Supporting Information
Generalizing Hildebrand solubility parameter theory to apply to one and twodimensional solutes and to incorporate dipolar interactions
J. Marguerite Hughes, Damian Aherne and Jonathan N Coleman*
School of Physics and CRANN, Trinity College Dublin, Dublin 2, Ireland.
*[email protected]
S1
Testing the accuracy of the lattice model
The lattice model is the simplest possible way to calculate the energetics of liquids.
However, its simplicity is only an advantage if it is reasonably accurate. It has a number of
elements which might be considered too simple. For example, the idea of placing molecules
on lattice sites was disliked by Hildebrand1,2 as he considered it a poor reflection of reality. In
addition, lattice models consider only nearest neighbour interactions which may result in loss
of precision. We can assess the accuracy of a lattice model by comparing its predictions to
those obtained using a slightly more sophisticated method of calculation. Here we calculate
the cohesive energy density for a liquid using a lattice model and compare it to the result
found by integrating over the interaction between all pairs of molecules.
The lattice model
In a lattice model, the cohesive energy density can be found from the sum of the
binding energies between all pairs of nearest neighbour lattice sites divided by the volume of
the system:
EC 
NzS  1
z
 S
2 NvS 2vS
(S1)
Here N is the total number of lattice sites (molecules), z S is the number of nearest
neighbours per liquid lattice site (=6 in a cubic lattice model),  S is the volume per lattice
site (molecular volume) and  is the intersite binding energy. This binding energy can be
thought of as the energy minimum of the pairwise intermolecular potential energy function
(see below).
Integration over all atom pairs
We can calculate the cohesive energy using a method first described by Hildebrand.2
We will study a system of N molecules in a liquid of volume V . Consider a reference
molecule. The potential energy of this molecule is the sum of the potential energies due to all
1
of the pairwise interactions with all other atoms in the liquid. We will assume the molecules
are small and we can model them as hard spheres. The number of other molecules with
centres in a spherical shell with inner radius r and outer radius r  dr is
dN  4 r 2 dr
N
g (r )
V
(S2)
where g (r ) is the radial distribution function. This function approaches 1 for large r but
deviates from 1 at small r to describe the local structure of the liquid near the reference
molecule. We can work out the potential energy of the reference molecule by multiplying
dN by the potential energy of interaction,  , of the reference molecule with each molecule
between r and r  dr and integrating over all space (all atoms). Furthermore, we can extend
this to the potential energy of all atoms by multiplying by N / 2 where the factor of 2 is to
avoid double counting. Dividing by V gives the total interaction energy per volume i.e. the
cohesive energy density:

EC 
N
 dN 2V
(S3)
r d
Here the lower limit of integration is d , the molecular diameter. This is because, if we treat
the molecules as hard spheres, the shortest possible distance between the centre of any
molecule and the centre of the reference molecule is d .
We can approximate the intermolecular potential as the van der Waals potential
(ignoring the repulsive part):
d 
    
r
6
(S4)
In this scenario, for two nearest neighbour molecules (separated by a molecular diameter, d ),
the potential is  , i.e. the intermolecular separation is defined by the bottom of the potential
well.
This means we can write the cohesive energy density as

EC  

r d
d 
 
6
N
N
N
 
   4 r 2 dr g (r )
 2 d 6  
r
V
2V
V
2 

d
g (r )dr
r4
(S5)
3
4 d 
We can make the approximation that the liquid volume is given by V  N    leading
3 2
to
EC  
72



