PARTICLES AS MOLECULES
C. M. S ORENSEN
3.5 INTERPARTICLE INTERACTIONS
alpha-omega dithiols can link to two metal nanoparticles (20, 44). Then nanoparticle molecules can bind together into roughly spherical clumps, as reported in Reference 44. With such dithiol linking, Sidhaye et al. (45) were able to reversibly expand and contract the spacing between linked nanoparticles with an optically induced cis/ trans conformation change in the linking molecule. Another tack is to put matching functional groups on the nonligating ends of the ligands, for example, hydrogen bond donors and acceptors (46). Gold nanoparticles have been ligated with mercap-toalkyl oligonucleotides that can detect, via binding, complementary nucleotides bound to other gold nanoparticles (47).
almost exclusively as solutions and the precipitation of the NP from the solution into two- and three-dimensional superlattices. The melting and vaporization aspects of solid or liquid systems have seen far less study likely because the thermal stability of most NP molecules is not good and decomposition could result before the phase transition. There are some melting and liquid studies, however. I start with a general description of colloidal solutions and then I describe in some detail the current understanding of NP interactions.
3.5.1 Colloidal Solutions
It is well known that colloidal suspensions can share many features with simple mol-ecular systems such as gas, liquid, and solid crystalline and amorphous glass phases. This is particularly true when the colloid is nearly monodisperse for then the interparticle interactions, which are usually size dependent, are nearly all the same and hence the phase boundaries, which depend on the interactions, are distinct.
Indeed, as the size distribution of a colloid narrows, one could claim that the colloidal suspension transforms into a solution, just as the different particles, by becoming alike or even identical, are transforming to molecules. Unlike simple molecular systems, which by their definition have no variety and are not dissolved in a medium, particle colloids and solutions can vary the interactions via changing size, surface groups, solvent, etc., and thereby change the phase diagram.
Figure 3.4, which is borrowed (somewhat modified) from a recent review (48), shows the possible phase diagrams that can occur for a colloidal system. On the left
Figure 3.4 (Left) Phase diagram for purely hard sphere interaction potential, which shows only fluid (F) and crystalline (C) phases. (Middle) A short-range (relative to “particle” size) interaction is added to the hard sphere potential. Then a dilute gas (G) phase can appear, as well as a metastable liquid-liquid coexistence (Lþ L). (Right) The attractive interaction is long range and a complete phase diagram occurs, with gas, liquid, and crystalline solid phases. The triple line (TL) temperature increases with increasing attractive interaction strength.
3.5 INTERPARTICLE INTERACTIONS 47
is the phase diagram for a purely hard sphere system with no attractive interactions, which shows only fluid (F) and crystalline (C) phases. The transition from liquid to crystal is driven by entropy. For volume fractions greater than 0.545 there is more translational freedom, hence greater entropy, for hard spheres in a close packed solid than the amorphous glass. The fluid to crystalline equilibrium for a solution is the saturated dissolved solute, which could be single nanoparticles of the SPC, in equilibrium over the precipitated solid, which for SPC could be a superlattice.
Achieving a solid to liquid transition at high volume fractions can be difficult, however, because dense systems can experience kinetic slowing and arrest.
Addition of short-range attractive interactions brings on a metastable liquid-liquid (L) coexistence comprised of coexisting high and low concentration colloidal sol-utions. One can imagine that as the attractive interaction is turned on the metastable coexistence curve rises up from low temperature and pushes the fluid to the fluid-crystal region to the left in Figure 3.4. This is metastable relative to the fluid-fluid-crystalline equilibrium, the fluid phase of which develops a low concentration gaseous phase (G) at low temperature. As the attractive interaction increases in range relative to the size of dissolved colloidal entity, the liquid-liquid coexistence rises to higher temperature, into an equilibrium regime, and a triple line (TL) appears. Now the phase diagram looks like that of a simple atomic system with three-phase equilibria, the phase diagram on the right.
The relative range of the attractive interaction between the colloidal particles is a key parameter that affects the phase diagram. It is likely that one can control this key parameter in solutions of nanoparticles by adjusting combinations of the particle size and ligand shell depth (i.e. ligand length, see Fig. 3.2) and hence range throughout the possibilities of Figure 3.4. Another key parameter is the strength of the attractive interaction (i.e. the depth of the interparticle potential well) which controls the effective temperature, hence the position of the triple line. For particles the strength of the interaction depends on both the particle and ligand shell Hamaker constants relative to the solvent. The triple line goes to higher temperature the greater the interaction strength. We have included these properties as global parameters along the margins of Figure 3.4.
The phenomena displayed in Figure 3.4 are for spherically symmetric potentials, and once this symmetry is relaxed, the complexity, hence opportunities, in the phase diagram expand significantly. One might say that Figure 3.4 is the argon atom limit.
