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J. Phys. Chem. B 2006, 110, 5936-5949
Fundamental Global Model for the Structures and Energetics of Nanocrystalline Ionic Solids Elena Bichoutskaia* and Nicholas C. Pyper UniVersity Chemical Laboratory, Lensfield Road, Cambridge CB2 1EW, U.K. ReceiVed: October 11, 2005; In Final Form: December 23, 2005
This paper presents a general theory elucidating the relationships between the structures and cohesive energetics of alkali halide nanocrystals consisting of small sections of bulk rocksalt structures with m1 and m2 rows but infinite along the z axis. The theory introduces the electrostatic interactions between the ions treated as point charges and the short-range repulsions between the closest ion neighbors with the latter terms written in the Born form Ar-n. Minimum energy structures are defined by the distances ae and be separating the closest ions perpendicular and parallel to the z direction. The ratio ae/be, defining the crystal shape, is independent of the strength A of the short-range repulsion, greater than the bulk value of unity, and increases with decrease of the crystal cross section or n. This ratio tends toward unity in the hard sphere limit of infinite n. Both be/R6:6 e 6:6 and ae/R6:6 e , with the bulk separation Re , are less than one, increase with increase of the crystal cross section or n, and are independent of A if this is independent of structure. The structural dependence of A increases 6:6 its value with a decreasing crystal cross section rendering closer to unity the ratios ae/be, be/R6:6 e , and ae/Re . Energy gains on relaxing the crystal toward equilibrium from its bulk separations decrease with increase of the crystal cross section or n, being about 60 kJ/mol for a one-dimensional chain with n ) 6 but 0.5 kJ/mol for m1 ) m2 ) 4 with n ) 12. The energy gained on relaxing to a structure with ae/be constrained at unity is about 10 times greater than the further energy gains consequent on removing this constraint. The present theory neglecting the interaction between ions and the encapsulating nanotube explains the experimentally 6:6 measured be/R6:6 e ratios. The observation that the ae/Re values are greater than one shows that ion-wall interactions are important in determining the values of ae.
1. Motivation Single walled carbon nanotubes (SWNTs) are ideal nanometric objects which facilitate the formation of low-dimensional ordered structures within their cavity. Filling materials deliberately introduced in SWNTs range from single elements, specifically, heavy metals such as Ru,1 Bi,2 Ag,3 Au, Pt, and Pd4 to molecules such as fullerenes,5,6 metal oxides,7 and halides.8-13 The main interests of filling SWNTs are to enforce the encapsulated material to adopt a low-dimensional morphology and to produce templates for atomically regulated lowdimensional crystal growth.14 Recently, the crystal growth behavior of ionic solids in SWNTs, and especially of single binary alkali halides such as KI, has been thoroughly investigated both experimentally10-12 and theoretically.15-17 The experimental studies revealed that the structures of the encapsulated crystals were systematically distorted from those of the bulk, while the theoretical studies, using mainly molecular dynamics simulations, yielded valuable insights into the possible mechanisms of crystal growth inside a SWNT. In this paper, a general analytical theory for the structures and energetics of nanocrystalline ionic solids is presented. The aim is not only to present a comprehensive mathematical formulation but also to provide physical insights into the resulting predictions. The paper is concerned with structures of the alkali halides encapsulated with the 〈001〉 direction parallel to the nanotube axis. Although such crystals are, in principle, infinite in extent along the z direction, a view along this direction will show a small plane consisting of m1 rows and m2 columns * Corresponding author.
Figure 1. Structures of 2 × 1 (top) and 2 × 2 (bottom) nanocrystals, infinite along the z direction, showing one unit repeat.
