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Analytical expressions for the disjoining pressure between particle-stabilizedfluid–fluid interfaces and composite materials†

Carolina Vannozzi*

Received 6th December 2011, Accepted 14th February 2012

DOI: 10.1039/c2sm07307b

A method based on a hybrid Hamaker–Lifshitz approach is developed to derive numerical and

analytical expressions of the disjoining pressure (i.e. van der Waals forces per unit area) of the thin film

between two drops having micro/nanoparticles straddling their interfaces, as a function of the particle

concentration, dimension and core material. This is useful to determine the ability of nanoparticles to

stabilize immiscible polymer blends against coalescence, thus in the design of effective nanoparticle

stabilizers. The system is modelled as two facing half spaces (i.e., two surfaces or walls with infinite

depth) with 2D-lattices of spherical particles straddling the interfaces, interacting through a matrix. The

method is also applied to the case of two compound half spaces with 3D-lattices of spherical inclusions

to validate the method by comparison with well known effective medium expressions. This is also

a model for drops with particles dispersed in their bulk or two interacting colloidal/nanocrystals. Our

approach is promising for its ability to deal with complex geometries and with the presence of an

intervening medium in a simple way, leading to analytical expressions that can be used both in

experiments and in numerical simulations of coalescence; and it is especially applicable to polymeric

systems having no strong dipole interactions. Multi-body interactions are only included by evaluating

the Hamaker constants through the Lifshitz approach.

1. Introduction

Despite the long established use of particles in emulsions and

foams as stabilizers, only recently particles adsorbed at the

interface of two immiscible polymers have been used to stabilize

polymeric blends.1–4 Indeed, new areas of research have been

focused on the use of nanoparticles either with uniform surface

properties3,5–12 or coated with polymer ligands of mixed type, i.e.

ligands of the dispersed and continuous phase type, the so-called

‘‘Janus nanoparticles’’.5,10,13–17 This wide interest is due to the

nanoparticle ability to give characteristic optical, electric and

magnetic properties to the resulting composite material.16,18,19

Additionally, nanoparticles can increase the blend dispersed

phase volume fraction because of their small size.

Current theories on coalescence suggest that particles can

stabilize blends through the formation of a steric barrier or by

producing Marangoni stresses due to their surface activity,

slowing down the drainage of the matrix phase, as in the case of

low molecular weight surfactants.5,10 On the other hand, particles

that have a core material different from the bulk materials will

Chemical Engineering Department, University of California SantaBarbara, Santa Barbara, CA 93106, USA. E-mail: carolina.vannozzi@gmail.com; Tel: +1 805893 3412

† Electronic supplementary information (ESI) available. See DOI:10.1039/c2sm07307b

5214 | Soft Matter, 2012, 8, 5214–5224

change the magnitude of the van der Waals (VDW) forces

between two drops, influencing their coalescence stability.

In fact, in the usual description of coalescence, VDW inter-

actions determine the final drop fusion. In clean interface

systems, they are usually incorporated in the normal stress

boundary conditions as a disjoining pressure (P), which is

commonly approximated as that between two parallel infinite

half spaces (i.e., two surfaces or walls with infinite depth) per unit

area:20–23

Phs;hs ¼ � Ah

6pD3(1)

where the superscripts stand for the two interacting bodies, i.e.

hs, hs indicate two infinite half spaces, Ah is the Hamaker

constant, and D is the local distance between the two interfaces.

Ah can be measured or calculated through Lifshitz theory, i.e. via

the dielectric functions of the bulk materials, as will be explained

further. P is negative for attractive interactions and positive

otherwise.

Since it iswell known thatmetals in void are characterized byAh

roughly two orders of magnitude higher than dielectric materials,

it is likely that metal nanoparticles stabilizing blends can accel-

erate film drainage and favour droplet coalescence compared to

dielectric particles or other stabilizers. This seems to be confirmed

byBorrell andLeal’s experiments in the four-rollmill,10where two

drops, compatibilized by polymer-coated Janus gold nano-

particles, undergoing a head-on collision, showed a dramatic

This journal is ª The Royal Society of Chemistry 2012

reduction in the drainage time of about 70%compared to the same

system compatibilized by block-copolymers, while they still

considerably increased the drainage time by one order of magni-

tude compared to a clean interface system. Therefore, remark-

ably, this type of nanoparticles were less effective stabilizers than

surfactants, despite being more surface active. An opposite

behaviour was found using similar polymer coated Janus nano-

particles, having a polymeric core instead, in the blend

morphology study of M€uller and coworkers.5

Lacking a quantitative evaluation of VDW interactions in the

case of interfacial micro/nanoparticles, the objective of the

present paper is to derive a simple analytical expression to

account for the change inP due to the presence of these particles

in polymer blends to guide the design of effective stabilizers, as

a function of the particle size, concentration and type of core

material. This is achieved through a hybrid Hamaker–Lifshitz

approach,24,25 which hinges on the assumption of additivity of

the interaction potentials of the bodies that make up the system.

This assumption is an oversimplification of the complex induced-

dipole multi-body interactions present in real systems, but it is

often used to evaluate P. It usually provides the correct quali-

tative and quantitative behaviour of the interactions26 and it is

well suited in systems without strong dipole interactions, like the

polymeric systems of interest here. Moreover, it gives simple

analytical expressions that can be used in both computational

studies of coalescence26 and experimental ones.

In the present model, the two drops are approximated as half

spaces, separated by a matrix phase, with particles arranged in

2D-lattices straddling the interfaces (see Fig. 1a). This model is

valid in describing the coalescence of drops as long as the thin

lubrication film between two approaching drops is relatively

large compared to the particle dimension or interparticle spacing.

In flow induced coalescence theory, the film width is a fraction of

the drop radius R, i.e. O(Ca0.5R),27 where Ca is the capillary

number of the flow defined as Ca h RGmm/s, with G being the

shear rate, R the drop radius, mm the matrix viscosity and s the

interfacial tension of the system. This means that for Ca ¼O(10�4 to 10�2), typical of flow-induced coalescence experiments,

Fig. 1 Sketch of the model systems. (a) Section representing two infinite

compound half spaces of material 1, with spherical inclusions of material

2 straddling its interfaces, interacting across medium 3. The particles are

arranged in a 2D-simple cubic lattice, with lattice constant a, and the two

interfaces at distanceD apart. (b) The same system as in (a), with particles

dispersed in a 3D-SC lattice of spherical particles in the bulk of medium 1.