Comparison to gas-liquid-solid systems is very useful, but it must be remembered that a colloid is much more complex. Recall that in a real gas the molecules move in straight lines between collisions; that is, they move ballistically. In a solution the par-ticles move diffusively. The pressure of a real gas is replaced by the osmotic pressure of the particles in the solution. Given this, it is not surprising that since the real gas pressure has a virial expansion so does the osmotic gas of the solution as expressed by the van’t Hoff equation:
P¼ kTc 1 þ Bð 2cÞ (3:2)
PARTICLES AS MOLECULES 48
where P is the osmotic pressure, k is Boltzmann’s constant, c the concentration, and B2 the second virial coefficient. B2is equal to 4 for a hard sphere potential and in general is positive for repulsive and negative for attractive potentials.
As an example, it is useful to recall protein solutions. Protein molecules can be quite large; for example, a lysozyme molecule is an ellipsoid with dimensions 3 4 5 nm, similar to nanoparticles (they are in fact nanoparticles). Protein aque-ous solutions have seen significant study in the recent past because it is desirable to form protein crystals from solution for structural analysis. Such a system contains not only the protein molecule and the water solvent but usually dissolved ions which dissociate and a variable pH. Moreover, the protein molecule may have a variety of surface states that affect its interaction with other protein molecules as well as the water. The lesson here is that often the system is too complex and an effective inter-particle interaction potential must be prescribed. Such a procedure often works because the number concentration of the large protein molecules or nanoparticles is far less than that of the other constituents.
3.5.2 Solubility of Nanoparticle Molecules
There appear to be no quantitative studies of nanoparticle molecule solubility, although essentially every researcher knows that the ligand end groups greatly affect the type of solvent that will keep the colloid stable. The old rule of “like dissolves like” applies. Thus, alkane-coated nanoparticles will not be soluble in polar solvents, for example, water, but will be in another alkane. Conversely, if the alkane ligands are terminated with carboxylic acid groups, the nanoparticles will be soluble in water but not in the alkane.
It is interesting that here we encounter the double identity of nanoparticle mol-ecules. Are they suspended particles in a colloid or dissolved molecules in a solution?
We contend that if they satisfy the condition of near stoichiometry discussed above to classify them as molecules, then their stable suspensions are more than that, they are solutions as well. We remark here that gravity can now get in the way, for even a monodisperse, hence stoichiometric, system of particles will settle out if the thermal energy, kT, is not large enough to keep the monomers suspended. This happens for particles on the order of 50 nm.
We presented qualitative observations of solubility of 5-nm gold nanoparticles ligated with dodecanethiol dissolved in toluene (49). A rough temperature versus concentration phase diagram showed the expected greater solubility with increasing temperature. In other work (28) we showed that 5-nm gold ligated with alkylthiols ranging over C8, C10, C12, C16 became more soluble with increasing chain length at room temperature; the C8 thiol being essentially insoluble. Calculation of the van der Waals attractive potential for two gold cores separated by one ligand length yielded values of 5 kT, 2.2 kT, 2 kT, and 0.6 kT, for the different chain lengths, respectively, at room temperature.
Nucleation and growth of 3d superlattices from nanoparticle solution has seen recent study (50–52). Superlattice cluster size versus time was studied after the system was destabilized via either place exchanging to a less soluble ligand, creating
3.5 INTERPARTICLE INTERACTIONS 49
the nanoparticles in a poor solvent, or quenching temperature from a one-phase to two-phase regime. The classic diffusion limited theory of LaMer and Dinegar (53) was not successful in describing the growth kinetics. Roughly spherical superlattices did nucleate from the solution, as shown in Figure 3.5 (50). Work currently in progress in our laboratory (52) has shown that the size of the nanoparticle molecule clusters decreases with depth of quench, as shown in Figure 3.6. This is similar to precipitation of molecular and ionic solids.
3.5.3 Nanoparticle Interactions
The discussion above can be simply summarized to say that solution phase behavior is much more complex than pure component solid-liquid-gas phase behavior because the interparticle interactions have a much greater variety. The effective interparticle interaction in solution can have:
1. Excluded Volume Effects. These could be effective hard sphere potentials representing the finite size of the particles. Such potentials are important for the crystallization transition; the left side of Figure 3.4 is an example. For ligated nanoparticles, the finite size is likely better represented by a soft sphere poten-tial, the softness due to the steric interactions of the ligands (see below).
2. London – van der Waals Attractive Forces. These forces depend on the dielec-tric constants of the particle materials and the surrounding solvent via the Figure 3.5 Gold nanoparticle molecule clusters formed from precipitation of the system of monomers.
PARTICLES AS MOLECULES 50
Hamaker constant. The Hamaker constant for a metal is about 2 eV and about two orders of magnitude larger than for nonmetals. For two spheres of radius R¼ d/2 the van der Waals potential is (54)
Vrdw(r)¼ A 12
1 x2 1þ1
x2þ 2 ln 1 1 x2
(3:3)
where A is the Hamaker constant of the particle and x¼ r/d where r is the center-to-center distance. This has limits of
Vrdw(r) s1 for r& d (3:4)
r6 for r d (3:5)
In Equation (3.4), s is the separation between the two particles surfaces, see Figure 3.7.
