# Direct Measurement of Sub-Debye-Length Attraction between Oppositely Charged Surfaces

**Abstract**

Using a surface force balance with fast video analysis, we have measured directly the attractive forces between oppositely charged solid surfaces (charge densities sigma(+), sigma(-)) across water over the entire range of interaction, in particular, at surface separations D below the Debye screening length lambda(S). At very low salt concentration we find a long-ranged attraction between the surfaces (onset ca. 100 nm), whose variation at D<lambda(S) agrees well with predictions based on solving the Poisson-Boltzmann theory, when due account is taken of the independently-determined surface charge asymmetry (sigma(+) not equal to |sigma(-)|).

Direct Measurement of Sub-Debye-Length Attraction between Oppositely Charged Surfaces

Nir Kampf,

1

Dan Ben-Yaakov,

2

David Andelman,

2

S. A. Safran,

1

and Jacob Klein

1,

*

1

Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot 76100, Israel

2

Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel

(Received 15 May 2009; published 11 September 2009)

Using a surface force balance with fast video analysis, we have measured directly the attractive forces

between oppositely charged solid surfaces (charge densities

þ

,

) across water over the entire range of

interaction, in particular, at surface separations D below the Debye screening length

S

. At very low salt

concentration we ﬁnd a long-ranged attraction between the surfaces (onset ca. 100 nm), whose variation at

D<

S

agrees well with predictions based on solving the Poisson-Boltzmann theory, when due account is

taken of the independently-determined surface charge asymmetry (

þ

j

j).

DOI: 10.1103/PhysRevLett.103.118304 PACS numbers: 82.70.y, 07.10.Pz, 82.45.Mp

The Debye-Hu

¨

ckel, and more generally the Derjaguin-

Landau-Verwey-Overbeek approach, based on lineariza-

tion of the Poisson-Boltzmann (PB) theory, are used clas-

sically to calculate the repulsive interactions between

uniform symmetrically-charged surfaces, and their predic-

tions have been amply veriﬁed [1,2]. In 1972, Parsegian

and Gingell [3] extended this to interactions between un-

equally-charged surfaces in the low surface potential limit

(as may result from strong electrostatic screening at high

salt concentration c

b

). In that limit (where the theory may

be linearized) the range of the interaction scales with the

Debye screening length

S

¼ð""

0

k

B

T=2e

2

c

b

Þ

1=2

; here "

0

is the permittivity of free space, " is the dielectric constant,

k

B

the Boltzmann constant, T the absolute temperature,

and e the electronic charge. For unequally-charged sur-

faces the forces may be attractive or repulsive [3], but in

the case of surfaces whose charge densities are equal and

opposite (i.e., antisymmetric),

þ

¼j

j, the forces are

monotonically attractive as a function of D.ForD>

S

,

such attraction is predicted to decay exponentially with D

[3]. Qualitatively, this attraction may be viewed [4] as due

to the entropy gain arising through release of counterion

pairs from the intersurface gap into the surrounding reser-

voir, as the surfaces approach. Direct measurements [5–9]

of forces between approaching, oppositely charged sur-

faces in the Parsegian-Gingell regime reveal the onset of

measurable attraction at D ð1:5–3Þ

S

, followed by

jumps into contact from these separations due to mechani-

cal instability of the measuring devices [10]; the magnitude

of the attraction during the jumps could not be measured.

Recently the case of oppositely charged surfaces across

water at very low bulk salt concentration c

b

was treated

theoretically [4,11,12], where the high surface potential

c

necessitates solution of the PB equation:

r

2

c

¼ð2ec

b

=""

0

Þ sinhðe

c

=k

B

TÞ: (1)

This low-salt regime is of special interest because the

attraction between the surfaces can be signiﬁcant over a

surface separation range of order 50 nm or more, which is

nonetheless smaller than the screening length

S

(recalling

S

/ c

1=2

b

). In this regime the attraction predicted by the

nonlinear theory [Eq. (1)] scales approximately as (1=D

2

)

[4]. Here we measure directly the forces between two

oppositely charged surfaces as a function of their separa-

tion across water with no added salt. Our results reveal a

long-ranged attraction, which at its onset (at D ¼ ca:

100 nm) is many orders of magnitude larger than van der

Waals (vdW) attraction. Using fast video recording we are

able to characterize the attraction between the surfaces

continuously as they come into contact. We ﬁnd that the

measured forces agree well with the variation given by the

PB theory [Eq. (1)] [12–14] at all surface separations D,

particularly at D<

S

where its predictions diverge from

those of the linearized treatment [3], when proper account

is taken of the (independently-determined) asymmetry in

the surface charge density.