In both models medium 1 represents the bulk of two drops and medium 3

the thin film of matrix phase separating them.

This journal is ª The Royal Society of Chemistry 2012

and micron-sized drops, our model can be used for particle sizes

of the order of nanometres. We also studied the case of particles

dispersed in the drop phase, modelled as a 3D-lattice of spheres

(Fig. 1b). These situations can occur if ligand-stabilized particles

dispersed in the drops self-assemble in lattice structures, similarly

to what is observed for block-copolymer micelles28 or in the case

of two facing nanocrystals.

This paper is organized as follows: after a short discussion on

the issues arising from other approaches, considered in Section

1.1, Section 2 describes a brief review of the original Hamaker

summation method and its adaptation to the present cases. In

Section 3, results from the full pairwise summation and the

analytical formulae are compared for facing 2D-lattices of

spheres interacting across void. The method is also validated by

comparing the expression derived for 3D-lattices of spherical

inclusions dispersed in two infinite half spaces (i.e. nanocrystals)

with effective medium theory. Finally, the derived expressions

for particles straddling the interface are simulated and the error

introduced with our approximation is evaluated.

1.1. Issues with other approaches

Other approaches could be used to evaluate the effects of inter-

facial particles on P; however, each has some flaws that are

briefly discussed here.

As already stated, a hybrid Hamaker–Lifshitz approach is

employed here for its ability to give simple analytic expressions

despite the complex geometry of our discrete systems, regardless

of the particle size. Vice versa, the Lifshitz theory (LT) alone,

which is a quantum mechanical treatment of the interactions,

taking into account collective or multi-body effects, yields

analytical expressions only for a few simple geometries, for

instance two facing half spaces.29,30 Moreover, LT is a continuum

based theory, thus, it is not well suited to describe the discrete

nature of the interfacial region with particles partially immersed

in both materials (see Fig. 1a). An additional problem with LT is

that a theoretical description of the size dependent dielectric

function of nanoparticles, which has to be known in order to

apply LT to the present case, is still the target of active experi-

mental and theoretical research.31–38

Another possible approach to the present problem is the use of

the Effective Medium Theory (EMT) to calculate the effective

dielectric constant of the interfacial layer, considered as a sepa-

rate phase and then use it to calculate an effective Ah through

Lifshitz theory. However, EMT is a well established method to

treat 3D bulk periodic materials or random interfacial multi-

layers of polymer-coated nanoparticles at a liquid/air inter-

face,31,32 but here only one layer of particles is considered. Thus,

EMT cannot be used in its established formulation and has to be

adapted to include the appropriate boundary conditions for the

interfacial monolayer, complicating considerably the problem.

Moreover, EMT approaches are not applicable to distances of

the order of the lattice constant, because they ignore the spatial

arrangement of the system; on the contrary, this is possible with

the discrete approach here developed.

Another possibility to calculate an effective Ah of the interfa-

cial layer is via phenomenological mixing rules,39–41 where the

volume averaged dielectric function of each side of the interface

is obtained taking into account the particle and the surrounding

Soft Matter, 2012, 8, 5214–5224 | 5215

fluid. However, this method has only been experimentally tested

for straddling dielectric nanoparticles39–41 and it is not straight-

forward to apply it to metal particles, thus it is not pursued here.

2. Theory

2.1. Theoretical background: interaction of two bodies across

vacuum and a medium within the Hamaker approach

We now describe the basic features of the Hamaker and of the

Hamaker–Lifshitz approach, which form the theoretical basis of

our study. In his seminal work Hamaker42 describes a method to

calculate the total interaction energy between two macroscopic

bodies as a volume integral, taken over the interior volume of the

two bodies V1 and V2, of the distance dependence of the inter-

action energy of each atom in one body with each atom in the

other body:29

EðrÞ ¼ �ð

V1 ;V2

q1q2l12

jr1 � r2j6d r1d r2; (2)

where qi is the density of the atoms in the i-th body and l12 is the

London–van der Waals constant, which is a function of the

dielectric permittivity of the two media.31 The final expression of

the interaction energy between the two bodies is the product of

one part dependent only on material properties of the body with

another part dependent only on the distance between the two

interacting bodies, a function of their geometry. Eqn (2) leads to

the definition of Ah within the Hamaker approach as Ah ¼q1q2l12p

2.42 Thus, the interaction can be expressed as: E(D) ¼AhE

bb(D), where Ebb is the geometric dependence of the inter-

action energy between the two bodies andD is the distance of the

two interfaces.

Eqn (2) enables us to calculate the distance dependence of the

total interaction potential for several simple geometries, such as

two infinite half spaces, two spheres, a sphere and a half space, an

atom and a half space, etc.32,43 The retardation effect, which is

important for distances greater than 10 nm, can be considered in

(2) by incorporating correction factors to the atom pair poten-

tial,44 but it is neglected here for simplicity. From eqn (2), the

disjoining pressure is calculated as the negative gradient of the

total interaction energy with respect to D:

P ¼ � dE

dD: (3)

The presence of an intervening medium is easily incorporated

within the additivity assumption. In fact, the total interaction

energy of two bodies made of materials 1 and 2 interacting in

a medium of material 3 is given by:32,42,43,45,46

Etot ¼ E12 + E33 � E13 � E23 ¼ (A12 + A33 � A13 � A23)Ebb,(4)

where Eij and Aij are the interaction energy and the Hamaker

constant of the bodies i and j interacting across vacuum,

respectively. The interaction between two bodies, both of mate-

rial 1, is simply given by substituting subscript 2 with 1 in eqn (4).

