17 Electrostatic Repulsion BetweenTwo Parallel Soft Plates AfterTheir Contact

17.1 INTRODUCTION

We have so far considered the interaction between soft particles for particle separa-

tions before their surface charge layers come into contact. In this section, we con-

sider the interaction after the surface charge layers come into contact. We consider

the interaction forces or energies between colloidal particles covered with graft

polymers (i.e., polymer brush layers). When these particles approach each other,

steric repulsion is acting between the surface charge layers on the respective parti-

cles. A theory of de Gennes [1] on the interaction between particles coated by

uncharged polymer brush layers assumes that when the two brushes come into con-

tact, they are squeezed against each other but they do not interdigitate. If brushes

are charged, then electrostatic repulsion between brushes must also be taken into

account in addition to the steric repulsion. In this section, we extend the theory of

de Gennes [1] to the case of charged polymer brushes and obtain an expression for

the electrostatic repulsion between the surface charge layers of the plates after they

come into contact. We restrict ourselves to the case where the brush layers are

squeezed against each other but they do not interdigitate. We will thus not consider

electrostatic interactions for the case where the polymer layers on interacting parti-

cles penetrate each other.

17.2 REPULSION BETWEEN INTACT BRUSHES

Consider two parallel identical plates coated with a charged polymer brush layer of

intact thickness do at separation h immersed in a symmetrical electrolyte solution of

valence z and bulk concentration n as shown in Fig. 17.1a [2]. We assume that dis-

sociated groups of valence Z are uniformly distributed in the intact brush layer at a

number density of No. We first obtain the potential distribution in the system when

the two brushes are not in contact (h> 2do). We take an x-axis perpendicular tothe brushes with its origin 0 at the core surface of the left plate so that the region

Biophysical Chemistry of Biointerfaces By Hiroyuki OhshimaCopyright# 2010 by John Wiley & Sons, Inc.

381

do< x< h� do is the electrolyte solution and the regions 0< x< do and h� do< x< h are the brush layers. We also assume that the brush layers are penetrable to

electrolyte ions as well as water molecules. As a result of the symmetry of the sys-

tem we need consider only the regions do< x< h/2 and 0< x< do.The electrostatic interaction force Pe(h) per unit area between the two charged

brush layers at separation h when they are separated (h� 2do) is given by the

osmotic pressure at the midpoint between the plates x¼ h/2 minus that in the bulk

solution phase, namely,

Pe(h) ¼ 4nkT sinh2[y(h=2)=2]; h � 2do ð17:1Þ

where y(h/2) can be obtained by solving Eqs. (32) and (33).

17.3 REPULSION BETWEEN COMPRESSED BRUSHES

Next we consider the situation after the two brushes come into contact. In this case

the thickness of each compressed brush layer, which we denote by d (�do), is al-ways half the separation h, namely,

(a) Before contact (b) After contact

ψo

ψDON

ψDON

ψ(x)

h

d

h0 0 hx x

ddo

do h−do

do

h

FIGURE 17.1 Interaction between two identical parallel plates covered with a charged

polymer brush layer and potential distribution across the brush layers. (a) Before the two

brushes come into contact (h� 2do). (b) After the two brushes come into contact (h< 2do).do is the thickness of the intact brush layer. From Ref. 2.

382 ELECTROSTATIC REPULSION BETWEEN TWO PARALLEL SOFT PLATES

h ¼ 2d � 2do ð17:2Þ

Since the density of dissociated groups increases as h decreases, it is now a function

of h. We denote by N(h) the density of dissociated groups after contact. Following

de Gennes [1], we assume that when the two brushes come into contact, they are

squeezed against each other but they do not interdigitate (Fig. 17.1b). We also as-

sume that the brushes are contracted in such a way that the density of dissociated

groups is always uniform over the compressed brush layers. That is, the product of

