Supplementary Information
1) Small Ion Model
For a weakly charged spherical [1], prolate, or oblate ellipsoidal particle [2,3], the
electrophoretic mobility, nr, is given to a good approximation by [4]
 nr

e Z g ' ( x)
6   Rh (1  x)
(1)
x  B Rh I
B

2 e 2 N Av
 0  r k BT
(2)
 3.286
1
M
1/ 2
nm
(3)
Above, e is the fundamental charge (1.602 x 10-19 C), Z is the net valence charge of the particle,
 is the solvent viscosity, Rh is the hydrodynamic radius of the particle, g’(x) is discussed below,
I is the ionic strength of the BGE, 0 is the permittivity of free space (8.8542 x 10-12 C2/(J m)), r
is the relative permittivity of the solvent (BGE), kB is the Boltzmann constant (1.3806 x 10-23
J/K), and T is absolute temperature. The last term on the right hand side of Eq. (3) is for an
aqueous solvent at 298.15 K ( = 0.89 cp = 0.00089 kg/(m sec), r = 78.54). A factor ignored in
Eq. (1) is the relaxation effect [4-9] which refers to the distortion of the ion atmosphere around
the particle by the application of and external electric or flow field. For weakly charged
particles, ignoring this effect is a good approximation. The “nr” subscript on  emphasizes this
“no relaxation” limit. In Eq. (1), g’(x) is closely related to the “Henry function” that goes to 1 as
x goes to 0 and 3/2 as x goes to infinity [1, 2]. For the comparatively small particles of interest
1
in the present work, we can set g’(x) = 1 to a good approximation. The hydrodynamic radius, Rh,
is related to the translational diffusion constant, DT, by

DT
k BT
6   Rh
( 4)
In the special case of a spherical particle under standard “stick” hydrodynamic boundary
conditions, we can equate Rh to the actual particle radius, a. For prolate and oblate ellipsoids, Rh
can expressed in terms of the major and minor ellipsoidal axes, and for nonspherical particles, it
is useful to think of Rh as an effective particle size [3]. For species j, we shall represent this
hydrodynamic radius as aj. For species j, we can write the “no relaxation” mobility,
 j nr
 j0
 j0 

eZ

6  a j
B2


eB
6 
B2 z j I
1 B aj
 zj
aj
(5)
I
2
z

9 m nm  j

  9.549 x 10
V sec  a j

 31.38 x 10 9
m2
V sec M 1/ 2
(6)
(7)
Eq. (5) follows directly from Eq. (1) setting g’(x) = 1. j 0 represents the mobility of particle j in
the limit of zero ionic strength. The physical significance of the last term on the right hand side
of Eq. (5) is hydrodynamic backflow as counterions stream past particle j and drag solvent with
them. This is called the “electrophoretic effect” [1].
We can correct particle mobility for the “relaxation effect” by writing
j

 j nr (1   j )
(8)
2
Since the ion atmosphere tends to lag behind the direction in which the particle is moving in, this
ion atmosphere distortion produces additional drag on the particle making |j|  |j nr| and hence
the relaxation correction, j, is a positive quantity. This relaxation correction depends on T,
properties of solvent (, r), properties of particle (aj, Z), and properties of the background
electrolyte (ion radii and concentration). The earliest model was formulated by Onsager and
Fuoss [4] and works best for small, weakly charged particles
 OF
j


B1
B1 I S j
(9)
B3
24  N Av
 0.7817
1
M 1/ 2
(10)
The last term on the rhs of Eq. (10) is appropriate for an aqueous solvent at 298.15 K. The Sj
term in Eq. (9) depends on the mobilities (or equivalently the aj values) of the ions of the BGE,
their valence charges, and their concentrations. General expressions for Sj can be found
elsewhere [4, 5] and shall not be reproduced her. It is straightforward to write an Excel
spreadsheet to compute Sj. Using Eqs. (5) and (9) in Eq. (8)
j


 j 0   B2 

B1 S j  j 0  z j I

1 B a I
zj
j

(11)
With the use of Eq. (6), this can be rearranged to yield
1 B a j
I

| zj |
1  B1 S j I
aj | j |


(12)
The right hand side of Eq. (2) shall be called h1(aj, I) and the left hand side shall be called h2(aj,
I), respectively. Within the framework of the “small ion” model, these quantities should be
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equal, but it should be emphasized that this is not true in general. For an assumed value of aj, h2
can be computed for a range of ionic strength. For the same assumed value of aj and a set of j
values measured experimentally over a range of ionic strength, h1 can also be computed. If the
small ion model adequately describes the particle mobility over the ionic strength range, then it
should be possible to find a value of aj that make the h1 and h2 values coincident over the entire
ionic strength range. Consequently, these two sets can be used to determine both aj and also the
suitability of the small ion model in describing the mobility of a peptide.
2) Relaxation Correction for the BMM-NLPP Approach
The relaxation effect is a difficult problem to deal with at a fundamental level due to the
coupling of the differential equations for fluid flow, electrodynamics, and local ion transport [49]. The free solution electrophoresis of “hard” spherical particles with a centrosymmetric charge
distribution has been thoroughly studied [6-9] and the methodology is not difficult to implement
[9]. For rigid particles of arbitrary shape, Boundary Element procedures have been developed
and applied to a variety of problems [3, 10-12], but these procedures are computationally
demanding and not widely available. However, for prolate ellipsoidal particles, the relaxation
correction is very similar to that of a sphere under identical conditions of the same BGE with the
same hydrodynamic radius, Rh, and same “zeta potential”, , averaged of the particle surface
[12]. As in earlier studies [13-18], we shall approximate the relaxation correction of a model
peptide with the corresponding relaxation correction of a sphere with the same Rh and reduced
zeta potential, y = e/kBT, as our model peptide. These quantities, Rh and y, are readily
computed in the BMM-NLPB procedure.
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For a range of x = BRhI1/2 values, y values, and in particular BGEs, the O’Brien and
White procedure [9] is used to calculate the relaxation correction. As in the past, this data is fit
to interpolation polynomials that make it straightforward to compute the relaxation correction for
our model peptides [13-18]. However, in the present work, these interpolation polynomials are
more complex and more accurate than in previous studies. First of all, symmetrical electrolytes,
such as NaCl, are easier to parameterize than asymmetrical electrolytes, such as Na2SO4. We
shall denote symmetrical electrolytes with the parameter IOPEL = 2 and asymmetrical
electrolytes with IOPEL = 1. We write
j
  j nr 1  0.001 y  
 