This approach works for bodies of finite size as well as infinite

half spaces. When making use of the combining relations,23,24,47

namely A132 ¼ A12 + A33 � A13 � A23 and A131 ¼ A11 + A33 �2A13, the total interaction energy derived in eqn (4) can be

5216 | Soft Matter, 2012, 8, 5214–5224

written as Etot ¼ A132Ebb. Using this approach the interaction

energy of composite bodies interacting in a medium can be

derived. For example, in the Appendix is derived via the

Hamaker method the well known interaction energy of a five

layer system, often used in colloidal science to model particles

interacting with a coating layer.24

The accuracy of eqn (4) can be improved by evaluating A132

through the Lifshitz method, to partially account for multi-body

interactions, in the so-called Lifshitz–Hamaker approach.24,25

When the two bodies are metals (i.e. with an infinite static

dielectric constant) in a dielectric medium, Ah is generally eval-

uated by considering the complex dielectric function of the

metal.32 In our analysis, for the sake of both simplicity and lack

of data, we used the semi-empirical method described by

Bargeman and Vanvoors.46 This method is very often used in the

literature for estimating the interaction energy of metal nano-

particles in organic solvents.48 It enables us to evaluate the

Hamaker constant of two bodies interacting across a medium

from the experimentally measured Aij of the constituent pairs of

materials interacting across void.

The Hamaker method is valid as long as the particles do not

interact with each other and the surrounding material does not

preferentially orient itself due to the presence of the interacting

bodies.42 In the case of surface functionalized particles, the first

condition is reasonable, because the polymer coating prevents

the gold cores or the dielectric spheres from getting too close to

each other, this in turn keeps the area fraction low. The second

condition is also reasonable in polymer blends, where no strong

dipole moments are present.

2.2. Interaction energy per unit area of two half spaces with

3D-lattices of spheres as inclusions

The interaction energy per unit area of two composite half spaces

with 3D-lattices of spheres as inclusions (Fig. 1b) interacting in

void was determined by decomposing the total interaction

energy, Etot, in the sum of the interaction energies of sub-systems,

expressed in terms of analytical expressions well known or easily

determinable via eqn (2):

Etot ¼ Ehs;hs11 � 2E3D-l;hs

11 þ E3D-l;3D-l11|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

1

þ 2�E3D-l;hs

21 � E3D-l;3D-l21

�|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}2

þE3D-l;3D-l22|fflfflfflfflffl{zfflfflfflfflffl}

3

; (5)

the superscripts refer to the geometry of the pair of bodies

involved in the interaction, i.e. 3D-l stands for 3D-lattice and hs

for half space, while the subscripts refer to the materials involved,

as represented in Fig. 1b.

Specifically, each bracket in eqn (5) represents the interaction

energy of each sub-system:

I. The three terms in the first bracket represent the interaction

energy of two infinite half spaces containing cavities. This is

obtained by subtracting, from the interaction energy of two half

spaces of material 1 Ehs,hs11 , twice the interaction energy of a 3D-

lattice of particles of material 1 interacting with a half space of

material 1, E3D-l,hs11 , and by adding the interaction energy of two

facing ghost 3D-lattices of particles of material 1, E3D-l,3D-l11 . The

This journal is ª The Royal Society of Chemistry 2012

last addition is necessary to avoid an underestimation of the total

interaction energy of two infinite half spaces with cavities;

II. The two terms in the second bracket correspond to the

interaction energy of a 3D-lattice of material 2 interacting with

a half space of material 1 containing cavities. This is obtained by

subtracting, from the interaction energy of a 3D-lattice of

material 2 interacting with a half space of material 1 E3D-l,hs21 , the

interaction energy of a 3D-lattice of material 2 interacting with

a ghost 3D-lattice of material 1, E3D-l,3D-l21 ;

III. The last term E3D-l,3D-l22 represents the interaction energy of

two facing 3D-lattices of particles of material 2.

Following the classical procedure outlined in Section 2.1 to

incorporate the presence of the intervening medium 3, using the

Lifshitz–Hamaker approach to express the energies in terms of

the Hamaker constants and the combining relations, the total

interaction energy is:

Etotmedium ¼ A131E

hs,hs(D) + A212E3D-l,3D-l(D + a)

+ 2(A231 � A131)E3D-l,hs(D + a/2), (6)

where E3D-l,3D-l is calculated by adding up the interaction energy

of each couple of 2D-layers present in the two facing bodies and

E3D-l,hs by adding up the interaction energy of each 2-D layer with

the opposite half space:

E3D-l;3D-l ¼XNi; j¼1

E l;lðDþ xði � jÞÞ; E3D-l;hs

¼XNi¼1

El;hsðDþ a=2þ xði � 1ÞÞ; (7)

where N is the number of layers in each body and x is the

interparticle distance, which depends on the lattice type, x ¼ a,

x ¼ affiffiffi2

p=3, x ¼ a=

ffiffiffi3

p, x ¼ a=

ffiffiffi2

p, respectively for SC or HCP,

BCC, and FCC lattices.

The interaction energy of a compound system interacting in

void is obtained from eqn (6) by making the substitution:A131/

A11, A212 / A22 and A231 / A21. The decomposition in sub-

systems in eqn (6) is also sketched in Fig. 2.

The expressions for the interaction energies used in this study

will be derived in Section 2.5. Using eqn (6) and (2), the disjoining

pressure of composite half spaces interacting across medium can

be derived:

Fig. 2 Model decomposition in different sub-systems of two composite

half spaces interacting across a medium. The simplified picture is

obtained by applying the combining relations. Top row, from left to

right: two half spaces of material 1 interacting across medium 3 and two

3D-lattices of particles of material 2 interacting in medium 1. Bottom

row, from left to right: 3D-lattices of material 2 interacting with a half

space of material 1 across medium 3 and 3D-lattice of material 1 inter-

acting with a half space of material 1 across medium 3.

This journal is ª The Royal Society of Chemistry 2012

PtotmedðDÞ ¼ � A131

6pD3

�XNj;i¼1

pA212

3rlirlj h ln

h2 � 4R2

p

h2

!þ4R2

p

�h2 � 2R2

p

�h�h2 � 4R2

p

�0@

1A

�XNi¼1

rliðA231 � A131Þ

3

Rp�

j � Rp

�2 þ Rp�j þ Rp

�2� 2Rp�

j � Rp

��j þ Rp

�!;(8)

where h ¼ D + a(i + j) and j ¼ D + a/2 + a(i � 1) for simple cubic

lattices and rli is the 2D number particle density of lattice i, i.e.

rli ¼ 2=a2ffiffiffi3

pfor a 2D-hexagonal-lattice or rli ¼ a�2 for a square-

lattice. In the following we assume that both interacting bodies

have the same particle density in each lattice, i.e. rli ¼ rl. As for

the interaction energy Etotmedium, the DP for two compound half

spaces interacting in void can be obtained from eqn (8) by

making the substitution A131 / A11and A231 / A21.

