20 DLVO Theory of ColloidStability

20.1 INTRODUCTION

The stability of colloidal systems consisting of charged particles can be essentially

explained by the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory [1–7].

According to this theory, the stability of a suspension of colloidal particles is de-

termined by the balance between the electrostatic interaction and the van der

Waals interaction between particles. A number of studies on colloid stability are

based on the DLVO theory. In this chapter, as an example, we consider the inter-

action between lipid bilayers, which serves as a model for cell–cell interactions

[8, 9]. Then, we consider some aspects of the interaction between two soft spheres,

by taking into account both the electrostatic and van der Waals interactions acting

between them.

20.2 INTERACTION BETWEEN LIPID BILAYERS

Lipid bilayer membranes have been employed extensively as an experimental

model of biological membranes. We take the dipalmitoylphosphatidylcholine

(DPPC)/water system as an example to calculate the intermembrane interactions.

On the basis of the DLVO theory [1–7], we consider two interactions, that is, the

electrostatic repulsive double-layer interaction and the van der Waals attractive in-

teraction. In addition to these two interactions, we take into account the steric inter-

action between the hydration layers formed on the membrane surface. DPPC forms

bilayers when it is dissolved in water. Inoko et al. [10] have made X-ray diffraction

studies on the effects of various cations upon the DPPC/water system. They have

found remarkable effects of CaCl2 and MgCl2: depending upon the concentration

of CaCl2, four different states appear, which they term I, II, III, and IV, and three

states I, III, and IV appear in the case of MgCl2. Figure 20.1 illustrates the four

states in the case of CaCl2. In pure water, the system consists of a lamellar phase

with a period of 6.45 nm and excess water (state I). Addition of 1.3mM CaCl2 de-

stroys the lamellar structure and makes it swell into the excess water (state II). The

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

420

lamellar phase, however, reappears when the concentration of CaCl2 increases

(state III): a partially disordered lamellar phase with a repeat distance of about

15 nm comes out at a concentration of about 10mM, and then the degree of order

of the lamellar phase is improved and the repeat distance decreases with increasing

CaCl2 concentration. Above 200mM, the repeat distance becomes constant at

6.45 nm (state IV), the same value as in state I. When the concentration is reduced,

the transitions IV! III and III! II occur at the same concentration as mentioned

above, but the transition II! I takes place at a concentration close to 0mM. In

other words, there is a hysteresis in the transition I! II. As shown below, the ap-

pearance of the above four or three states in the DPPC/water system by addition of

CaCl2 or MgCl2 can be explained well by the DLVO theory [8, 9].

Consider the potential energy V between two adjacent bilayer membranes per

unit area, which may be decomposed into three parts:

V ¼ Ve þ Vv þ Vh ð20:1Þ

where Ve is the potential of the electrostatic interaction caused by the adsorption of

Ca2+ or Mg2+ ions, Vv is the potential energy of the van der Waals interaction, and

Vh is the potential energy of the repulsive interaction of hydration layers. The DPPC

membrane is expected to be electrically neutral in pure water. It will be positively

charged by adsorption of divalent cations (Ca2+ or Mg2+) on its surfaces. We

assume that there are Nmax sites available for adsorption of cation per unit area and

FIGURE 20.1 Schematic representation of the four states I, II, III, and IV of the DPPC/

water system observed by Inoko et al. [10]. Values in units of mM give the CaCl2 concentra-

tion at which the transition between two states occurs.

INTERACTION BETWEEN LIPID BILAYERS 421

that the cation adsorption is of the Langmuir type. Then, the number N of adsorbed

cations per unit area is given by

N ¼ NmaxKn exp(� 2eco=kT)

1þ Kn exp(� 2eco=kT)ð20:2Þ

where n and K are, respectively, the bulk concentration of Ca2þ or Mg2þ and the

binding constant of Ca2þ or Mg2þ ions, and co is the surface potential of the bilayer

membrane. The surface charge density s¼ 2eN on the bilayer membrane is thus

given by

s ¼ 2eNmaxKn exp(� 2eco=kT)