N(h) and d is constant and always equal to Nodo, namely,

N(h)d ¼ N(h)h

2¼ Nodo ð17:3Þ

The potential inside the brush layer is equal to the Donnan potential with the charge

density ZeN(h). We denote by cDON(h) the Donnan potential in the compressed

brush layer, which is a function of h. Since the potential inside the compressed

brush layers is constant independent x (but depends on the separation h)(Fig. 17.1b), the Poisson–Boltzmann equation in the brush layer becomes

d2y

dx2¼ k2 sinh y� ZN(h)

2zn

� �¼ 0 ð17:4Þ

which is equivalent to the condition of electroneutrality in the brush layer, namely,

rel(h)þ rfix(h) ¼ 0 ð17:5Þ

where rel(h) is the charge density of fixed charges and rfix(h) is the charge densityresulting from the electrolyte ions, both being functions of h, and they are given by

rfix(h) ¼ zeN(h) ð17:6Þ

rel(h) ¼ �2zen sinh[yDON(h)] ð17:7Þ

In Eq. (17.7), yDON(h)¼ zecDON(h)/kT is the scaled Donnan potential in the com-

pressed brush layer. We thus obtain from Eq. (17.7)

sinh(yDON(h)) ¼ZN(h)

2zn¼ ZNodo

znhð17:8Þ

or equivalently

yDON(h) ¼ lnZN(h)

2znþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZN(h)

2zn

� �2

þ 1

s24

35 ¼ ln

ZNodoznh

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZNodoznh

� �2

þ 1

s24

35

ð17:9Þ

REPULSION BETWEEN COMPRESSED BRUSHES 383

where Eq. (17.3) has been used in the last step in Eqs. (17.8) and (17.9). Note that

the scaled Donnan potential yDON(h) of the compressed brush layer is related to that

of the intact brush layer as

sinh(yDON(h)) ¼ sinh(y(0)DON)

2doh

� �ð17:10Þ

The interaction force Pe(h) per unit area between the two brush layers after con-

tact may be calculated from Eq. (17.1) by replacing y(h/2) with yDON(h), namely,

Pe(h) ¼ 4nkT sinh2[yDON(h)=2]; h � 2do ð17:11Þ

which gives, by using Eq. (17.8),

Pe(h) ¼ 2nkT

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ZNodo

znh

� �2s

� 1

24

35; h � 2do ð17:12Þ

Equation (2.136) may be rewritten in terms of the Donnan potential of the intact

polymer brush layers y(0)DON with the help of Eq. (17.8) as

Pe(h) ¼ 2nkT

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ (sinh y

(0)DON)

2 2doh

� �2s

� 1

24

35; h � 2do ð17:13Þ

In Fig. 17.2, we give some results of the calculation of the interaction force Pe(h)acting between two compressed brush layers as a function of reduced separation khfor several values of ZNo/zn at scaled brush thickness kdo¼ 10. Calculation was

made with the help of Eq. (17.12) for h� 2do and Eq. (17.1) for h� 2do. We see

that the interaction force is always repulsive before and after the two brushes come

into contact.

It is interesting to note that in contrast to the case of h� 2do, where the interac-tion force is essentially an exponential function of kh, the interaction force after

contact of the brush layers (h� 2do) is a function of the inverse power of h. In orderto see this more clearly we consider the following limiting cases.

a. For highly charge brushes, that is, jyDONj� 1 or jZNo/znj� 1, Eq. (17.13)

becomes

Pe(h) ¼ 2nkT jsinh(y(0)DON)j2doh

� �¼ 2jZjNodokT

zhð17:14Þ

That is, the interaction force is proportional to 1/h. It must be stressed that the

interaction force in this case becomes independent of the electrolyte concen-

tration n.

384 ELECTROSTATIC REPULSION BETWEEN TWO PARALLEL SOFT PLATES

b. For weakly charge brushes, that is, jyDONj� 1 or jZNo/znj� 1, on the other

hand, Eq. (17.13) becomes

Pe(h) ¼ nkT(y(0)DON)

2 2doh

� �2

¼ (ZNodo)2kT

z2nh2ð17:15Þ

which is proportional to 1/h2 and to 1/n.

c. In the limit of small separations h! 0, we again have Eq. (17.14), namely,

Pe(h) ! 2jZjNodokT

zh; h ! 0 ð17:16Þ

If, further, the total charge amount s per unit area, namely

s ¼ ZNodo ¼ ZNd ð17:17Þ

(which is independent of h) is introduced, then we have that

Pe(h) ! 2jsjkTz eh

; h ! 0 ð17:18Þ

This result agrees with the limiting behavior of the interaction force per unit

area between two parallel identical plates with constant surface charge den-

sity s [3, 4].