2.3. Comparison with EMT: pair additivity approach

validation

To test our method, eqn (8) was compared to eqn (1), in whichAh

was substituted with an effective Hamaker constant (Aeffh) eval-

uated using the effective dielectric functions of the composite

materials within LT. In fact, EMT is a well-established technique

for obtaining analytic and numerical effective-properties of

composite periodic materials.49,50 The ability of the proposed

method to match the EMT results, as a function of particle

concentration and the dielectric properties of the three media,

was evaluated. This could also give an indication of the relevance

of multi-body effects with respect to pairwise interactions in the

system under investigation. For a quantitative evaluation of eqn

(1) and (8), we chose two half spaces of polybutadiene (PBd)

interacting across polydimethylsiloxane (PDMS), described in

the experiments of drop coalescence with Janus polymer-coated

gold nanoparticles as stabilizers in ref. 10.

The effective electrical and optical properties of half spaceswith

spherical inclusions, necessary to calculate Aeffh, were evaluated

both for conducting particles and for dielectric particles. For the

sake of simplicity, only the static dielectric constant was consid-

ered, although for nanoparticles the Drude model offers a better

description of the dielectric function and its dependence on

particle size. We used the ‘‘exact’’ (i.e. including higher order

multi-poles) numerical data for the static dielectric constant of

a material with conducting spherical inclusions derived by

McPhedran andMcKenzie51,52 and the asymptotic expressions for

dielectric spherical inclusions derived by Sangani andAcrivos.53,54

Lastly, to evaluate eqn (8), knowledge of A12 ¼ APBd/Au and

A231 ¼ AAu/PDMS/PBd was necessary. Due to the lack of available

experimental data, they were determined by fitting the data

calculated using our method to the EMT approach.

2.4. Particles at interfaces

In the case of interfacial particles (Fig. 1a), the interface can be

modelled as a surface with zero thickness, where the properties of

Soft Matter, 2012, 8, 5214–5224 | 5217

the bulk phases 1 (the drop) and 3 (the matrix) undergo a step

change, having particles of material 2 arranged in a 2D-lattice of

inclusions straddling it. We can speculate that the Janus particle

immersion in the drop phase is determined by the volume frac-

tion of the two types of coating ligands. Thus, a particle with 50%

volume fraction in one type of ligand will have the core centre

positioned right at the interface. In our model we consider

different degrees of immersion (g) of the particles in phase 1 with

respect to the interface, i.e. the particle centre of mass is shifted

with respect to phase 1 of a distance gRp, thus for g ¼ 0 the

particle centre is at the interface, while for g¼ 1 it is immersed in

phase 1 and for g ¼ �1 in phase 3. In the case of bare particles,

whose position at the interface is given by contact angle

considerations, our method still applies.

The interaction energy of the composite system is derived

using the procedure outlined in Section 2.2, modified to take into

account the 2D-lattice of particles straddling the interface. As

sketched in Fig. 3, the total interaction energy of the system is

given by the sum of the interaction energies of sub-systems

interacting through medium 3, namely:

I. The interaction energy of the same system with holes in place

of the interfacial particles (see the first row of Fig. 3). This is given

by: twice the energy of one bi-material 2D-lattice [i.e. composed of

spheres made up half of material 1 (the drop phase) and half of

material 3 (the thin film of matrix phase)] interacting with a half

space of material 1 subtracted from the interaction energy of two

half spaces. We also need to add to the above the interaction

energy of two facing 2D-lattices of bi-material particles, for

reasons similar to those explained in Section 2.2 point I.

II. Twice the interaction energy of the bi-material 2D-lattice

interacting with the 2D-lattice of gold particles subtracted from

Fig. 3 Model decomposition of a system with interfacial particles. The

distance between the two interfaces is D. Top row, interaction energy of

the same systemwith holes in place of the interfacial particles, from left to

right: two interacting 2D-lattices of bi-material particles—twice a 2D-

lattice of bi-material particles interacting with a half space of material 1 in

a medium of material 3 + two half spaces of material 1 interacting across

material 3. Bottom row, from left to right: twice a 2D-lattice of material 2

particles interacting with a 2D-lattice of bi-material particles + twice

a 2D-lattice of particles of material 2 interacting with a half space of

material 1 across medium 3 + two interacting 2D-lattices of material 2.

5218 | Soft Matter, 2012, 8, 5214–5224

twice the energy of a 2D-lattice of gold particles interacting with

the facing half space;

III. The interaction energy of the two facing gold lattices.

The decomposition of the system in the single constituents is

summarized in the following equation:

Eint(D) ¼ �2El,lbi-material,2(D) + El,l

bi-material,bi-material(D)

+ A22El,l(D) � 2A131E

Tsl,hs(D) + 2A231El,hs(D) + A131E

hs,hs(D) (9)

where Tsl stands for a 2D-lattice of truncated spheres. In fact, as

can be noticed from Fig. 3, the bi-material particles interacting in

medium 3 with a half space (the first sketch in the second row) are

simply equivalent to a lattice of truncated spheres (in the picture

hemispheres are represented) of material 1 interacting in medium

3. Eqn (9) can also be derived rigorously by assuming that the

interface is a separate composite phase, as described in

Appendix, and applying eqn (4) to include the intervening

medium.

We were not able to determine simple analytical expressions to

evaluate the first and second term of eqn (9). Thus, we assume

that the bi-material particles are all made of either material 1 or

material 3. In this case, the interaction energy of the two bi-

material particle lattices lies between the interaction energies of

these two extreme cases; thus, the error associated with these two

approximations can be estimated. If we assume that the particles

are all made of material 1, the first three terms of eqn (9) are

simplified, via eqn (4), to the interaction of two facing 2D-lattices

of material 2 spheres interacting through material 1. Thus, we

obtain:

Eint(D) ¼ A212El,l(D) � 2A131E

Tsl,hs(D) + 2A231El,hs(D)

+ A131Ehs,hs(D). (10)

By substituting the analytic interaction energies (derived in the

next section) in eqn (10) and taking its negative gradient, we

derive the following disjoining pressure expression for interfacial

particles:

Pl;lðDÞ ¼ �A212r2lp

3

�Dþ 2gRp

�ln

�Dþ 2gRp

�2�4R2p�

Dþ 2gRp

�2!