3. Solvent-Mediated Forces. These break into two classes:
a. Electrostatic. Dissolved ions form counter ion layers around charged par-ticles to yield a screened Coulomb potential. This combined with the van der Waals potential yields the classic DLVO potential (55).
b. Ligand – Solvent Interactions. Simply said the rule of “like dissolves like”
qualitatively describes these interactions. To estimate the free energy of mixing of the ligands in the presence of the solvent when ligand layers Figure 3.6 Radius of gold nanoparticle molecule clusters formed by temperature quenching stable solutions of the molecules in a mixture of 2-butanone and tert-butyltoluene to various depths below the saturated solution temperature.
3.5 INTERPARTICLE INTERACTIONS 51
from two different nanoparticles start overlapping, one needs to consider two different regimes (56). In the first regime, the ligand chains undergo inter-penetration, and in the second regime the chains undergo interpenetration and compression. These two regimes are shown schematically in Figure 3.7a and 3.7b, respectively. These two regimes can be distinguished as:
Regime 1: 1þ ‘ , x , 1 þ 2‘ (Interpenetrations only)
Regime 2: x, 1 þ ‘ (Interpenetration and Compression) where‘ is the scaled contour length of the ligand chains, that is, ‘ ¼ L/d Free energy of mixing in both regime 1 and regime 2 are known in the literature in terms of the Flory x-parameter between the solvent and the
Figure 3.7 (a) Ligand interdigitation. (b) Ligand steric repulsion.
PARTICLES AS MOLECULES 52
tethered chains. In terms of our scaled variables, one can write this in regime 1 as
V2
kT¼pd3 2vmf2av 1
2 x
[x (1 þ 2‘)]2; 1þ ‘ , x , 1 þ 2‘ (3:6) wherevmis the volume of a solvent molecule and favis the average volume fraction of the ligand segments in the tethered layer.
V3
kT¼pd3 vm f2av 1
2 x
3 ln L x 1
þ 2 x 1
‘
3 2
; x, 1 þ L (3:7) Note that when x¼ 1 þ ‘; V2¼ V3as expected.
4. Ligand – Ligand Interactions. An elastic contribution to the potential due to loss of conformational entropy as the ligands begin to overlap, a steric repulsion, is also known in the literature (57, 58). In terms of the scaled variables it can be written as
V4
kT ¼1 2
pvd2
‘ [x (1 þ ‘)]2; x, 1 þ ‘ (3:8) wherev is the number of ligands per unit area of the nanoparticles.
5. Dipolar Forces. These can be either electric or magnetic and in each case signi-ficantly larger for nanoparticles than for atoms or normal-sized molecules.
Figure 3.8 Various components of the interaction potential between two 5-nm gold nanopar-ticle molecules with dodecanethiol ligands.
3.5 INTERPARTICLE INTERACTIONS 53
Figure 3.8 shows the interaction potential between two 5-nm gold nano-particle molecules with dodecanethiol ligands, the sum of Equations (3.3), (3.6), (3.7), and (3.8).
I now describe how these various interactions affect nanoparticle solutions and the phases that can be obtained from these solutions.
Figure 3.9 Results from 2d random aggregation simulations in which the depth of the poten-tial of interaction was either 7 kT (a) or 4 kT (b). At 4 kT the aggregates are roughly circular crystallites in equilibrium with a monomer phase, that is, a dissolved phase. At 7 kT the aggre-gates are fractal, with a fractal dimension of 1.4, the DLCA value, over large length scales, but they retain a dense crystal packing over small length scales. These are called fat fractals.
PARTICLES AS MOLECULES 54
The hard sphere interaction causes a liquid to solid transition at a solution volume fraction of 0.545, as described above with reference to Figure 3.4. The resulting solid has an fcc lattice. This is slightly, about 0.001 kT, more stable than its close packed counterpart at 0.74, the hcp lattice. Binary hard sphere systems can form five different lattices of different stoichiometry depending on the size ratio.
Addition of the van der Waals interaction can cause formation of condensed phases at much lower volume fractions than 0.545. If strong relative to the thermal energy kT, the phases are ramified aggregates, usually fractal, and these can lead to gels. Variation of the relative strength of the interaction can yield more compact structures like
“fat fractals” and lattices, illustrated in Figure 3.9 (59). Ohara et al. (60) showed that since the van der Waals interaction is size dependent, bigger nanoparticles will nucleate to superlattices first when the solution is destabilized and size segregation can occur. Korgel and Fitzmaurice (61) showed that van der Waals interaction with a substrate when the solution is dried can compete with the nanoparticle-nanoparticle interaction. Then depending on the relative strength of the two interactions, the morphology of the resulting dried layers can be controlled.
Inclusion of screened Coulomb electrostatic interaction along with the van der Waals and hard sphere interactions can engender the entire description of Figure 3.4. This has been realized in many ways in protein solutions, but not yet for nanoparticle solutions.