d
g (r )dr
r4
(S6)
2
For simplicity we can approximate g ( r )  1 , and approximating vS 
EC  
4 d 3
( ) we get
3 2
4
vS
(S7)
which is within a numerical factor (of order 1) of the lattice model result in equation S1.
This result clearly shows that while the lattice model is crude, it gives results that are
very close to those found using more sophisticated models. Thus we believe that its
simplicity is a significant advantage and more than justifies its use here.
S2
Applying Hildebrand parameters to 2-dimensional platelets
We can use the lattice model to calculate the enthalpy of mixing of platelets in a
solvent. We consider the initial state of the system as N2D identical platelets stacked to form a
single crystal and a large volume of solvent. We assume N2D is very large, and that the
platelets are big enough that the interactions between edges and solvent can be ignored. We
consider each platelet as divided into n×n×1 sites arranged in a cubic lattice. Thus the total
number of platelet lattice sites is n 2 N 2 D . Within the crystal, each platelet has z2D =2
neighbours. The solvent, meanwhile, consists of NS molecules arranged in a cubic lattice of
NS sites.
The final state is a uniform mixture of the two components consisting of
N  n 2 N 2D  N S lattice sites in a cubic arrangement. The overall volume of the mixture is
expressed as V  vS N  vS (n 2 N 2 D  N S ) where vS is the lattice unit (or solvent molecule)
volume. From this, we can also see that the volume fraction of platelets in the mixture is
given by 2 D 
v N
vS n 2 N 2 D
, and the volume fraction of solvent by S  S S .
V
V
The enthalpy of mixing is defined as the enthalpy change going from the initial state
of the system to the mixed state. We can compute H Mix as the sum of 4 distinct terms:
H Mix  H f ( S )  H f (2 D )  H i ( S )  H i (2 D )
(S8)
each of which we will consider in turn.
H i (2 D ) represents the energy required to separate all of the platelets to infinite
separation and can be thought of as the cohesive energy of the crystal:
H i (2 D ) 
n 2 N 2 D z2 D
 2 D 2 D
2
(S9)
3
where the factor of 2 is to avoid double counting, z2D =2 is the number of van der Waals
bonded nearest neighbours per platelet lattice site and  2 D 2 D is the binding energy at the
interface between 2 platelet lattice sites. We take  2 D 2 D (and all other intersite energies) to
be positive.
H i ( S ) represents the energy required to separate all of the solvent molecules to infinite
separation and can be thought of as the cohesive energy of the solvent:
Hi(S ) 
N S zS
 S S
2
(S10)
where z S =6 is the number of nearest solvent neighbours per lattice site,  S  S is the binding
energy at the interface between 2 solvent lattice sites and the factor of 2 accounts for double
counting.
Next we consider the energy released on condensation of solvent and platelets from
infinity to form the final mixture of platelets within the lattice. We split this into two terms
relating to platelets ( H f (2 D ) ) and solvent molecules ( H f ( S ) ). To calculate the first term, we
consider that each platelet will be surrounded by a combination of other platelets and solvent
molecules, the average proportions of which are determined by their respective volume
fractions 2D and S , where 2 D  S  1 :
H f (2 D )  
n 2 N 2 D z2 D
( 2 D 2 D2 D   S 2 DS )
2
(S11)
Similarly, the environment experienced by an average solvent molecule is also
dependent on both S and 2D . However, in this case there is an additional constraint. Whilst
the number of nearest neighbours any solvent molecule has is z S (=6), only a maximum of
z2D (=2) of those may ever be occupied by platelets, due to the platelet dimensions. This is
illustrated in the schematic S1.
In other words, ( zS  z2 D ) neighbouring sites will always be occupied by solvent
molecules, and z2D neighbouring sites by a combination of platelets and solvent molecules
(the average proportions of which are given by S and 2D ). Hence we can write
H f (S )  
N S ( z S  z2 D )
N z
 S  S  S 2 D ( S  SS   S 2 D2 D )
2
2
(S12)
Summing all of these terms gives us
4
N S ( z S  z2 D )
N S z2 D
n 2 N 2 D z2 D
H mix  
 S S 
( S  SS   S  2 D2 D ) 
( 2 D 2 D2 D   S 2 DS )
2
2
2
N z
n 2 N 2 D z2 D
 S S  S S 
 2 D2 D
2
2
which rearranges to
H mix 
N S z2 D
n 2 N 2 D z2 D
( S  S   S  SS   S 2 D2 D ) 
( 2 D 2 D2 D   S 2 DS   2 D 2 D )
2
2
and may be further simplified to
H mix 
N S z2 D
n 2 N 2 D z2 D
( S  S (1  S )   S 2 D2 D ) 
( 2 D 2 D (1  2 D )   S 2 DS )
2
2
Letting   2 D  (1  S ) yields
H mix 
N S z2 D
n 2 N 2 D z2 D
( )( S  S   S 2 D ) 
(1   )( 2 D 2 D   S 2 D )
2
2
Finally, if we remember that   2 D 
H mix 
v N
vS n 2 N 2 D
, and 1    S  S S , we find that
V
V
z2 D V
z V
(1   )( )( S  S   S  2 D )  2 D
( )(1   )( 2 D  2 D   S  2 D )
2 vS
2 vS
This means the enthalpy of mixing per unit volume is given by:
z2 D
 H Mix 
 V   2v  (1   )  S S   2 D 2 D  2 S 2 D 