FIGURE 17.2 Scaled repulsion Pe/64nkT per unit area between two parallel plates cov-

ered with a polymer brush layer as a function of scaled separation kh for zNo/vn¼ 1, 2, 10,

and 100 (i.e., yDON¼ 0.4812, 0.8814, 2.312, and 4.605, respectively) at kdo¼ 10. Solid lines,

after contact (h< 2do). Dotted lines, before contact (h� 2do). From Ref. 2.

REPULSION BETWEEN COMPRESSED BRUSHES 385

According to de Gennes’ theory [1], the repulsive force Ps(h) per unit area be-

tween two uncharged brushes of intact thickness do at separation h is given by

Ps(h) ¼ kT

s32doh

� �9=4� h

2do

� �3=4" #; h � 2do ð17:19Þ

where each grafted polymer is considered as a chain of blobs of size s. Note that

Eq. (17.19) is correct to within a numerical prefactor of order unity. The first term

on the right-hand side of Eq. (17.19) comes from the osmotic repulsion between

the brushes and the second term from the elastic energy of the chains. When brushes

are charged, it is expected that the steric force Ps(h) and electrostatic force Pe(h) areboth acting between the charged brush layers under compression. That is, the total

repulsive force P(h) per unit area between the charged compressed brush layers is

given by the sum of Pe(h) and Ps(h), namely,

P(h) ¼ Pe(h)þ Ps(h) ð17:20Þ

Now we compare the magnitudes of Pe(h) and Ps(h) with the help of Eqs. (17.13)and (17.19). We can expect that No and 1/s3 are of the same order of magnitude.

Indeed, if each brob in the polymer chain has one elementary electric charge, then

we exactly have No¼ 1/s3 (with jZj ¼ 1). It can be shown that for highly charged

brushes with jZNo/znj� 1, Pe(h) can be comparable in magnitude to Ps(h), whileotherwise Pe(h) gives relatively small contribution. The condition jZNo/znj� 1 is

FIGURE 17.3 Scaled repulsive force per unit area between two parallel plates covered

with compressed polymer brush layers after they come into contact. P�e ¼ Pe=NokT is the

scaled electrical repulsion, P�s ¼ s3Ps=kT is the scaled steric repulsion and their sum

P�e þ P�

s . From Ref. 2.

386 ELECTROSTATIC REPULSION BETWEEN TWO PARALLEL SOFT PLATES

fulfilled for most practical cases. As a typical example, we may put s� 1 nm and

thus No� 1/s3� 1027m�3. Thus, if, for instance, n¼ 10�2M¼ 6� 1024m�3, then

we have N/n� 1.7� 102 so that Pe (h) may be approximated by Eq. (17.14), which

is independent of the electrolyte concentration n. In Fig. 17.3, we compare scaled

repulsive forces P�s (h) and P�

e(h) and their sum P�s (h)þ P�

e(h) as a function of h/2dofor jZj ¼ 1 and z¼ 1, where P�

e(h) ¼ Pe=NokT and P�s (h) ¼ s3Ps=kT . We see that Pe

can be comparable in magnitude to Ps and can even exceed Ps especially when the

brushes are weakly compressed.

REFERENCES

1. P. G. de Gennes, Adv. Colloid Interface Sci. 27 (1987) 189.

2. H. Ohshima, Colloid Polym. Sci. 277 (1999) 535.

3. D. McCormik, S.L. Carnie, and D. Y. C. Chan, J. Colloid Interface Sci. 169 (1995) 177.

4. J. N. Israelachvili, Intermolecular and Surface Forces, 2nd edition, Academic Press,

New York, 1992.

REFERENCES 387