þ4R2

p

��Dþ 2gRp

�2�2R2p

��Dþ 2gRp

���Dþ 2gRp

�2�4R2p

�!

� A231rl

3

Rp�

Dþ Rpðg� 1Þ�2 þ Rp�Dþ Rpðgþ 1Þ�2

� 2R2p�

Dþ Rpðg� 1Þ��Dþ Rpðgþ 1Þ�!

� A131rl

3

R3

pDð1þ gÞ2ðg� 2Þ þ Rpðg2 � 1ÞD3�Dþ Rpð1þ gÞ�2

!

� A131

6pD3:

(11)

Eqn (11) is the main result of our study. It can be used both in

numerical codes to evaluate the influence of interfacial particles

in inhibiting coalescence or for simple estimates in experiments.

As in the previous section Pl,l for the interfacial particle system

This journal is ª The Royal Society of Chemistry 2012

interacting in void is obtained from eqn (11), making the

substitution A231 / A21, A212 / A22 and A131 / A11.

Fig. 4 The reference system for the integration of the pair-potential of

the reference particle, positioned at z ¼ 0, with the 2D-lattice of spheres

belonging to the facing layer.

2.5. Derivation of El,hs, El,l, and ETsl,hs

In the following section, we derive the analytical expressions for

the interaction energies of the sub-systems, necessary to evaluate

the system total energy in eqn (7) and (10), unavailable in the

literature, namely: a 2D-lattice of spheres interacting with a half

space El,hs, two facing 2D-lattices of spheres El,l, and a 2D-lattice

of truncated spheres interacting with a half space ETsl,hs.

Numerical simulations were also employed to consider the

discrete nature of the problem and the effect of different spatial

arrangements. The numerical values were also compared with the

analytical formulae eqn (8) and (11) to evaluate their range of

validity.

The derived analytical expressions of the sub-system energies,

together with Es,s32,45 already available in the literature, are

summarized in Table 1.

2.5.1. Analytical method. The analytical expression of El,l is

here derived by integrating the well-known interaction potential

of two spheres Es,s,32,45 reported in Table 1. To do so, we start by

calculating the interaction energy of one sphere with a 2D lattice

of spheres, where the facing sphere is positioned at a centre-to-

centre distance z ¼ D (see Fig. 4), through the Hamaker pairwise

integration:24

Es;lðzÞ ¼ 2p

ðN0

E s;sðrÞrlx d x; (12)

where r ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ z2

pis the particle centre-to-centre distance. The

integral approach is limited to separation distances bigger than

the interparticle distance.55

The integration is finite since at large distances the asymptotic

form of Es,s(r) is like the interaction between two molecules,

fr�6.24 In the present case the retardation due to the finite speed

of light, usually present for distances above 10 nm,33 was

Table 1 Interaction potentials used in the paper. The energies are non-dime

E(z) D

Ehs;hs ¼ � 1

12pz2Intw

Es;s ¼ � 1

6

2R2

p

z2 � 4R2p

þ R2p

z2þ ln

z2 � 4R2

p

z2

!!T

El;l ¼ rlEs;l ¼ �p

6rl

2

4R2

p þ z2 � 2R2p

� �ln

z2 � 4R2

p

z2

!!To

El;hs ¼ rlEs;hs ¼ � rl

6

�Rp

z� Rp

þ Rp

zþ Rp

þ ln

�z� Rp

zþ Rp

Ab

ETsl;hs ¼ rlETs;hs

¼ � rl

12

�2Rp

zþ Rpð1þ gÞ þ�2Rpzgþ Rpð�1þ g2Þ

z2

þ 2ln

�z

zþ Rpð1þ gÞ

AthT

This journal is ª The Royal Society of Chemistry 2012

neglected for the sake of simplicity, but it could be incorporated

by modifying the Hamaker ‘‘constant’’ with a spatially varying

correction factor.26,31,44,45,56 Retardation will only make the

calculation cumbersome, but the integral (12) remains finite. The

final result is:

Es;l ¼ �p

6rl

4R2

p þ�z2 � 2R2

p

�ln

z2 � 4R2

p

z2

!!: (13)

The interaction energy of two facing 2D-lattices of particles

per unit area is simply: El,l ¼ rlEs,l. This is true when the two

layers have the same lattice constant and are symmetrical or

stagger, so that each particle in the layer has the same interaction

energy with the facing layer. The same is true for an array of

particles interacting with a half space, i.e. El,hs ¼ rlEs,hs, where

Es,hs is the interaction energy of a sphere with a half space,32

reported in Table 1.

ETsl,hs(D), i.e. the interaction of a 2D-lattice of truncated

spheres with a half space, is calculated through a volume inte-

gration of a molecule/half space interaction potential24 inside the

truncated sphere domain:

nsionalized by Ah; rl is the 2D particle density

escription

teraction of two half spaces per unit area: z is the distance between theo interfaces.

wo spheres: z is the centre-to-centre distance.

wo facing 2D-lattices of spheres: z is the centre-to-centre distance of twopposite spheres.

2D-lattice of spheres interacting with a half space: z is the distanceetween the centre of the sphere and the half space interface.

2D-lattice of truncated spheres interacting with an infinite half space: z ise distance between the centre of the sphere and the half space interface.he particles are immersed in phase 3 of a distance gRp.

Soft Matter, 2012, 8, 5214–5224 | 5219

Fig. 5 Dimensionless P between two facing 2D-lattices of spheres

arranged in a square lattice, for different a ¼ [2.4; 3; 4; 6; 12]Rp. The

dotted curves are for the stagger configuration, the solid curves are for

symmetric layers.

ETsl;hsðDÞ ¼ � rl

6

ðz¼2Rp

z¼R1

z�2Rp � z

�ðD� R1 þ zÞ3 dz; (14)

whereR1¼Rp(1� g) is the particle immersion in phase 3, e.g. for

hemispheres, R1 ¼ Rp. The resulting analytical formula is given

in Table 1.

Analytic expressions for two interacting 2D-lattices of hemi-

spheres or truncated spheres were not derived, because the

interaction potential depends on the relative position of the

spheres, making the integration analytically intractable. Similar

complications were experienced by Vold57 for the determination

of the analytical interaction potential of two anisotropic particles

via the Hamaker approach.