2 D
S
(S13)
We note that this is very similar to the standard expression for enthalpy of mixing for
small molecule mixtures:
H Mix z0 D

 (1   )   S  S   N  N  2 S  N 
V
2vS
(S14)
We can express equation S13 in terms of Hildebrand solubility parameters by noting
that this parameter is defined as the square root of the cohesive energy density. This allows us
to write the cohesive energy density of the solvent as
EC , S 
zS  S  S
  T2, S
2 vS
(S15)
where the factor of 2 is to avoid double counting. The cohesive energy of the initial platelet
crystal can be written as
EC ,2 D 
z2 D n 2 N 2 D 2 D  2 D z2 D  2 D  2 D

2
n 2 N 2 D vS
2
vS
(S16)
5
We can use this expression to define the Hildebrand solubility parameter of platelets
to be
z2 D  2 D  2 D z2 D 2

T ,2 D
2 vS
zS
EC ,2 D 
(S17)
We note that this new definition is allowed because the significant difference in geometry
between a platelet and a solvent molecule which occupies a single lattice site makes using the
same definition in each case inappropriate.
If we assume that the interactions between all lattice sites are dominated by the
London interaction, we can use the geometric mean approximation (see below) to estimate
 S 2 D :
 S 2 D   S S  2 D2 D 
2vS
 T , S  T ,2 D
zS
(S18)
Inserting these expressions into equation S14 gives
2
2
z2 D
1
 H Mix 
 V   z  (1   ) T ,S  T ,2 D   3  (1   ) T ,S  T ,2 D 

2 D
S
(S19)
This is very similar to the standard expression
2
H Mix
  (1   )  T , S   T , N  .
V
(S20)
for solute N .
S3
Applying Hildebrand parameters to 1-dimensional rods
We can follow a similar procedure to derive the enthalpy of mixing for rigid rods in a
solvent.
We consider the initial state of the system as N1D identical rods cubic close packed to
form a single crystal and a separate large volume of solvent. We assume N1D is very large.
We consider each rod as divided into n×1×1 sites arranged linearly. Thus the total number of
platelet lattice sites is nN1D . Within the crystal, each rod has z1D (=4) nearest neighbours. The
solvent consists of N S molecules arranged in a cubic lattice of N S sites. The final state is a
uniform mixture of the two components consisting of nN1D  N S lattice sites in a cubic
arrangement.
As before, we can divide H Mix into 4 distinct terms, using equation S8:
H Mix  H f ( S )  H f (1D )  H i ( S )  H i (1D )
6
H i (1D ) represents the energy required to separate all of the rods to infinite separation
and can be thought of as the cohesive energy of the crystal:
H i (2 D ) 
nN1D z1D
1D 1D
2
(S21)
where 1D 1D is the binding energy at the interface between two rod lattice sites and the 2 is to
avoid double counting.
As before, H i ( S ) represents the energy required to separate all of the solvent
molecules to infinite separation and can be thought of as the cohesive energy of the solvent:
Hi(S ) 
N S zS
 S S
2
(S10)
where z S (=6) is the number of nearest neighbours per solvent lattice site,  S  S is the binding
energy at the interface between two solvent lattice sites and the factor of 2 accounts for
double counting.
The energy released going from the state of infinite separation to the final state of
rods and solvent mixed within the lattice is expressed the same way as for the platelets, with
volume fractions 1D and S in this instance:
nN1D z1D
(1D 1D1D   S 1DS )
2
(S22)
N S ( zS  z1D )
N z
 S  S  S 1D ( S  SS   S 1D1D ) .
2
2
(S23)
H f (1D )  
and
H f (S )  
Similarly to the platelet case, only a maximum of z1D (=4) of a solvent molecule’s nearest
neighbour sites may ever be occupied by rods, due to the rod length, as shown in schematic
S2.
Summing as before, we find the enthalpy of mixing of rods and solvent molecules to
be
z1D
 H Mix 
 V   2v  (1   )  S S  1D1D  2 S 1D 