We use a surface force balance (SFB) [15] (see sche-

matic inset to Fig. 1), with high sensitivity in measuring

both normal forces, F

n

ðDÞ, and especially shear forces,

F

S

ðDÞ, between two molecularly smooth, curved mica

surfaces (curvature radius R 1cm) as a function of their

absolute separation D. The high sensitivity, necessary for

detailed examination of the forces, requires a weak force-

measuring spring K

n

, which in turn leads to a spontaneous

jump-in of the surfaces [10] as in previous studies [7,9].

We measure forces during the jump-in via a dynamic

measurement, using fast video recording of the SFB optical

interference fringes which, with frame grabbing and solu-

tion of the optical equations, reveals the variation with time

t of the separation DðtÞ of the surfaces. From DðtÞ the

resulting normal force F

n

ðDÞ between the surfaces is eval-

uated from the instantaneous balance of forces [16], using

Eq. (2) below:

F

n

ðDÞ¼m

d

2

D

dt

2

K

n

ðDðtÞÞþ6R

2

dD

dt

DðtÞ

;

(2)

where m is the total mass of the (moving) lower surface, K

n

PRL 103, 118304 (2009)

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0031-9007=09=103(11)=118304(4) 118304-1 Ó 2009 The American Physical Society

is the normal spring constant (150 N=m), and is the

viscosity of the liquid in the gap ( ¼

water

¼

0:85 10

3

Pa:s). DðtÞ is the deﬂection of the spring,

given by DðtÞ¼D

t¼0

DðtÞþv

app

t, where v

app

is the

inward applied velocity (in the range 10–20 nm= sec , see

also inset to Fig. 1), and is determined from the video

recording at large separations where F

n

ðDÞ0. The ﬁrst

and second terms on the right-hand side of Eq. (2) are the

inertial and spring-bending contributions to the force,

while the third is the Reynolds-Taylor hydrodynamic lu-

brication force between the moving surfaces [17].

Force proﬁles between mica surfaces in pure water

(Milli-Q Gradient A10 puriﬁcation system, water resistiv-

ity 18:2M and total organic content 3–4 ppb) were

measured as previously detailed [16]. Mica loses surface

K

þ

ions in water to become negatively-charged, with

consequent repulsive forces F

n

ðDÞ between two similar

surfaces (the K

þ

ions disperse throughout the volume

and contribute negligibly to the effective salt concentra-

tion). The effective salt concentration is evaluated via the

far-ﬁeld decay length

S

of the forces, while the effective

charge density

on each bare mica surface may be

extracted via a ﬁt to the F

n

ðDÞ proﬁles based on the PB

equation (1) with the condition of constant surface charge.

Following this, one surface was then removed and coated

with chitosan [18], a positively-charged polysaccharide.

The chitosan adsorbs from aqueous solution (followed by

washing to remove excess), as previously described [18], to

form a thin layer which reverses the charge on the mica,

through charge overcompensation, from net negatively

charged to net positively charged, as seen at once from

the resulting forces. Atomic force microscopy (not shown)

was used to characterize the chitosan layer topography,

revealing a uniformly coated surface smooth to 2 0:4nm

rms over a 1 1 m

2

area.

Figure 1 shows normalized force proﬁles F

n

ðDÞ=R be-

tween the bare mica surfaces measured across pure water

in the usual quasistatic way, i.e., via monitoring of the

spring bending during stepwise changes in D [15]. A

long-ranged repulsion due to counterion osmotic pressure

is observed (star symbols), with measurable forces setting

on from D ¼ ca: 200 nm, with a Debye length

S

¼ 68

8nmevaluated from the far-ﬁeld regime (D>

S

). This

value corresponds to an effective salt concentration of ð2

0:5Þ10

5

M 1:1 electrolyte (attributed to dissolved at-

mospheric CO

2

, the extent of which may be evaluated from

the slightly acidic pH of our solutions, pH ¼ 5:8 0:2,

and residual ions leached from the glassware [19]). Using

this value for c

b

we extract the effective bare mica surface

charge density j

j¼e=ð66:5nm

2

Þ through a best ﬁt of

the data using the PB equation [Eq. (1)] with constant

surface charge (augmented by a vdW attraction term [2]

