3544
J. Phys. Chem. 1994,98, 3544-3549
Nanocrystal Microemulsions: Surfactant-Stabilized Size and Shape Robert L. Whettent and William M. Gelbart' Department of Chemistry, University of California, Los Angeles, California 90024- I569 Received: November 19, 1993; In Final Form: January 31, 1994'
We formulate an analogy between thermodynamically stable water/oil microemulsions and surfactant-stabilized nanocrystal dispersions, with the goal of accounting for how specific sizes and shapes of nanocrystals might become dominant in equilibrated solutions of, for example, alkali halides or semiconductors in organic solvents. Qualitative estimates of the energetics-Le., of the dependence of lattice energy on crystal dimensions-suggest that the size and shape distributions can be quite narrow, with their average properties controlled primarily by the solution composition,in particular the ratio of surfactant to crystalline species.
I. Introduction Over the past decade, the investigation of nanometer-scale crystallites-"nanocrystals"-of elemental and compound metallic,l semiconductor,2and other materials has greatly intensified. This developmenthas been stimulated by a wide range of potential applications identified in the areas of catalysis, microelectronics, and electrooptics.3~4Advances in bulk-scale preparation and characterization methodologies have led to significant narrowing of the size and shape distributions of these special particles. On this front, an important step has been an appreciations of the role of surface-active (surfactant) molecules in retarding the growth and coalescence of nanocrystals, and perhaps also in stabilizing certain particular sizes and morphologies. A most remarkable consequence of this effort is a series of reports describing highyield formation of single-size nanocrystals in nearly ideal lattice and morphological forms. These include the noble-metal colloids found by the groups of Schmid6and Moiseev7-which are a series of monodisperse fcc particles with cuboctahedral structure, the giant CuzSe compounds of Fenske and co-workers: and a hierarchy of CdS compounds with tetrahedral morphology found by Wang et al.? including one, C ~ ~ ~ S S O ( C that ~ H Shas )~~, undergone complete structural characterization.10 We emphasize here that these preparations are all of the "self-assembly" type, in which the particles are formed from much smaller units in a single stage, rather than from any multistep building-up implied by rational chemical synthesis: much uncertainty remains about the sources of kinetic and thermodynamic stability of the special structures involved. Stabilization of monodispersed clusters of the above kind is normally discussed in the context of 'arrested growth" and other kinetic schemes.ll In the present paper we explore the possibility that dilute dispersions of these nanocrystals may correspond to equilibrium solutions of self-assembled structures. Specifically, we argue that the situation is analogous to that of the more familar dispersion of oil, water, and surfactant, Le., to the case of fluidfluid microemulsions.12Accordingly,all possibleclusters (nanocrystal aggregates) are assumed to be in statistical thermodynamic equilibrium with one another, just as the micelles of various sizes and shapes in an aqueoussurfactant solution or the microemulsion droplets in a stable oil-water-surfactant system. In section I1 we outline briefly an equilibrium theory of droplet size distributions in fluid-fluid microemulsions, developing a formulation which is especially naturally suited for application to the nanocrystal situation. We feature there both the curvature dependence of the interfacial energy and the competition of the droplets with bulk-phase separation, since these will prove to be f Present address: School of Physics, Georgia Institute of Technology, Atlanta, GA 30332-0430. Abstract published in Aduance ACS Abstracts, March 1 , 1994.