1D
S
(S24)
Defining the Hildebrand solubility parameter of rods to be
EC ,1D 
z1D 1D 1D z1D 2

 T ,1D
2 vS
zS
(S25)
7
and assuming that the interactions between all lattice sites are dominated by the London
interaction, we can use the geometric mean approximation (see below) to estimate  S 1D :
 S 1D   S  S 1D 1D 
2vS
 S 1 D
zS
(S26)
Inserting these expressions into equation S24 gives
2
2
z1D
2
 H Mix 
 V   z  (1   ) T ,S  T ,1D   3  (1   ) T ,S  T ,1D 

1D
S
(S27)
This is again very similar to the standard expression
H Mix
2
  (1   )  S   N  .
V
(S20)
Hence, for any nanomaterial N with dimension d  3 , we can extract two general
definitions:
2
 H Mix   d 
 V   1  3   (1   ) T ,S  T , N 


d 
(S28)
and
EC , N 
zN  N  N  d  2
 1   T , N
2 vS
 3
(S29)
Figure S1: A) Initial platelet crystal; each platelet has z2 D  2 nearest neighbours B) Initial
solvent crystal ( zS  6 nearest neighbours) C) A platelet in the solvent-platelet mixture may
be entirely surrounded by solvent molecules D) A solvent molecule in the final mixture must
always have zS  z2 D  4 solvent nearest neighbours
8
Figure S2: A) and B) Initial rod and solvent crystals C) A rod may be surrounded by solvent
molecules on z1D  4 sides in a mixture D) A solvent molecule must always have zS  z1D  2
nearest neighbour solvent molecules.
S4
Deriving Mixing Enthalpy from First Principles as a function of Dispersive and Polar
Solubility Parameters
Derivation of an expression for the enthalpy of mixing within a lattice model involves
the calculation of the bracketed energetic term in equation S14. For simplicity we will label
this as
E   S S   N  N  2 S  N 
(S30)
To calculate E , it is necessary to calculate  S  S ,  N  N and  S  N . We will assume
that each of these interaction energies is the sum of a dispersive and a dipolar term. For two
small molecules (or lattice sites) A and B , we write this in the general form
 A B   D , A B   P , A B where D and P represent dispersion and polar respectively.
For the dispersion interaction, the energy of interaction is given by3
 D , A B  
3  A B I1 I 2 1
 k1 A B
2 (4 0 ) 2 I1  I 2 r06
(S31)
Here,  A is the polarisability of a molecule of type A , I A its ionization potential, r0 the
equilibrium molecular separation (lattice site size) and k1 is a constant.
Similarly, for two small molecules (or lattice sites) interacting by dipolar interactions, the
energy of interaction is given by
 P , A B  
2  A2  B2 1 1
 k2  A2  B2
2
6
3 (4 0 ) kT r0
(S32)
9
where  A is the dipole moment of a molecule of type A and k2 is a constant. We can see that
dispersive interactions are controlled by respective molecular polarisabilities, but dipoledipole interactions by the square of the dipole moment.
Inserting these expressions into equation S30 we get:

E  2k1 A B  k1 A2  k1 B2  2k2  A2  B2  k2  A4  k2  B4

which may be simplified as:
E  k1 ( A   B ) 2  k2 (  A2   B2 ) 2
(S33)
This is effectively a solubility parameter equation for E , where the  A and  A2
terms may be thought of as solubility parameters in their own right. However, not only do
they have different units from each other, but it is also desirable to recast them into new
variables which have the same units as existing (Hildebrand or Hansen) solubility parameters,
in order that comparisons may be easily made. Therefore, by combining equations S15 and
S33 we can write3-5
 D , A A 
2vS 2
 D , A  k1 A2
zS
(S34)
2vS 2
 P , A  k2  A4
zS
(S35)
and
 P , A A 
for A, and similar for B. Here  D , A and  P , A are Hansen’s dispersive and polar parameters for
material A and  S is the lattice site (or solvent molecule) volume. Substituting these values
into the equation for  changes it to
E 
2 S
( D , A   D , B ) 2  ( P , A   P , B ) 2 
zS
(S36)
Converting this to enthalpy of mixing as described above and writing this specifically in the
case of a nanomaterial-solvent system, this becomes
 H Mix   d 
2
2