2.5.2. Pairwise summation. The discrete nature of the

problem was taken into account using a direct pairwise

summation of the interaction energy per unit area of two facing

2D-lattices of spheres positioned on a simple square lattice.

Using Matlab7, the total interaction energy of the system was

calculated for each D, and P was obtained by taking its

numerical gradient. All length-scales were non-dimensionalized

by Rp and the energies by Ah. In our model, for fixed particle

geometry, the lattice constant a (or a* ¼ a/Rp in dimensionless

terms) is a function of the ligand length and the degree of

interpenetration of the ligands of two adjacent particles.

The total interaction energy of the system per unit area was

obtained as the product of the particle area density in one

interface and the total interaction energy of a reference particle

with the opposite lattice. The latter was obtained by direct

pairwise summation of the interparticle potentials, i.e. Es,s, of the

reference particle with each particle in the 2D facing lattice

contained within the cut-off radius Rc; the remaining interaction

up to infinity was calculated by integrating eqn (12) with Rc as

the lower limit of integration and it was added to the result of the

direct pairwise summation. This method to calculate the total

interaction energy is similar to what is usually done in molecular

dynamic calculations of the total energy of a molecular system.49

Rc was chosen to contain 100 � 100 particles, so that the inte-

gration was an accurate approximation of the sum; 120 � 120

and 50� 50 systems were also tested, with no appreciable change

in the results. The deviation of the analytical formula (16) from

the numerical simulations was evaluated. Additionally, the effect

of the different relative position of the two facing grids (i.e. in

register vs. stagger lattices) was probed by changing the relative

position of the reference particle with respect to the facing lattice.

3. Results and discussion

3.1. Pl,l for two facing 2D-lattices and multilayers of SC

spheres–analytic expressions versus numeric simulations

In this section, Pl,l for two facing 2D-lattices of spheres inter-

acting in void is calculated by both the analytical formula, i.e. the

first bracket in eqn (11) with A232 / A22 and D ¼ z, and the

discrete numerical approach. Since the interaction energies are

non-dimensionalized by Ah, the effect of geometry is here tested

regardless of the type of the interacting media. Fig. 5 shows for

different a the numerical simulations of the dimensionless Pl,l

5220 | Soft Matter, 2012, 8, 5214–5224

between two facing 2D-lattices of SC spheres for the two extreme

cases of symmetric (head-on) and stagger configurations; in the

latter configuration, the lattice site is positioned in the centre of

the facing unit cell. For each lattice spacing, Pl,l in the stagger

configuration case is several orders of magnitude smaller than in

the in register configuration for z < a, while they are coincident

otherwise (see Fig. 5).

It can also be noted that, for a $ 4Rp, Pl,l in the stagger

configuration is finite as z / 0, because the two interfaces can

interpenetrate when the lattice spacing is greater than the particle

diameter. Moreover, Fig. 5 implies that the interaction energy

per unit area in the stagger configuration is higher than in the

head-on configuration for z < a. This implies that the head-on

configuration is more stable than the stagger one (unlike the case

of ionic particles or vertical dipoles, which show the opposite

behaviour24), in agreement with the study of ref. 36 on the self-

assembly of nano-colloids. Thus, the system in a stagger

configuration tends to restore the head-on symmetric configu-

ration, with a force on the xy plane, given by the negative surface

gradient of the total interaction energy that depends on the

relative position of the two lattices (see Fig. S1 of the ESI†, where

we show the vector plot of the force experienced on a particle,

occupying various positions in the unit cell relative to the facing

lattice).

The maximum lateral force on two gold lattices interacting

across PDMS is reported in Fig. 6 as a function of z for different

lattice spacing. The force magnitude is of the order of pN only

when z is of the order of nanometres. In this case, it might be

comparable to other surface forces acting parallel to the inter-

face, such as the viscous forces. For example, if we consider two

coalescing drops, the order of magnitude of the viscous forces

can be estimated from scaling arguments27 as:

s ¼ mm

up

D¼ mm

�Dp=af

�D2

mmD¼ D

ðsþ RPÞafR

(15)

where af z RCa0.5 is the film radius, mm is the film viscosity, i.e.

the matrix viscosity, up is the parabolic part of the velocity

evaluated at the centre of the thin film, and Dp is the pressure

drop between the film pressure (the sum of the disjoining pressure

and the capillary pressure � s/R) and the pressure outside the

This journal is ª The Royal Society of Chemistry 2012

Fig. 6 Maximum lateral force per gold particle for different a: a ¼ 3Rp

(solid curves); a ¼ 4Rp (dashed curves); a ¼ 12Rp (dotted curves), Rp ¼1.5 nm and AAu/PDMS/Au ¼ 2.99 � 10�19 J.

thin film, which is assumed to be zero. From eqn (15), the order

of magnitude of the viscous force acting on each particle can be

estimated as: Fvisc ¼ sR2p ¼ (s + RP)R2

pDR�2Ca�0.5.

Our analysis suggests that, since the head-on configuration

is the preferred configuration, coalescence stability might be

favoured. Indeed, the actual fusion of the two drops happens

when the facing fluid–fluid interfaces meet, but this might not

be possible due to steric hindrance of the two facing particles,

that keep the interfaces at a distance of 2Rp. Thus, in the

head-on configuration, flocculation might be favoured, unless

interfacial instabilities arise. On the other hand, if the drops

approach when the particles are in a stagger configuration,

there is the possibility that the two lattices interpenetrate. This

can lead to drop fusion, unless particle-induced bridging

occurs.1,58

Fig. 7 also showsPl,l for multilayer systems as a function of the

number of layers in each body. Note that the slope of Pl,l

increases considerably for the first three added layers. It is clear

that the analytical formula, eqn (8), matches numerical

Fig. 7 Dimensionless P between two 2D square lattices of spheres with

stagger and in-register configurations (solid curves) and a multilayer

system made of an SC lattice of spheres (dotted curves);N ¼ [2; 6; 10; 40]

is the number of layers. The dashed curve is the interaction of two infinite

half spaces per unit area. The analytical and numerical calculations agree

after a distance comparable with a.

This journal is ª The Royal Society of Chemistry 2012

simulations very well even for multilayers. Thus, the possibility

of using the analytical formula results in a great reduction in

computational costs.