0022-3654/94/2098-3544%04.50/0
key ingredients in our discussion of the nanocrystal dispersions. For this latter case, we apply a similar statistical thermodynamic theory in section I11 to the case of alkali halide clusters stabilized by fatty acid surfactants in organic solvents. We choose this particular system because of our previous experience with alkali halide nanocrystals in the gas phase,I3 and because of the simple crystallite structures involved. We argue, however, that our theoretical approach is equally valid and feasible for the technologicallymore important examplesinvolving semiconductor clusters. Finally, we discuss an experimental approach which we believe will facilitate incisive probes of the thermodynamic equilibriumpropertiesof nanocrystal size and shape distributions. 11. Microemulsion Analogy
In order to motivate and provide a convenient underpinning for our treatment of nanocrystal dispersions,we consider first the case of oil/water microemulsions. This latter situation-a thermodynamicallystable, dilute suspension of, for example,water droplets in oil-has been addressed many times in the literature:I2we formulate and stress here those aspects which are most directly relevant to the nanocrystal case. We consider water-in-oil droplets, each consistingof a spherical drop of water surrounded (and thereby stabilized) by a monolayer of surfactant. Assumingaconstant thickness (6) forthesurfactant monolayer, each droplet is completely specifiedby a single length, e.g., its outer radius R . (Indeed, experiments confirm that the area per surfactant (and hence the film thickness) is essentially independent of the size of the microemulsion droplet and, further, that this remains a good approximationeven in the lamellar phases of the microemulsion.'2) For example, the numbers of water and surfactant molecules in a spherical droplet of outer radius R are given by the geometric relations
and
with uW and v, the volumes of the individual water and surfactant molecules, respectively. Note that, for R large compared to the "molecular scale" 6, n, and n, vary as R3 and R2, respectively, as they must. Now, the overall free energy of the system can be written as (with n, henceforth abbreviated by n )
where Nnis the number of droplets consistingof n water molecules, 0 1994 American Chemical Society
Nanocrystal Microemulsions
The Journal of Physical Chemistry, Vol. 98, No. 13, 1994 3545
and p,,({Nn])is the chemical potential of such a droplet. Furthermore, since the (dilute) solution of droplets has been assumed to be ideal, we have also that
X”
(3)
4x16’
where pi is the corresponding standard chemical potential, Nt = N , Ns No is the total number of (water, surfactant, and oil) molecules in the solution, and all energies are measured in units of keT. Finally, note that the distribution (Nn)is subject to two constraints,arising from the conservationof water and of surfacant molecules, respectively:
2x16’
+ +
I
0 30
31
32
WA
33
34
X n N , , = N, and
Minimizing the free energy (2), subject to the constraints (4A) and (4B), gives the equilibrium distribution of droplet sizes for the arbitrary composition defined by the mole fractions X, = N,/N, and X, = Ns/Nt. (The mole fraction of oil, X,, follows of course from 1 - X, -Xu.) Specifically, we find
Xn/n = exp[-pi
+ An + ~*n,(n)]
(6)
with y(R) theinterfacial freeenergy per unit area (in the presence of surfactant) for a droplet whose radius is R. (Recall that (1A) provides an explicit mapping between R and n, so that we can use them interchangeably here.) y ( R ) can in turn be written in the form14 y(R)=?*+-
0
k 2
2 R
2 2 R,
70
2x1O8
(5)
where X,,fn = Nn/N,,and X and A* are the Lagrange multipliers associated with the constraints (4A) and (4B). It is straightforward to show that A and X* are precisely the chemicalpotentials of the water and surfactant molecules in a solution having composition X,, X,. Note that the n-dependence of n, is given directly by (1A) and (1B); thus the size distribution X,, follows as soon as we specify how the standard chemical potential p: varies with n. Now, the only interesting (Le., nonlinear) n-dependence of p i is that which arises from the surfactant monolayer. Indeed, the linear terms in p i are simply npi, where p i is the chemical potential of a water molecule in bulk water: we can thus absorb p i into A, with the understanding that henceforth the chemical potential X is defined with reference to that of bulk water. For the nonlinear part of p i , we can write pi = y(R) 4 r R 2
0
(7)
where k is the bending elasticconstant of the surfactant monolayer, Ro is its spontaneous radius of curvature, and y * is the interfacial free energy per unit area for the optimum curvature (l/Ro). Typically, k is of the order of tens of keT, Le., k ranges from 5 to 50,’s and Ro is hundreds of angstroms. Figure 1A-C shows the evolution of size distribution upon increasing the ratio of Xw/X, for fixed X, = 0,001, for a microemulsion system characterized by k = 50, y* = 1/20 A2 (approximately 20 dynfcm), and Ro = 100 A (with 6 = 10 A, vw = 30 A’, and v, = 300 A3). The distributions involve relatively little polydispersity, and the average size therefore follows closely the value-R* (see arrow in figures)-which would be required (by the “volume-to-surface” ratio Xw/X,) ifonly one size were present. That is, R* is the value of R which allows the “local” (single droplet) ratio of n,(R) to q ( R ) from (1A) and (1B) to equal the “global” ratio of N, to Ns (defining the particular composition of interest).