1þ Kn exp(� 2eco=kT)ð20:3Þ

Adsorption of cations causes the electrostatic repulsion between the bilayer

membranes. The electrostatic repulsive force Pe(h), which is given by Pe(h)¼�dVe(h)/dl, acting between two adjacent membranes is directly expressed as (see

Eq. (9.26))

Pe(h) ¼ nkT�exp(� 2ecm=kT)þ 2 exp(ecm=kT)� 3

� ð20:4Þ

where c(h/2) is the potential at the midpoint between the two adjacent bilayers and

is obtained by solving the following Poisson–Boltzmann equation for the electric

potential c(x):

d2cdx2

¼ � 2e(nCa þ nMg)

ereoexp � 2ec

kT

� �� exp

eckT

� �� �ð20:5Þ

where the x-axis is taken perpendicular to the bilayers with its origin at the surface

of one bilayer so that the region 0< x< h corresponds to the electrolyte solution

between the two adjacent bilayers. Because of the periodic structure of the lamellar

phase, we have

dcdx

����x¼0þ

¼ 0 ð20:6Þ

dcdx

����x¼0þ

¼ � sereo

ð20:7Þ

dcdx

����x¼h=2

¼ 0 ð20:8Þ

422 DLVO THEORY OF COLLOID STABILITY

We assume that the van der Waals interactions of one membrane with the others

are additive. Then, the expressions for Vv are given below.

Vv(h) ¼X1m¼0

vv(hþ m(hþ d)) ð20:9Þ

with

vv(h) ¼ � A

12p1

h2þ 1

(hþ 2d)2� 2

(hþ d)2

� �ð20:10Þ

where v(h) is the potential energy of the van der Waals interaction per unit area

between two parallel membranes of thickness d at a separation h and A is the

Hamaker constant. The van der Waals force Pv(h)¼�dVv/dh is given by

Pv(h) ¼X1m¼0

(mþ 1)pv(hþ m(hþ d)) ð20:11Þ

with

pv(h) ¼ � A

6p1

h3þ 1

(hþ 2d)3� 2

(hþ d)3

� �ð20:12Þ

We assume that each membrane has adsorbed water layers on its surface. Con-

tact of these hydration layers will cause a repulsive force, whose potential Vh is

approximately given by

Vv(h) ¼ 1; h < ho0; h � ho

�ð20:13Þ

The critical concentration for the I! II�III transition is determined by

d(Ve þ Vv)

dh

����h¼ho

¼ 0 ð20:14Þ

or

Pe(ho)þ Pv(ho) ¼ 0 ð20:15Þ

Here, we put ho¼ 2.25 nm [8, 9]. In Ref. [9], the condition (20.15) was applied to a

mixed aqueous solution of CaCl2 and MgCl2 for various concentrations and

the three unknown parameters, that is, the Hamaker constant A for the DPPC/water

INTERACTION BETWEEN LIPID BILAYERS 423

system, and KCa and KMg, which are the adsorption constants for Ca2þ and Mg2þ

ions, respectively, were determined. The results are

A ¼ (3:6� 0:8)� 10�21 J; KCa ¼ 21� 9M�1; KMg ¼ 2:5� 0:7M�1

ð20:16ÞUsing these values, we have calculated the potential energy V of interaction per

mm2 per pair of adjacent membranes as a function of membrane separation h and

CaCl2 concentration n. Results are shown in Fig. 20.2. The curve for n¼ 0mM in

Fig. 20.2 shows that V becomes minimum at h¼ ho in accordance with the observa-

tion. The minimum of V at h¼ ho disappears in the curve for n¼ 1.3mM (i.e., VeþVv has a maximum at h¼ ho). This curve has a secondary minimum at larger h. Itsdepth is too shallow to maintain the observed arrangement of membranes in the

lamellar phase. Therefore, the curve for n¼ 1.3mM corresponds to the disordered

dispersion of membranes as was observed experimentally at n¼ 1.3mM, that is,

the state II. The minimum of V is deepened and correspondingly the position of

potential minimum shifts to smaller h with increasing n as seen in the curves for

n¼ 50mM and n¼ 150mM, which therefore correspond to state III. With a further

increase of n, the curve again has a sharp minimum at h¼ ho as shown by the

curve for n¼ 500mM, which should correspond to state IV. The curve for n¼ 0.05

mM has a special character: it has a sharp minimum at h¼ ho and a very shallow

minimum at very large h and a maximum between them. Here, we assume that the

FIGURE 20.2 The interaction energy V per pair of two adjacent DPPC membranes as a

function of h at various CaCl2 concentrations, calculated with A¼ 3.6� 10�21 J, KCa¼ 21

M�1, and T¼ 278K.