Additionally, we found that the analytical expression of Pl,l

agrees very well with the mean interaction energy (see Fig. 7),

where the arithmetic mean is taken over all the possible positions

of the reference particle with respect to the facing lattice.

Moreover, they both match the numerical pairwise summation

data for z $ a.

3.2. Two interacting half spaces with SC lattices of spheres as

inclusions—comparison with EMT

The system studied here is composed of two composite half

spaces of PBd (n1 ¼ 1.5, 31 ¼ n12 ¼ 2.25), with 3D simple cubic

lattices of gold spheres as inclusions, interacting across void.

Fig. 8 shows P evaluated by eqn (8) with D ¼ z compared to P

evaluated by eqn (1), where the value of Ah was estimated using

the effective dielectric functions of the composites, as described

in Section 2.3. Similar calculations for the same system inter-

acting across PDMS (n3 ¼ 1.4, 33 ¼ n32 ¼ 1.96) and for systems

with dielectric particles interacting both in void and PDMS are

reported for completeness in Fig. S2–S6 of the ESI†. The

parameters needed to calculate the Hamaker constant of the

PBd/PDMS/PBd system are given in ref. 22.

In the case of dielectric particles, with dielectric constants

intermediate (3p2 ¼ 2.1) or greater (3p3 ¼ 6) than those of the

bulk phases, the pair additivity assumption works remarkably

well in void (Fig. S2 and S3 of the ESI†) and in a medium up

to a minimum lattice spacing amin ¼ 2.2Rp (see Fig. S4 and S5

of the ESI†), for distances bigger than O(10*Rp). Thus, the

present method, which takes into account the geometry of the

problem, agrees very well with the EMT approach, which, on

the contrary, considers a unit cell with effective properties

smeared out in the cell. In the case of metal particles in void

(Fig. 8), the two methods are almost superimposable up to

amin ¼ 2.4Rp, using AAu/PBd ¼ 4 � 10�19 J (determined as

Fig. 8 P (expressed in J)between twohalf spaces ofPBd interacting across

void, with Au particles as inclusions arranged in an SC lattice, a ¼ [6Rp;

4Rp; 3Rp; 2.4Rp], the solid curves are obtainedusing theEMTapproach, the

dashed dotted curves are obtained using the proposed approach; the same

system, with no inclusions (dotted curves); and two infinite half spaces of

gold interacting across void (dashed curves). AAu/PBd ¼ 4 � 10�19 J was

determined by fitting our method with EMT results.

Soft Matter, 2012, 8, 5214–5224 | 5221

Fig. 10 Effect of particles radius Rp ¼ [1.5 nm; 5 nm; 10 nm; 20 nm; 50

a fitting parameter due to the lack of data in the literature).

This is surprising considering that the additivity approach is

not a good approximation24 for a high dielectric mismatch. The

discrepancy for smaller a is obvious, i.e. the composite material

becomes a conductor with a static dielectric constant infinite in

value. Moreover, the same system interacting across PDMS,

using AAu/PDMS/PBd ¼ 9.69 � 10�20 J(determined as a fitting

parameter) coincides with the EMT method up to a ¼ 3Rp (see

Fig. S6 of the ESI†). Additionally, we note that in our model

and in effective medium theories it is not taken into account the

presence of a metal-to-insulator transition, found in nano-

particle thin films when the interparticle distance is 5 �A29,48,59

and that for distances smaller than 12 �A, quantum effects start

to appear.

The present model gave very positive results compared to the

EMT approach, despite the approximations used.

nm] (solid curves), for a* ¼ 4; clean interface system (PBd–PDMS–PBd)

(dashed dotted curve).

3.3. P for particle stabilized thin films

Eqn (11) with g ¼ 0 and D ¼ z is employed to estimate Pl,l in

a system composed of two PBd drops with polymer coated

Janus gold nanoparticles (having PBd and PDMS ligands in

equal amount) at their interfaces interacting across a PDMS

matrix; this is the system also studied experimentally by

Borrell and Leal.10 The Hamaker constants necessary to

evaluate eqn (11) are: AAu/PDMS/PBd ¼ 9.69 � 10�20 J evaluated

by the method outlined in the previous section, AAu/PBd/Au ¼2.99 � 10�19 J calculated with Bargeman’s method46 and

APBd/PDMS/PBd ¼ 3 � 10�21 J from ref. 10. In Fig. 9 the effect of

particle concentration is presented at fixed Rp ¼ 1.5 nm. The

particle concentration is changed from a ¼ 6Rp to a ¼ 2.4Rp.

The VDW interactions are clearly enhanced by the presence of

this thin layer of interfacial gold nanoparticles. Despite this, for

z ¼ 50 nm there is an almost two order of magnitude difference

between the disjoining pressure of a system composed of half

spaces of gold interacting in PDMS and one with interfacial

Fig. 9 Effect of particle concentration. Two infinite half spaces of PBd

interacting across PDMS with Au particles (Rp ¼ 1.5 nm) at the inter-

faces, for a* ¼ [8; 6; 4; 3; 2.4] (solid curves); two half spaces of Au

interacting across PDMS (dashed curves); composite five-layer system

(see Appendix) composed of two PBd half spaces with two thin Au layers

(each 3 nm thick), interacting across PDMS (dotted curves); and two PBd

half spaces interacting across PDMS (dash dotted curves).

5222 | Soft Matter, 2012, 8, 5214–5224

particles with a ¼ 2.4Rp. This difference increases dramatically

with increasing D, making Pl,l closer to the one of a five layer

system.

The effect of particle size is reported in Fig. 10, where Rp is

changed from 1.5 nm to 50 nm at fixed a ¼ 4Rp. The effect of

particle size is dramatic in enhancing the VDW interactions in

the film. However, when considering bigger particles, the

question is to what extent is the continuum approach appli-

cable to describe coalescence? As mentioned in the Introduc-

tion, if the particle size is of the same order of D or af,

a discrete description of the interface might be necessary. Given

the complexity of the coalescence process, further studies,

incorporating eqn (11) in numerical codes to simulate coales-

cence, are needed in order to estimate the impact of metal

particle concentration and radii.

Applying our method to a specific system enabled us to

estimate the error introduced by the approximation used to

calculate eqn (11), as mentioned in Section 2.4. In the present

case, as shown in Fig. 11 and 12, the relative error (E2 � E1)/E2

of the two extreme cases, i.e. ghost particles composed all of

PBd (E1) and ghost particles composed all of PDMS (E2), is

very small (<10%) both when changing concentration and

particle radius.