72
E
FJ
74
/
70
80
\
0 125
130
135
140
RIA Figure 1. Size distributions Vr,) for water-in-oilmicroemulsion droplets at a water mole fraction of 0.001, with water-to-surfactant mole ratios of 4.6 (A), 19 (B), and 38 (C). In all cases, the are plotted vs outer-radius R (related to water-aggregation number n via q 1A)and refer to a choice of k = 50, y* = (1/20)/&, & = 100 A, 6 = 10 A, v, = 30A3,andv, = 300A3. (Seetext for energyunitsandfortheexplanation
vR]
of R*.)
Note that, in the case where k = 0, Le., no bending energy contribution to y(R), the nonlinear part of the standard chemical potential pi for an n-drop (see (6,7),and (1B)) becomes simply y*4rR2, with 4rR2 = ns(v,/6) for large R (>>a). In this limit, then, the energy y*47rR2can be written as y*a*n, (where a* = v s / 6 is a molecular area) and can be “absorbed” accordingly (see ( 5 ) ) into the X*nsterm just as the linear portion pin was absorbed into An. That is, the chemical potentials for both water (A) and surfactant (A*) are simply referenced to new zeros of energy. Consequently, the size distribution X,,/n can be written as exp[Xn X*n,], with X = X - p i and A* = A* - y*a*, and it becomes independent of both p i and y*a*. That is, the equilibrium state of the microemulsion is determined completely by composition. Indeed, even when realistic nonzero estimates of k and & are incorporatedso that p i involves terms with n-dependence other than n and %(n)-we have seen that the preferred size (R*) is still controlled largely by the value of X,/X, alone. The final distribution (C) in Figure 1 corresponds to the point where “emulsification failure”16occurs, Le., where a bulk phase of water begins to coexist with the water-in-oil microemulsion: further increase in the water-to-surfactant ratio (X,/X,) leads simply to more bulk water, in equilibrium with the fixed microemulsion size distribution given in Figure 1C. Note that this is the point where the Lagrange multiplier h = X -pi has reached zero, from negativevalues, Le., where the water chemical potential in the microemulsion dispersion is no longer lower than that of bulk water. More explicitly, Xw/Xs= 38 is the point in the phase diagram (for these values of y*, k, and &) where a two-phase region first appears. Recall from the discussion immediately above that a consequence of the conservation condition (4B) is that the interfacial energy per molecule ( y * a * )
+
3546 The Journal of Physical Chemistry, Vol. 98, No. 13, 1994
is simply absorbed into the chemical potential A*, so that its value is irrelevant. In the absence of surfactant, however, there is no A*, and bulk-phaseseparation occurs directly from a solution of monomeric water molecules if the interfacial tension y* is too large: this corresponds to the usual solubility limit of water in oil, in the absence of surfactant, where the price paid for creating droplet surface is too large to be compensated by the entropy of dispersion.