F

vdW

ðDÞ=R ¼ðA=6D

2

Þ, with A ¼ 2 10

20

J), shown

as the solid curve in the upper part of the ﬁgure. As in

earlier studies [16] the surfaces jump under vdW attraction

from a separation 3 0:5nminto adhesive contact. After

coating the upper surface with chitosan to reverse its

charge a monotonic attraction, larger than the scatter in

the data, sets on at D 100 nm, with the surfaces jumping

[10] from D ¼ ca: 60 nm into an adhesive contact at D ¼

2 0:3nm(a measure of the thickness of the compressed

chitosan coating). The dashed curve is the predicted attrac-

tion based on solving the PB equation, Eq. (1) (for c

b

¼

2 10

5

M), for antisymmetric surface charge densities,

i.e.

þ

¼j

j ( ¼ e=ð66:5nm

2

Þ), neglecting any vdW

contribution. We see that indeed the onset of attraction

and its magnitude are reasonably predicted by this ap-

proach, and in particular that the jumps-in to contact

(arrows in Fig. 1) occur roughly where the slope of the

predicted F

n

ðDÞ curve equals K

n

, as expected [10]. We

0100200300

-3000

-2000

-1000

0

1000

2000

σ

= e/66.5 nm

2

(c

b

=2x10

-5

M)

F

n

/R (

µ

N/m)

D (nm)

J

λ

s

D

K

n

K

s

D

K

n

PZT

K

s

v

app

FIG. 1. Normal force proﬁles F

n

ðDÞ=R between curved mica

surfaces (mean radius R) across water with no added salt. w, q:

interactions between bare mica surfaces (from independent ex-

periments, i.e., different pairs of mica sheets). The solid curve is

the prediction based on the PB equation [Eq. (1)] with constant

surface charge density

¼ e=ð66:5nm

2

Þ,a1:1 electrolyte

concentration c

b

¼ 2 10

5

M (derived from the Debye screen-

ing length at D

S

) and a vdW attraction F

vdW

ðDÞ=R ¼

A=6D

2

, with A ¼ 2 10

20

J. Solid black symbols (from

different experiments and contact points): interactions between

a chitosan-coated and a bare mica surface; the arrows indicate

the jump-in (J) at the point of mechanical instability, expected

for ð@F

n

=@DÞK

n

. Broken curve: predicted interaction [4,12]

based on antisymmetric surface charge densities,

þ

¼j

j¼

e=ð66:5nm

2

Þ. Dotted curve: vdW attraction F

vdW

ðDÞ=R ¼

A=6D

2

, with A ¼ 2 10

20

J. (asterisks and diamonds: 2nd

approach of surfaces at 2 different contact points in different

experiments; circles and triangles: 1st and 6th approaches,

respectively, at a given contact point in a different experiment).

The inset shows the schematic of the SFB with the two surfaces

facing each other in a crossed-cylinder conﬁguration; K

n

and K

s

are the constants of the normal and shear springs, respectively,

PZT is the sectored piezocrystal, and v

app

the applied velocity

of the normal spring mount (bottom) in the dynamic mode [see

Eq. (2)].

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118304-2

emphasize that there is negligible, if any, transfer of chi-

tosan between the bare and coated mica surfaces when they

come into contact; this is demonstrated by the fact that 1st,

2nd and 6th approaches (Fig. 1 caption) of the surfaces at

given contact points show very similar attractive proﬁles.

To proceed, we conﬁrm that the onset of the measured

long-ranged attraction is not due to large, positively-

charged polymer loops extending from the adsorbed chi-

tosan layers, which are known to lead to long-ranged

attractive bridging forces [20] (bearing in mind the large

radius of gyration of the chitosan, R

g

100 nm [18]). To

do this we measure the shear forces between the bare mica

and the chitosan-bearing mica. The resulting shear force

traces are shown in Fig. 2, and reveal that there is little

shear force between the surfaces as they approach down to

the jump-in separation. By clamping the normal force

springs, we are also able to suppress the jump-in, and

thus to show that shear forces are within the scatter down

to D<ca: 3 nm. Since shear forces are very sensitive to

any bridging by the polymer chains [21,22], this conﬁrms

that loops from the chitosan layer do not extend beyond a

few nm from the mica surface, and that the long-ranged

attraction is due purely to electrostatic effects.