424 DLVO THEORY OF COLLOID STABILITY

transition between two states cannot take place beyond the maximum or the poten-

tial barrier. The potential barrier disappears at 1mM as shown by curve for 1.3mM

(as seen from the side of h¼ ho) and thus the transition I! II occurs at this con-

centration when n increases. On the other hand, it disappears at about 0mM (as

seen from the side of large h) and the transition II! I occurs at this concentration

when n decreases. This behavior of V explains the hysteresis observed in the I! II

transition.

In the case of MgCl2, there appear only three states I, III, and IV and the transi-

tion I! III occurs at n¼ 5mM. This can be explained as follows. Since KMg is

much smaller than KCa, V(H) for n¼ 5mM, where a maximum at h¼ ho, also has a

minimum at large h. This minimum is much deeper than that for CaCl2 and is deep

enough to preserve the order of the lamellar structure. That is, the disordered state II

does not appear in the case of MgCl2.

20.3 INTERACTION BETWEEN SOFT SPHERES

In the preceding section, we have treated the interaction between lipid bilayers,

where we have regarded a lipid bilayer as a hard plate without surface structures. In

this section, we consider the total interaction energy between two identical soft

spheres of radius a at separation H (Fig. 20.3). Each particle is covered with a layer

of polyelectrolytes of thickness d. Let the density and valence of ionized groups are,respectively, Z and N, so that the density of the fixed charges within the surface

charge layer is rfix¼ ZeN. With the help of Eq. (15.45), the total interaction energy

V(H) is given by

Ve(H) ¼ 2par2fix sinh2(kd)

ereok4ln

1

1� e�k(Hþ2d)

� �ð20:17Þ

FIGURE 20.3 Interaction between two similar spherical soft particles.

INTERACTION BETWEEN SOFT SPHERES 425

We treat the case in which the number density of molecules within the surface

layer is negligibly small compared to that in the particle core. The van der Walls

interaction energy is dominated by the interaction between the particle cores, while

the contribution from the surface charge layer can be neglected. Thus, we have

(from Eq. (19.31))

Vv(H) ¼ � Aa

12(H þ 2d)ð20:18Þ

where A is the Hamaker constant for the interaction between the particle cores in

the electrolyte solution. The total interaction energy V(H)¼Ve(H)þVv(H) is thus

V(H) ¼ 2par2fix sinh2(kd)

ereok4ln

1

1� e�k(Hþ2d)

� �� A

12(H þ 2d)ð20:19Þ

which is a good approximation provided that

H � a and ka � 1 ð20:20Þ

We rewrite Eq. (20.19) in terms of the unperturbed surface potential co (see Eq.

(1.27)), which is given by, under condition given by Eq. (20.20),

co ¼s

ereok1� e�kd

2kd

� �ð20:21Þ

Then, Eq. (20.19) becomes

Ve(H) ¼ 2pereoac2oe

2kd ln1

1� e�k(Hþ2d)

� �ð20:22Þ

The total potential energy V is thus given by

V(H) ¼ 2pereoac2oe

2kd ln1

1� e�k(Hþ2d)

� �� A

12(H þ 2d)ð20:23Þ

At a maximum or a minimum of a plot of V(H) versus H, we have dV(H)/dH¼ 0,

namely,

�k � 2pereoac2oe

2kd e�k(Hþ2d)

1� e�k(Hþ2d)

!þ A

12(H þ 2d)2¼ 0 ð20:24Þ

which can be rewritten as

f (kH) ¼ Z ð20:25Þ

426 DLVO THEORY OF COLLOID STABILITY

where f(kH) and Z are defined by

f (kH) ¼�k(H þ 2d)

�2ek(Hþ2d) � 1

ð20:26Þ

Z ¼ Ak

24pereoac2oe

2kdð20:27Þ

We plot F(kH) as a function of kH for several values of kd in Fig. 20.4, showingthat F(kH) takes the maximum value of 0.6476 at kH¼ 1.5936� 2kd. (Discussionsbased on a plot of F(kH) versus kH were found in Refs. [11] and [12].)