Whetten and Gelbart
= Ncr) to replace (9) by the “sum rule”
electroneutral
Note that (10) guaranteesthat NNa+= Ncl-(electroneutrality). Finally, the conservation of surfactant can be expressed as n odd
n even
111. Alkali Halide Case
NK+
In the case of dispersions of an alkali halide in an organic solvent, which (we will maintain) are stabilized by ionic surfactant, the statistical thermodynamics is somewhat complicated by the facts that (1) instead of a single component (e.g., water, in our previous microemulsion example) comprising the interior of the self-assembledcolloidaldroplet, we have two, oppositely charged, species (e.g., Na+ and Cl-), consequently,(2) we must distinguish between even and odd aggregation numbers, accounting explicitly for the positive and negative charges carried by the latter, and finally, (3) because of the particular crystal structure associated with the “droplet interior”, only specific shapes and sizes need to be considered. From this point, for sake of concreteness,we shall take Na+Clas our alkali halide and the potassium fatty acid salt K+RCOOas our surfactant. Each Na+ (Cl-) on the nanocrystal surface will be assumed to form an ionic bond with a RCOO- (K+) surfactant ion. Recognizing the special role of cuboidal (Le., rectangular parallelepiped) structures17of the simple-cubiclattices involved, we treat a X b X c structures, with a-c all (not-toolarge) integers. We denote by n = abc the total number of Na+ and C1- ions in the a X b X c nanocrystal, by n, = abc = (a - 2)(b - 2)(c - 2) the corresponding number of surfactant counterions K+ and RCOO- (n, is, simultaneously,the number of alkali halide (Na+ and Cl-) ions sitting on the “outside”, or surface, of the cluster), by Nnthe number of n-nanocrystals (Le., of aggregation number n, containing a total of n Na+ and C1- ions), and by Nt the total number of molecules and ions in solution (solvent Na+ C1- K+ RCOO-). Note that when n is even, the nanocrystal will be neutral, comprised of an equal number of Na+ and C1- ions. For each odd n, however, the nanocrystal can carry a charge of either + 1 or -1, according to whether it contains an extra Na+ or C1-. For even n clusters, then, there will be n/2 Na+’s (and RCOWs) and n/2 C1-’s (and K+’s), while for n odd there will be (n 1)/2 Na+’s(Cl-3) and (n- 1)/2 Cl-’s (Na+’s)for net positive (negative) charge. Finally, the correspondingnumbers of K+’sand surfactant anions-RCOO-’s-are (n,- 2)/2 and (n, + 2)/2 for positive odd-n clusters (henceforth denoted “n+”) and (n, + 2)/2 and (n, - 2)/2 for negative ones (“r”). Consider now the various mass conservation and electroneutrality constraints on Nn,Nn+,and N,. In analogy with the microemulsion case treated earlier, we can suppress the various “monomeric” species (Le., the individual ions which are present in solution outside of the n-aggregates) and write
+
+
+
+
+
C (n/2)Nn + gd[((n + 1 ) / 2 ) ~ , + ((n- 1 ) / 2 ) ~ , 1 =
n even
n
“a+
(8)
where is the total (fixed in any given experiment) number of Na+ ions. Note that we can also write, for the total number of C1- ions,
C (n/2)Nn + z [ ( ( n- 1 ) / 2 ) ~ ++ ((n + 1 ) / 2 ) ~ , 1 =
n even
n
NCl-
(9)
But we find it useful to subtract this latter relation from (8) and to explicitly use the fact that the solution is overall
(11)
Of course, a similar relation can be written for the surfactant anion-RCOO-but the “sum rule” (IO) automatically guarantees the conservation of this species since NK+= N R C C ~ . It remains only to write the free energy of the solution in terms of N,, Nn+,and Nr and to minimize it with respect to these variables subject to the mass conservation and electroneutrality constraints given by (8), (lo), and (11). More explicitly, minimizing
((PJJvd - N,)1 (12) with respect to N., Nn+,and N,, subject to these three constrains-with associated Lagrange multipliers A, A’, and A*-we find, for the equilibrium values of the mole fractions of (even and odd) n-nanocrystals (compare these expressions with that given earlier in ( 5 ) ) ,
N,,+/N,= exp[-pL
+ A+:
n
~ *+ fr ]