To obtain the F

n

ðDÞ proﬁle over the entire range of the

interaction—including, in particular, the jump-into-contact

regime—we carried out dynamic measurements using fast

video recording and frame-grabbing analysis described

above [Eq. (2)]. The results are shown in Fig. 3. For the

symmetric (mica-mica) repulsive regime (large open

circles and squares) the dynamic data are within the scatter

of the quasistatic force proﬁles from Fig. 1, and very close

to the predicted variation (dashed top curve, taken from

Fig. 1). For the case of attracting oppositely charged sur-

faces, the dynamic proﬁles all fall within a small range

(diamonds and empty circles show proﬁles at the extremes

of the range, with the striped region including 3 other

dynamic proﬁles, not shown for clarity). The ﬁve attractive

proﬁles summarized in Fig. 3 include interactions both on a

ﬁrst approach as well as on subsequent approaches: We

found no systematic differences between a ﬁrst approach

and second or subsequent approaches at the same contact

point, indicating that little if any transfer of the polyelec-

trolyte was occurring between the surfaces. The range of

0

100 200 300

0.0

0.1

0.2

0.3

5

10

15

20

F

s

( N)

D (nm)

D

J

X

0

X

0

= 30nm

2 sec

F

s

=10 N

J

FIG. 2. Shear force F

s

ðDÞ proﬁles between a chitosan-coated

surface and a bare mica surface sliding past it at separation D,

where X

0

is the applied lateral motion. The cartoon illustrates the

adsorption on the lower mica surface of the positively charged

chitosan, interacting with the negatively charged upper mica

surface. The right inset shows the actual traces, the top one

being the back-and-forth sliding motion of the top surface, and

the lower trace being the shear force transmitted between the

surfaces. When the lower surface is free (ﬁlled circles) there is

little shear force above the noise until the surfaces jump into

contact at J. On clamping the lower surface (squares) to suppress

any jump-in, F

s

is measurable—revealing little shear force

above the noise—right down to D 3nm.

0100200

-3000

-2000

-1000

0

1000

2000

F

n

/R ( N/m)

D (nm)

s

Chitosan

vs. chitosan

Chitosan

vs. mica

Mica vs. mica

0 100 200

100

1000

FIG. 3. Normal force proﬁles F

n

ðDÞ=R between two mica

surfaces (either bare or coated with chitosan) measured dynami-

cally via fast-rate video recording across water with no added

salt. Repulsive proﬁles between bare mica surfaces are shown by

the open circles and squares (from independent experiments, i.e.,

different pairs of mica sheets), and the dashed curve is the

theoretical prediction from Fig. 1. All attractive proﬁles between

a bare and a chitosan-coated mica surface fall in the striped

region shown: diamond and empty circles are data for dynamic

proﬁles at the two extremes of this spread, which include 3

additional dynamic proﬁles (not shown for clarity). The grey

band is the range of measured quasistatic proﬁles from Fig. 1.

Dotted curve: predicted forces for antisymmetric case

þ

¼

j

j¼e=ð66:5nm

2

Þ as in Fig. 1 but augmented by vdW

attraction F

vdW

ðDÞ=R ¼A= 6D

2

, with A ¼ 2 10

20

J.

Solid curve: predicted forces (based on solution of the PB

Eq. (1) with constant charge condition) for the asymmetric

case

þ

¼ e=ð90 nm

2

Þ, j

j¼e=ð66:5nm

2

Þ and c

b

¼

2 10

5

M, augmented by F

vdW

ðDÞ=R as above. Inset:

Quasistatically measured F

n

ðDÞ=R proﬁles between two

chitosan-coated mica surfaces (different symbols are for differ-

ent contact points). The curve is the prediction based on solution

of the PB [Eq. (1)] with constant charge condition and c

b

¼

2 10

5

M and

þ

¼ e=ð90 nm

2

Þ on both surfaces, augmented

by F

vdW

ðDÞ=R as above.

