusion[13,14], but now can be controlled by physical method like electric field[13]. Ionic diffusion is difficult to controll, but ionic electromigration is easily controllable. This paper implies that ionic diffusional flux is very small in contrast to ionic electromigration flux. Thus, if a drug can be ionized, we can efficiently control its release by using some physical methods like electromigration of ionic drug, rather than drug diffusion. 5 CONCLUSIONS It is clear from the discussions above that, in the steady state of a moving boundary formed by different strong electrolytes, the values of ratios of fluxes the ionic diffusion divided by those of ionic electromigration are generally less than 10 percent, and in the steady state of a stationary, viz., concentration, boundary, the values of the ratios are all less than 1 per cent. Thus, the results directly demonstrate that the flux of ionic diffusion are small and can be approximately omitted as compared with that of ionic electromigration. Therefore, this paper shows why we can ignore the ionic diffusion in the studies of ionic electromigration in a MBS and ITP formed by strong electrolytes at steady state. In addition, this paper also reveals that the ionic diffusion in a stationary boundary is much weak as compared with that in a moving boundary. And the wide stationary boundary in contrast to the thin moving boundary in MBS is not only caused by ionic diffusion, but also by the un-existence of "self-correcting effect". 6 REFERENCES [1] Cao C-X, Li R-Z, Xu H-B, comparisons between ionic diffusion and electromigration in moving boundary system and isotachophoresis formed by strong electrolytes at steady-state: I. theory, Chem. Online, http://www.chemistrymag.org/col/1999.c99106.htm. [2] Thormann W, Mosher R A, Bier M, computer-aided analysis of the electrophoretic regulating function omega, Electrophoresis 1985, 6:78. [3] McInnes D A, Longsworth L G, transference numbers by the method of moving boundaries: theory, practice and results,Chem. Rev., 1932, 11: 171. [4] Longsworth L G, an application of moving boundary to a study of aqueous mixtures of hydrogen chloride and potassium chloride, J. Am. Chem. Soc., 1930, 52: 1897. [5] Radi P, Sechumacher E, numerical simulation of electrophoresis: the complete solution for three isotachophoretic systems, Electrophoresis, 1985, 6:195. [6] Moore W J, Physical Chemistry, 5th Ed, London, Longman Group Limited, 1976. [7] Reijenga J C, Kenndler E, computational simulation of migration and dispersion in free capillary zone electrophoresis: I. description of the theoretical model. J. Chromatogr. A 1994, 659:403. [8] Friedl M, Reijenga J C, Kenndler E, Ionic strength and charge number correction for mobilities of multivalent organic anions in capillary electrophoresis. J. Chromatogr. A 1995, 709:163. [9] Cao C-X, proton mobilities obtained by moving boundary method and an empirical equation in capillary electrophoresis. J. High. Resol. Chromatogr. 1997, 20(12):701. [10] Cao C-X, comparisons between the mobilities of small salt ions obtained by moving boundary method and two empirical equations in capillary electrophoresis. J. Chromatogr. A 1997, 771:374. [11] Longsworth L G, moving boundary studies on salt mixtures, J. Am. Chem. Soc., 1945, 67: 1109. [12] Everaerts F M, Beckers Jo L, Verheggen E M, Isotachophoresis, Amsterdam, Elsevier Sci. Pub. Co., 1976. [13] Neumann E., fundamentals of electroporative delivery of drugs and genes, Bioelectrochem Bioenerg., 1999,48(1):3. [14] Frohoff-Hulsmann M. A., aqueous ethyl celluose dispersions containing plasticizers of different water solubility and hydroxypropyl methylcellulose as coating material for diffusion pellets. I. Drug release rates from coated pellets, Int. J. Pharm., 1999, 177(1):69. [1] The object was supported mainly by the National Natural Scientific Foundation of China(No. 29775014) and by the Health Committee of China(98-2-334), and partly by the Education Committee of Anhui Province(No. 97JL154) and Wannan Medical College. * Correspondence to:Cheng-Xi Cao, Tel: 551-3601589(O),3648377(H) , Fax: 551-3631760, e-mail: cxcao@mail.ustc.edu.cn Received 3 Feb.1999，Revised 14 May.1999