2
 j nr (1  0.001 y )
( IOPEL  1) 

( IOPEL  2)
(13)
Above,  is the relaxation correction and for the sake of brevity, the j subscript has been
dropped. For symmetrical electrolytes, the leading relaxation correction term varies as y2 but for
asymmetrical electrolytes it varies as y [6]. , in turn, is fit to a polynomial in y
 
JRMX
 A ( x, R ) y
J 1
J
J 1
(14)
h
For IOPEL = 2, JRMX is set to 3 and for IOPEL = 1, JRMX is set to 5. Above, the AJ(x, Rh)
terms depend on x, Rh, and the composition of the BGE, but not y. To remove the dependence
on Rh, the coefficients, in turn, are fit to polynomials of the form
AJ ( x, Rh ) 
3

K 1
KJ
( x) Rh( K 1)
(15)
The reason for the expansion in Eq. (15) deserves some explanation. In the O’Brien and White,
OW, procedure [9], the Brownian motion of the central ion is ignored and the particle size and
5
ionic strength in the resulting differential equations enter as the reduced variable x (Eq. (2)). As
shown elsewhere [5], Brownian motion of the central ion can be incorporated in an approximate
way in the OW procedure by replacing the small mobile ion radius, ai, with
a
eff
i
1
1 

  
 ai Rh 
1
(16)
For several Rh values, (Rh = 0.3, 1.0, and  nm), the OW procedure is applied to a range of xvalues (x1, x2, etc.) with the correction for the mobile ion radii given by Eq. (16). These results
are then fit to Eq. (15). Let xi+1 > x > xi. We can then do a simple interpolation
 x x
 x  xi 
  AJ ( xi 1 Rh )

AJ ( xi , Rh ) i 1
 xi 1  xi 
 xi 1  xi 
 xi 1  x  3
  x  xi  3

   KJ ( xi ) Rh( K 1)   
   KJ ( xi 1 ) Rh( K 1) 
 
  xi 1  xi  K 1

 xi 1  xi  K 1
AJ ( x, Rh ) 
(17)
Once the {KJ(xi)} values are tabulated, they can be used in the BMM-NLPB program. First, Eq.
(17) is used to compute {AJ(x,Rh)} and then Eq. (14) is used to compute . This is then used in
Eq. (13) to calculate the reduced mobility. Tables of {KJ(xi)} for NaCl and Na2SO4 are
available from the author upon request.
6 References
[1] Henry, D.C., Proc. R. Soc. London Ser. A, 1931, 133, 106-129.
[2] Yoon, B.J., Kim, S., J. Colloid Interface Sci., 1989, 128, 275-288.
[3] Allison, S.A., Carbeck, J.P., Chen, C., Burkes, F., J. Phys. Chem. B, 2004, 108, 4516-4524.
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[4] Onsager, L., Fuoss, R.M., J. Phys. Chem., 1932, 36, 2689-2778.
[5] Allison, S.A., Wu, H., Twahir, U., Pei, H., J. Colloid Interface Sci., 2010, 352, 1-10.
[6] Overbeek, J. Th. G., Colloid Beih., 1943, 54, 287-364.
[7] Booth, F., Proc. R. Soc. London Ser. A, 1950, 203, 514-533.
[8] Wiersema, P.H., Loeb, A.L., Overbeek, J. Th. G., J. Colloid Interface Sci., 1966, 22, 78-99.
[9] O’Brien, R.W., White, L.R., J. Chem. Soc. Faraday Trans. 2, 1978, 74, 1607-1626.
[10] Allison, S.A., Macromolecules, 1996, 29, 7391-7401.
[11] Allison, S.A., Biophys. Chem., 2001, 93, 197-213.
[12] Allison, S.A., J. Colloid Interface Sci., 2005, 282, 231-237.
[13] Xin, Y., Hess, R., Ho, N., Allison, S.A., J. Phys. Chem. B 2006, 110, 25033-25044.
[14] Pei, H., Xin, Y., Allison, S.A., J. Sep. Sci. 2008, 31, 555-564.
[15] Pei, H., Allison, S., J. Chromatogr. A 2009, 1216, 1908-1916.
[16] Allison, S.A., Pei, H., Twahir, U., Wu, H., Cottet, H., J. Sep. Sci. 2010, 33, 2430-2438.
[17] Allison, S.A., Perrin, C., Cottet, H., Electrophoresis 2011, 32, 2788-2796.
[18] Wu, H., Allison, S.A., Perrin, C., Cottet, H., J. Sep. Sci. 2011, 35, 556-562.
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