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dynamic proﬁles is also similar to the quasistatic data from

Fig. 1, indicated as a lightly-shaded region in Fig. 3, where

they overlap prior to the jump-in. The dynamic proﬁles

show a rapid increase in attraction at lower D values (D<

ca: 15 nm), indicating the importance of accounting for

vdW attraction. This is implemented in the dotted curve,

which represents the antisymmetric prediction

þ

¼j

j

from Fig. 1 augmented by a vdW attraction term

F

vdW

ðDÞ=R as in Fig. 1. The agreement with the data is

then improved, but a discrepancy remains over the range

10 nm <D<60 nm where the antisymmetric prediction

is outside the range of the measured proﬁles. To account

for this we must consider the possibility that the opposing

charge densities may be asymmetric,

þ

j

j. This is

done by measuring independently, in a separate (quasi-

static) experiment, the normal force proﬁle between two

similarly chitosan-coated surfaces across water, inset to

Fig. 3. Fitting this proﬁle using the PB equation [Eq. (1)]

with constant surface charge, augmented with the same

vdW attraction as above, then yields a surface density

þ

¼ e=½ð90 10Þ nm

2

. This independently determined

value of

þ

for the chitosan-coated surface, together with

for the mica taken at its value determined from the

bare-mica/bare-mica proﬁles, and incorporating vdW at-

traction as above, is then used to calculate the proﬁle

shown as the solid curve in Fig. 3. Agreement of the theory

with experiment over almost all the range of attraction then

becomes much closer. We remark that the crossover from

attractive to repulsive electrostatic interaction expected

theoretically [3] for the asymmetric case

þ

j

j is

predicted to occur in the range D<ca: 10 nm already

dominated by the vdW attraction, and so is not seen in

the calculated curve.

In conclusion, we have measured directly the forces

between oppositely charged surfaces across water as a

function of their surface separation D, at low salt concen-

trations where the Debye screening length

s

70 nm

comprises a substantial part of the range of measurable

attraction (ca. 100 nm). Dynamic surface force balance

measurements, using a rapid video recording technique,

enabled a detailed examination of the attractive forces,

particularly in the regime D<

s

, inaccessible to earlier

studies. Our results, taking due account of the indepen-

dently determined charge asymmetry on the interacting

surfaces, are in close agreement with predictions [11–14]

of the Poisson-Boltzmann theory for interactions in this

regime.

We thank the Israel Science Foundation (grants to J. K.,

to D. A. and to S. A. S.), the Minerva Foundation and the

Schmidt Minerva Center for Supramolecular Architecture

at the Weizmann Institute (J. K.), and the US-Israel

Binational Science Foundation (grants to D. A. and to

S. A. S.) for their support of this work. D. A. thanks the

Weizmann Institute. This research was made possible in

part by the historic generosity of the Harold Perlman

Family.

*Jacob.klein@weizmann.ac.il

[1] S. A. Safran, Statistical Thermodynamics Of Surfaces,

Interfaces And Membranes (Addison-Wesley, New York,

1994).

[2] J. N. Israelachvili, Intermolecular And Surface Forces

(Academic Press Limited, London, 1992).

[3] V. A. Parsegian and D. Gingell, Biophys. J. 12, 1192

(1972).

[4] S. A. Safran, Europhys. Lett. 69, 826 (2005).

[5] K. Besteman et al., Phys. Rev. Lett. 93, 170802 (2004).

[6] K. Besteman, M. A. G. Zevenbergen, and S. G. Lemay,

Phys. Rev. E 72, 061501 (2005).

[7] M. Giesbers, J. M. Kleijn, and M. A. Cohen Stuart,

J. Colloid Interface Sci. 252, 138 (2002).

[8] P. G. Hartley and P. J. Scales, Langmuir 14, 6948 (1998).

[9] K. Lowack and C. A. Helm, Macromolecules 31, 823

(1998).

[10] Sensitivity in measuring forces FðDÞ between surfaces D

apart requires springs of low constant K

n

, leading to Euler-

like instabilities and jumps-in whenever dF=dD > K

n

.

[11] A. A. Meier-Koll, C. C. Fleck, and H. H. von Grunberg,

J. Phys. Condens. Matter 16, 6041 (2004).

[12] D. Ben-Yaakov et al., Europhys. Lett. 79, 48 002 (2007).

[13] S. H. Behrens and M. Borkovec, Phys. Rev. E 60, 7040

(1999).

[14] D. McCormack, S. L. Carnie, and D. Y. C. Chan, J. Colloid

Interface Sci. 169, 177 (1995).

[15] J. Klein and E. Kumacheva, J. Chem. Phys. 108, 6996

(1998).

