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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).
dab : 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.
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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
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 |
(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
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(2) |
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(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
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(4) |
By using the Einstein-Nernst's equation which
describes the relationship between the diffusion coefficient and mobility of an ion[44]
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(5) |
one can obtain from Eq. 4
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(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) |
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|
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 (for ion i+1) |
(8) |
| |
 (for corresponding ion) |
(9a) |
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(9b) |
and by using the electroneutrality condition[45] for Eq. 9b,
Eq. 9a can be expressed as
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(9c) |
In order to define the width of a boundary, Eq. 10 was
derived by McInnes and Longsworth[3]
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(10) |
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| Table 1. The different
widths of a boundary in MBS(or ITP) formed by strong electrolytes at steady-state* |
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w |
n |
width of a buondary |
Eq. |
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| * 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. |
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 (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]
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(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
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(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
| |
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(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
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(17a-f) |
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|
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(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
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(19a-f) |
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(20a-f) |
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(21a-f) |
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(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). Thirdly, an un-appreciating opinion is
present in the field of MBS and ITP[42], which believe that gradual widening of
a stationary boundary, viz., concentration boundary, in MBS is caused by ionic diffusion.
However, by comparing the ratio of ionic diffusion divided by electromigration in MBS, it
will be revealed that the ratio of the flux of ionic diffusion divided by that of
electromigration in stationary boundary are much smaller than those in a moving boundary,
as will shown in the accompanying report[43]. This indicates that, other reason
such as un-existence of ‘self-regulating effect’, rather than ionic diffusion completely, is responsible to the
gradual widening of stationary boundary.
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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
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