There are five possible situations (Fig. 20.5a, b, and c and Fig. 20.6a and b).

Consider first the situations in Fig. 20.5, that is,

Case (i) 0� 2kd< 1.5936

In this situation, F(kH) shows a maximum (F(kH)¼ 0.6476) in the region kH� 0. In case (i), we have further three possible cases.

Case (a) Z� 0.6476

In this case, Eq. (20.25) has no root, so V(kH) decreases monotonically as

kH decreases.

Case (b) F(0)� Z< 0.6476

Here, F(0) is given by

F(0) ¼ (2kd)2

e2kd � 1ð20:28Þ

FIGURE 20.4 Plot of F(kH) as a function of kH for kd= 0, 1, and 3.

INTERACTION BETWEEN SOFT SPHERES 427

In this case, Eq. (20.25) has two roots kH¼ a and kH¼ b (a< b), and V(kH) shows a maximum at kH¼ a and a minimum at kH¼ b.

Case (c) Z<F(0)In this case, Eq. (20.25) has one root kH¼ b and V(kH) shows a minimum

at kH¼ b.

FIGURE 20.5 F(kH) and V(kH) at 0� 2kd< 1.5936 for three cases (a), (b), and (c).

FIGURE 20.6 F(kH) and V(kH) at 2kd� 1.5936 for two cases (a) and (b).

428 DLVO THEORY OF COLLOID STABILITY

Next, consider the situations in Fig. 20.6, that is,

Case (ii) 2kd� 1.5936

In this situation, F(kH) shows no maximum, decreasing monotonically as kHincreases in the region kH� 0. In this situation, we have further two possible

cases.

Case (a) Z�F(0)In this case, Eq. (20.25) has no root, so V(kH) decreases monotonically as

kH decreases.

Case (b) Z<F(0)In this case, Eq. (20.25) has one root kH¼ b and V(kH) shows a minimum

at kH¼ b.

Figure 20.7 shows some examples of the calculation of the interaction energy V(H) between two similar soft spheres at separation H, calculated with a¼ 0.5 mm,

A¼ 4� 10�21 J, co¼�15mV, n¼ 0.1M, and T¼ 298K. The curve for d¼ 0 nm

corresponds to Fig. 20.5b and curves for d¼ 1 and 10 nm to Fig. 20.6b.

Note that we have considered only the region H� 0, in which the surface charge

layers of the interacting soft spheres are not in contact. We have to consider other

interactions after contact of the surface layers, as discussed in the previous chapter.

REFERENCES

1. B. V. Derjaguin and L. D. Landau, Acta Physicochim. 14 (1941) 633.

2. E. J. W. Verwey and J. Th. G. Overbeek, Theory of the Stability of Lyophobic Colloids,Elsevier/Academic Press, Amsterdam, 1948.

FIGURE 20.7 Potential energy V(H) between two similar soft spheres at separation H,calculated with A= 4� 10�21 J, co =�15mV, n = 0.1M, and T = 298K.

REFERENCES 429

3. B. V. Derjaguin, Theory of Stability of Colloids and Thin Films, Consultants Bureau,New York, 1989.

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

New York, 1992.

5. J. Lyklema, Fundamentals of Interface and Colloid Science: Solid–Liquid Interfaces,Vol. 2, Academic Press, New York, 1995.

6. H. Ohshima, in: H. Ohshima and K. Furusawa (Eds.), Electrical Phenomena at Interfa-ces: Fundamentals, Measurements, and Applications, 2nd edition, revised andexpanded, Dekker, New York, 1998, Chapter 3.

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10. Y. Inoko, T. Yamaguchi, K. Furuya, and T. Mitsui, Biochim. Biophys. Acta 413 (1975)

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11. B. Chu,Molecular Forces, Wiley, New York, 1967.

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1963, p. 30.

430 DLVO THEORY OF COLLOID STABILITY