  1    (1   ) ( D , S   D , N )  ( P ,S   P , N ) 
V
3

 Mix d 
(S37)
Comparing it to Hansen's expression for the enthalpy of mixing
2
2
2
H Mix
1
1

  (1   )  D , S   D , N    P , S   P , N    H , S   H , N  
V
4
4


(S38)
we see that the form is identical, except for the presence in Hansen's equation of an
empirically obtained hydrogen - bonding term, and the factor of 1/4 in the P and H terms.
10
S5
The effect of dipole-induced dipole effects
Of course, this model is still quite limited; for example, it ignores at least one
potentially important interaction: the dipole-induced dipole interaction.
These interactions may be described by an intermolecular (i.e. intersite) interaction
energy:3
 DID , A B  
 A2 B 1
 k3  A2 B
2 6
(4 0 ) r0
(S41)
where the various constants are defined as in the previous formulation. We see that such an
interaction is controlled by the permanent dipole moment of one molecule and the molecular
polarisability of the other, and that the full description of such an interaction must then
include a term describing molecule A acting on B and molecule B acting on A .
Adding a term of type  DID, AB  k3 A2 B  k3B2 A into the expression for intersite
interactions to represent the dipole-induced dipole interaction, the resulting E is:


E  k1 ( A   B ) 2  k2 (  A2   B2 ) 2  2k3 ( A   B )(  A2   B2 )
(S42)
Defining  D , A and  P , A as before, it is evident that the dipole-induced dipole term can be
written as the product of these terms. This yields
E 

2 S 
2k3
2
2
( D , A   D ,B )( P , A   P ,B ) 
( D , A   D , B )  ( P , A   P , B ) 
zS 
k1k2

Letting
E 
k3
k1k2
(S43)
 A produces
2 S
( D , A   D , B ) 2  ( P , A   P , B ) 2  2 A( D , A   D , B )( P , A   P , B ) 
zS 
(S44)
As before, we can use this to express the enthalpy of mixing for a solute of dimension d:
 H Mix   d 
2
2

  1    (1   ) ( D ,S   D , N )  ( P ,S   P , N )  2 A( D ,S   D , N )( P ,S   P , N ) 
 VMix d  3 
(S45)
S6
Relating the dispersed concentration to enthalpy of mixing
11
We can use the expressions derived above to find the dispersed concentration of
solute using recent work which has shown that for small molecules and rod-like solutes the
maximum dispersed volume fraction is given by:6
v   H Mix / V  


 RT


  exp 
(S46)
where v is the volume per mole of the dispersed phase. Assuming we can model a platelet as
a very low aspect ratio rod, we can apply this to 0D, 1D and 2D solutes. For simplicity, we
will illustrate this first in the case where we consider only dispersive and dipole-dipole
interactions. Thus, inserting S37 into S46, and making the approximation of low volume
fraction ( 1    1 ), we obtain an expression for the maximum dispersed solute volume
fraction:
  d 