Fig. 11 Relative error ¼ (E2 � E1)/E2 changing a*, with a* ¼ [8; 6; 4; 3;

2.4] and Rp ¼ 1.5 nm.

This journal is ª The Royal Society of Chemistry 2012

Fig. 12 Relative error¼ (E2 � E1)/E2 changing Rp, with Rp ¼ [1.5 nm; 5

nm; 10 nm; 20 nm; 50 nm] and a* ¼ 4.

Fig. 13 (a) Five-layer model and its decomposition to incorporate

medium 3 using the pair additivity assumption. 2 is the coating layer and

(b) when 2 is substituted by the interfacial layer of particles, which is

a composite system itself, the overall system is obtained.

4. Summary and conclusions

Numerical and analytical expressions were derived to evaluate

the magnitude of the disjoining pressure per unit area of two half

spaces interacting across a medium, with particles at their

interfaces (function of the particle radius, interparticle distance

and the particle degree of immersion in the dispersed phase) or

dispersed in the bulk in a periodic array. The model, based on the

pair additivity of the interaction potentials of the constituents,

better applies to polymeric systems with metallic or dielectric

particles, or systems not containing water or strong dipoles. The

analytical expressions can be easily incorporated in numerical

simulations of drop coalescence with interfacial particles, as long

as the lubrication thin film radius is greater than the particle

dimensions or lattice constant and where curvature effects can be

neglected.

Our method was compared to the EMT approach in the case of

bulk materials with periodic inclusions, where the EMT is well

established for both conducting and dielectric particles. In this

way, the validity of the pair-additivity approach was thus inves-

tigated, regardless of the lack of experimental data. A very good

agreement between the two approaches was found for dielectric

particles in void and in a medium, for lattice spacing as small as

2.2Rp, i.e. almost touching spheres. For composites with inclu-

sions of SC lattices of metal spheres, the metal/polymer/polymer

Hamaker constant was estimated as a fitting parameter, due to the

lack of experimental or theoretical values for the system of

interest. Its value, consistent with other metal/polymer systems,

was later used to evaluate P of films with interfacial metal

This journal is ª The Royal Society of Chemistry 2012

particles. The fitting procedure showed a good agreement with the

EMT results for lattice spacings up to a ¼ 2.4Rp.

The effect of different spatial arrangements was also investi-

gated using direct pairwise summation. In the case of two facing

2D-lattices of particles in register configuration, it was found that

|P| is several orders of magnitude higher than the stagger

configuration, for separation distances comparable to the lattice

spacing. The difference in energy between stagger and in register

configurations also causes a lateral force parallel to the interface

which tends to move the particles from a stagger position to

a symmetric position with respect to the facing lattice. When the

distance between the two interfaces is of the order of the inter-

particle distance, the lateral force is O(pN) and might compete

with hydrodynamic forces, which also act on the particles during

the drainage process.

When the two interfaces are at a distance comparable or bigger

than the lattice spacing, there is a very good agreement between

the analytical expression and the numerical values. This enables

the use of the analytic formula in simulations of interacting

monolayers for coalescence studies or of interacting multilayers,

which would otherwise be computationally intensive. Examples

of multilayer systems are self-assembly materials, like nano-

crystals, which can provide a simple experimental validation for

our method.

Our method, applied to two facing half spaces of PBd inter-

acting across PDMS with gold particles at their interfaces

(system studied in the coalescence experiments of Borrell and

Leal10), showed the dramatic effect of particle radius and

concentration on P. Although the soft corona stabilizing the

particles can decrease considerably this effect, by limiting the

particle concentration, it cannot be disregarded to describe

correctly and quantitatively coalescence. Additionally, the rela-

tive error in eqn (11), which should be checked for each system of

interest, was calculated to be very small in this case.

The proposed method is general and helpful for a quick

determination of the magnitude of VDW interactions. For

nanocomposite materials, it can be easily improved using a more

accurate description of the nanoparticle dielectric function, or

experimentally measuredAh, or through the use of more accurate

interaction potential models, for example, by including

retardation.

Appendix

Fig. 13a shows the composite system C, made up of an infinite

half space of material 1 with a coating layer of material 2 and

thickness T, interacting across medium 3. Recalling eqn (4), the

interaction energy is:

EtotC3C(D) ¼ Ehs,hs

CC (D) + 2Ehs,hs33 (D) � 2Ehs,hs

3C (D). (A1)

The total interaction was further broken down into the single

component interactions:

EtotC3C(D) ¼ Elayer,layer

22 (D) + 2Elayer,hs21 (D + T) + Ehs,hs

11 (D + 2T)

¼ � 2Ehs,hs31 (D + T) � 2Elayer,ha

23 (D) + Ehs,hs33 (D), (A2)

where the superscript ‘‘layer’’ stands for coating layer, (i.e.

Elayer,layer22 is the interaction energy between two coating layers of

Soft Matter, 2012, 8, 5214–5224 | 5223

material 2 in void). If the coating layer is of a uniform compo-

sition, using the combining relations for the Hamaker constants,

eqn (A2) becomes:

EðDÞ ¼ � 1

12p

A232

D2� 2A123

ðDþ TÞ2 þA121

ðDþ 2TÞ2!: (A3)

Eqn (A3) is commonly used in colloidal science to evaluate the

stability of particles sterically stabilized by polymers.24 The

interfacial particles can be considered as a coating layer as shown

in Fig. 13b.

Acknowledgements

I acknowledge the partial financial support of NSF grant #

0624446 during 2007, L. Gary Leal, who wanted me to work on

the disjoining pressure problem in the presence of interfacial

particles in order to interpret his group’s experimental findings,

within my PhD in the framework Nanoparticles as Surfactant of

the IRG3 project of the MSERC centre at UCSB, Jacob Israel-

achvili’s classes for the clarity to derive well known interaction

potentials through integration, and the written private commu-

nication of Antonio Redondo, which was useful in the very early

stages of this work. This work was presented at the GRC on

Physics and Chemistry of Microfluidics 29 June–3 July 2009,

Barga, Italy, and won the Soft Matter Poster Prize at the 7th

AERC 10–14 May 2011 Suzdal, Russia.

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