Here we have again expressed all energies in units of kBT and have used the fact that each chemical potential for an n-cluster can be written in its ideal solution form, e.g., pn = pi f ln(Nn/ Nt). Now we need only establisheach standard chemicalpotential (e.g., p i ) as an explicit function of n and determine A, A*, and A’-and thereby the cluster size distributions (see (13))-for each solution composition (alkali halide and surfactant in organic solvent), i.e., for each = Ncr and NK+. Note that, again (in analogy with the microemulsion case described above), A is the chemical potential of Na+ and A* is that of K+,Le., of alkali halide and surfactant, respectively. The pi’s, Le., the total cluster energies in the presence of surfactant, are formulated as follows. Consider for concreteness an odd cluster, n = abc = 125 with a = b = c = 5, where n, = abc - ( a - 2)(b - 2)(c - 2) = 98; for n+, for example, we have (n 1)/2 = 63 Na+’s, (n- 1)/2 = 62 CYs, (q-2)/2 = 48 K+’s, and (n, 2)/2 = 50 R C O P s . More explicitly, all the Na+ and C1- ions in the cluster are positioned so as to occupy contiguous sites in a simple cubic lattice, with nearest-neighborspacing equal to the bulk NaCl value (2.81 A) (see Figure 2). Note that there are 54 “face” ions on the cluster, 36 “edge” ions, and 8 “corners” (adding up to the total number of surfactant ions, n, = 98). The surfactant ions are in turn arrayed over the surface of the alkali halide cluster, the K+ cations and RCOO- anions opposite their corresponding C1- and Na+ ions, respectively. Each K+ which “opposes” a C1- lying in a face of the cluster is positioned “straightout” (along the surface normal) a distance equal to the lattice spacing in bulk KCl (3.64 A); K+’s opposing edge and corner Cl-’s are set out by the same distance, but along the relevant square and cube diagonals, Le., symmetrically shared by the two (edge) or three (corner) faces involved. Similarly, each RCOO-
+
+
The Journal ofphysical Chemistry. Vol. 98, No. 13, 1994 3547
Nanocrystal Minoemulsions
t 1.6
nln,
1.4
ffipre 2. Rock-salt (simple-cubic) structure representative of the 5 X 5 X 5 (n = 125) minimum-size cation considered in our quilibration scheme for NaCl nanocrystals: specifically,NaaFsz+ is shown here (see ref 21).
is positioned a distance of 3.14 A either "stmightout" or squareor cube-diagonally from its opposing face, edge, or corner Nat, respectively. All nearest-neighbor pairs of ions are then allowed to interact with each other via the usual Born-Mayer potential
u(r) =
+ A,, exp(-r/p,,)
(14).
For the Na+-Cl- and K+-CI- interactions, we use the standard (bulk crystal) A and p values for NaCl and KCI (918 eV and 0.328 A and 2266 eV and 0.324 A, respectivelyIs). for NatRCOO- we use the NaF value for p (0.288 A), with A chosen (765 eV) to give the potential minimum at the sum of the ionic radii for Na+ and 0-,and for K+-RCOO- we use the KF value for p (0.300 A), with A chosen (I417 eV) to give rmin at the sum of the ionic radii for K+ and 0.. Finally, we include Coulomb interactions (QlQ2/r) between all pairs of ions. Since the alkali halide ions pack in the nanocrystal in a way which is necessarily different from in hulk,19 and more significantly, since we do not know a priori how to best arrange the surfactant ionson thecluster surface, thestructureand associated energy (&+, in this instance) should &determined via thermal equilibrium (e.&, Monte Carlo) simulation at the appropriate temperature, instead of fixing the geometry as outlined above. Nevertheless, we employthe much cruder, unoptimized,geometric model in order to provide a simple qualiratiue illustration of our assertion that thermodynamic stabilization ofspecific sizes and shapes of nanocrystals can be controlled by composition. Note also that only electrostatic interactions, and overlap repulsions, are included in our calculationssinceorderof magnitudeestimates of the solvent polarization energies indicate that they make a negligible contribution to pi. Recall from our discussion of the fluid (water-in-oil) microemulsion that the equilibrium size distribution of surfactantcoated (water) particles was determined completely by composition (X,/X,) in the limit where surface energy per unit areay-was independent of curuature. More explicitly, we can express the droplet energy from that context in the form pz = &*Ei, n , E , , + II,E,~~, with &. and the numbers of water molecules "inside" and on the "surface" ('outside") of an n-drop, and n,thecorresponding number ofsurfactant molecules; the E's are the associated three molecular energies. Noting that &. = n - hut and putting n,, = n, we have (upon adding and subtracting &E,,,i. and dividing by n)
+
&,
k / n = Ein+ ( n , / n ) [ ( L - E,) + (Esu,,- 4,JI
+
(15) Here E.bistheenergypermoleculein themicelleswhichcompete for surfactant with the microemulsion drops; it can conveniently (n,/n)E,,
n Figurr3. n/n.ratiosvs n forthreespxialsericsofcuboidalnanccrystals: 5 x5xarods(Rlledcirclcs),witha