N  exp   1   N ( D,S   D, N )2  ( P ,S   P , N ) 2 
  3  RT

(S47)
This predicts that the dispersed volume fraction will behave as a two-dimensional
Gaussian function in  D , S and  P , S space, i.e. the product of two individual Gaussians of
equal width, one a function of  D , S and the other a function of  P , S . We illustrate this for both
rods and platelets in figures S3 and S4.
12
Figure S3: Calculated dispersed volume fraction for rods. (A) Dispersed volume fraction (in
arbitrary units) in the case where only dispersive and dipole-dipole interactions are present
(equation 25). (B) A contour plot of the surface illustrated in A. (C) A contour plot plotted
from equation 26 where dispersive, dipole-dipole and dipole-induced dipole interactions are
present. Here, a nonphysical value of A = 0.8 is used in the dipole-induced dipole term. (D) A
contour plot plotted from equation 26 with a realistic estimate of A = 0.1. Note its similarity
to the case where only dispersive and dipolar interactions are considered (B). In all four
graphs, d  1 ,  N =50 L/mol (i.e. 50,000 g/mol assuming a density of 1000 kg/m3),  D , S =18
MPa1/2 and  P , S =9.3 MPa1/2 were used.
13
Figure S4 Calculated dispersed volume fraction for platelets. (A) Dispersed volume fraction
(in arbitrary units) in the case where only dispersive and dipole-dipole interactions are
present (equation 25). (B) A contour plot of the surface illustrated in A. (C) A contour plot
plotted from equation 26 where dispersive, dipole-dipole and dipole-induced dipole
interactions are present. Here, a nonphysical value of A = 0.8 is used in the dipole-induced
dipole term. (D) A contour plot plotted from equation 26 with a realistic estimate of A = 0.1.
Note its similarity to the case where only dispersive and dipolar interactions are considered
(B). In all four graphs, d  2 ,  N =500 L/mol (i.e. 500,000 g/mol assuming a density of 1000
kg/m3),  D , S =18 MPa1/2 and  P , S =9.3 MPa1/2 were used.
14
We can follow the same procedure for the case where we also consider dipole induced
dipoles. Using equations S45 and S46 the dispersed concentration is given by
  d 

N  exp   1   N ( D,S   D, N ) 2  ( P ,S   P , N ) 2  2 A( D ,S   D , N )( P ,S   P , N )  
  3  RT

(S48)
At first glance it would seem as though the last term, a product of the two different
parameters, would significantly change the behavior compared to equation S47. In fact, this
expression simply describes a specific case of a two-dimensional elliptical Gaussian function,
i.e. a Gaussian broadened in one dimension, narrowed in the other and rotated by a particular
angle (in this instance

) in the  D , S   P , S plane. The relative narrowing can be described
4
by the ratio of widths:
wnarrow
1 A

wbroad
1 A
(S49)
We note that A is determined by a set of physical constants, allowing us to write:
A
k3
k1k2

 1
1
 (4 ) 2 r 6 
0
0 

 kT
3
I1 I 2 1   2
1
1
1 1

2
6
2
6
 2 (4 0 ) I1  I 2 r0   3 (4 0 ) kT r0 
I1  I 2
I1 I 2
(S50)
Approximating I1  I 2  I gives
A
2kT
I
(S51)
As all real molecules have I>2kT, this means A<1. A crude estimate of the magnitude of
ionization potentials tells us that A  0.1, giving wNarrow  0.9wBroad . This narrowing is very
small, meaning that the incorporation of dipole-induced dipole forces hardly changes the
dependence of concentration on  D , S and  P , S compared to that predicted by equation S47.
S7 Universally positive enthalpy of mixing
The sign of our general expression for the enthalpy of mixing states (S45) depends on
the sign of E. The expression for E (S44) can be written as
zS E
 ( D , A   D , B ) 2  ( P , A   P , B ) 2  2 A( D , A   D , B )( P , A   P , B )
2 S
(S52)
which can be abbreviated to
15
zS E
 X 2  Y 2  2 AXY
2 S
(S53)
Let us assume that E<0
X 2  Y 2  2 AXY  0
(S54)
This can only be true if either X or Y are negative (A is positive by definition). This
expression can be rearranged to give
X Y
  2 A
Y X
(S55)
Here both X/Y and Y/X must be negative. Thus
1 X
Y 


 A
2 Y
X 
(S56)
The minimum value of the left hand side of this expression is 1 which means that the
enthalpy of mixing can only be negative if A>1.
(1)
Hildebrand, J. H. Annual Review of Physical Chemistry 1981, 32, 1.
(2)
Hildebrand, J. H.; Wood, S. E. The Journal of Chemical Physics 1933, 1, 817.
(3)
Israelachvili, J. Intermolecular and Surface Forces, Second Edition ed.;
Academic press, 1991.
(4)
Miller-Chou, B. A.; Koenig, J. L. Progress in Polymer Science 2003, 28,
1223.
(5)
Hansen, C. M.; Skaarup, K. J. Paint Techn. 1967, 39, 511.
(6)
Hughes, J. M.; Aherne, D.; Bergin, S. D.; Streich, P. V.; Hamilton, J. P.;
Coleman, J. N. Nanotechnology 2011, submitted.
16