[16] U. Raviv et al., Langmuir 20, 5322 (2004).

[17] D. Y. C. Chan and R. G. Horn, J. Chem. Phys. 83, 5311

(1985).

[18] The chitosan used and the adsorption procedure are iden-

tical to that described in N. Kampf, U. Raviv, and J. Klein,

Macromolecules 37, 1134 (2004); it has average molecu-

lar mass M ¼ 6 10

5

Da and 85% degree of deacetyla-

tion.

[19] While the level of residual ions is difﬁcult to control, there

is substantial reproducibility between different experi-

ments in water with no added salt (S. Perkin et al.,

Langmuir 22, 6142 (2006)); this is true particularly for a

given set of experiments using mica from the same batch

as in the present study, as seen in different force proﬁles in

Figs. 1 and 3.

[20] J. Klein and P. F. Luckham, Nature (London) 308, 836

(1984).

[21] E. Eiser et al., Phys. Rev. Lett. 82, 5076 (1999).

[22] U. Raviv, R. Tadmor, and J. Klein, J. Phys. Chem. B 105,

8125 (2001).

PRL 103, 118304 (2009)

PHYSICAL REVIEW LETTERS

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11 SEPTEMBER 2009

118304-4

- CitationsCitations21
- ReferencesReferences35

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[Show abstract] [Hide abstract]**ABSTRACT:**We theoretically study the rotation induced on a metal sphere immersed in an electrolyte and subjected to a rotating electric field. The rotation arises from the interaction of the field with the electric charges induced at the metal-electrolyte interface, i.e., the induced electrical double layer (EDL). Particle rotation is due to the torque on the induced dipole, and also from induced-charge electro-osmostic flow (ICEO). The interaction of the electric field with the induced dipole on the system gives rise to counterfield rotation, i.e., the direction opposite to the rotation of the electric field. ICEO generates co-field rotation of the sphere. For thin EDL, ICEO generates negligible rotation. For increasing size of EDL, co-field rotation appears and, in the limit of very thick EDL, it compensates the counter-field rotation induced by the electrical torque. We also report computations of the rotating fluid velocity field around the sphere.- "The initial mobile phase was 0.5 mmol L −1 ; however the fractograms of the " dumbbell " and the fully coated GNRs showed indistinguishable t R . The Debye- Hückel theory establishes that electrical double layer (DL), defined by the Debye screening length (κ −1 ), at low-salt concentration regimes (<1 mmol L −1 ) can be highly significant over a range of 50 nm or more353637. Therefore the concentration of NH 4 NO 3 was decreased to 0.1 mmol L −1 in order to increase the DL on the sample and membrane surface. "

[Show abstract] [Hide abstract]**ABSTRACT:**The development of highly efficient asymmetric-flow field flow fractionation (A4F) methodology for biocompatible PEGylated gold nanorods (GNR) without the need for surfactants in the mobile phase is presented. We report on the potential of A4F for rapid separation by evaluating the efficiency of functionalized surface coverage in terms of fractionation, retention time (t R ) shifts, and population analysis. By optimizing the fractionation conditions, we observed that the mechanism of separation for PEGylated GNRs by A4F is the same as that for CTAB stabilized GNRs (i.e., according to their AR) which confirms that the elution mechanism is not dependent on the surface charge of the analytes and/or the membrane. In addition, we demonstrated that A4F can distinguish different surface coverage populations of PEGylated GNRs. The data established that a change in Mw of the functional group and/or surface orientation can be detected and fractionated by A4F. The findings in this study provide the foundation for a complete separation and physicochemical analysis of GNRs and their surface coatings, which can provide accurate and reproducible characterization critical to advancing biomedical research.- [Show abstract] [Hide abstract]
**ABSTRACT:**Direct force measurements between oppositely charged latex particles in aqueous electrolyte solutions were carried out with a multiparticle colloidal probe technique based on atomic force microscopy. Force profiles between two dissimilarly charged surfaces can be only described when charge regulation effects are taken into account, while constant charge or constant potential boundary conditions are inappropriate. Surface potentials and regulation parameters are determined from force data obtained in symmetric systems with the Poisson-Boltzmann theory and constant regulation approximation. The resulting quantities are used to predict the force profiles in asymmetric systems, and good agreement between theory and experiment is found. These findings show that charge regulation is important to quantify double-layer forces in asymmetric systems.

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