of ions with alternating cations and anions in each row. The geometry of such a crystal is defined after specifying both the distance b between the successive planes encounted on increasing the z coordinate and the distance a between neighboring ions in the same plane. These coordinates are depicted in Figure 1. Understanding these structures first requires that one disentangles the effects which arise solely from the finite extent of the crystal in two of the directions from any further effects
10.1021/jp055800g CCC: $33.50 © 2006 American Chemical Society Published on Web 03/07/2006
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J. Phys. Chem. B, Vol. 110, No. 12, 2006 5937
TABLE 1: Coefficients Determining the Interchain Electrostatic and Repulsion Energies repulsion energy
Madelung energy
crystallite
nc
C
Mb(1)
q
nqcp
2×1 2×2 3×3 4×4
2 4 9 16
1 2 8 /3 3
0.116741 0.204910 0.259785 0.289080
1 1, 2 1, 2, 4, 5, 8 1, 2, 4, 5, 8, 9,10, 13, 18
bulk
∞
n1cp ) 1 n1cp ) 4, n2cp ) 2 n1cp ) 12, n2cp ) 8, n4cp ) 6, n5cp ) 8, n8cp ) 2 n1cp ) 24, n2cp ) 18, n4cp ) 16, n5cp ) 24, 13 18 n8cp ) 8, n9cp ) 8, n10 cp ) 12, ncp ) 8, ncp ) 2
4
0.361270
generated from explicit interactions between the encapsulated ions and the nanotube wall. The answering of this question raises the possibility that the nanotube plays no role beyond that of an essentially passive spectator which simply constrains the crystal geometry by defining the numbers (m1 and m2) of rows and columns in the planes. In this event, the wall would provide no more than a potential of the square well type in which the interaction with any ion is zero unless it directly encounters the wall in which event the interaction is sufficiently large as to expel the ion from the immediate vicinity of the wall. Disentangling the effects of ion-wall interactions from those generated solely by the reduced dimensionality of the crystals requires that one understands the structures and energetics of isolated crystals of the type shown in Figure 1, which, therefore, form the main subject of the present investigation. The overarching objective of this paper is therefore to develop a general theory which explains the inter-relationships between both crystals differing only in the number (m) of rows and columns in the planes and between crystals having the same m but chemically different ions. Such a theory must also expose the relationship between the nanocrystals and the bulk material having the rock-salt structure, thereby showing how the properties of the nanocrystals evolve toward those of the bulk as the number of rows and columns in the planes is increased. The initial step in generating a physically transparent yet general theory of all such nanocrystals is to consider only the dominant interactions. These are the electrostatic interactions between the ions considered as point charges and the short-range repulsive interactions between each cation and just its immediately neighboring anions. This approach not only ensures that the theory will be mathematically tractable and appear in a relatively simple form but is also provides the benchmark against which the effects of introducing further but smaller interactions can be gauged. Such a theory should ideally incorporate the effects of chemical variation of the ions through a very small number of parameters defining the short-range interactions. The present theory can therefore be regarded as the natural extension, to nanocrystals, of the classic Born description18 of bulk materials. 2. Analytic Deduction of Overall Structural Trends 2.1. Basic Definitions. The Born model,18 although semiempirical, provides the simplest satisfactory overall account of the cohesion of bulk ionic crystals. This emerges as the balance between the overall Coulombic attractions of the ions, treated as point charges, and the short-range repulsive forces between the closest pairs of ions. The latter interactions, maintaining the crystal at equilibrium against the attractive point coulomb term, can, conceptually, be taken to include all effects arising from the finite spatial extension of the ions. These include not only the overlap induced repulsion that is ultimately a purely quantum effect originating from the Pauli principle but also any modifications of the purely electrostatic interactions. These two interactions, namely, the point Coulombic and the short-range repulsion, will also make the largest contributions to the cohesive
energy, which in turn determines the structure of the nanocrystals in the absence of interaction with the encapsulating nanotube. The simplest yet still mathematically tractable overall account of the structures and cohesive energetics of such crystals is therefore developed by including only these two terms. The need for avoiding excessively complicated and possibly intractable mathematics dictates that the short-range repulsion between two ions separated by a distance r is taken to have the original Born form A/rn rather than the possibly more accurate BornMayer expression.18 It is now well established19 from measurements of the compressibilities of a wide range of essentially ionic crystals that the Born exponent n is the average of independent cation and anion contributions. The contribution from an ion having a 2p6 outermost electronic configuration is 7; those from ions having such configurations of 3p,6 4p6, and 5p6 are 9, 10, and 12, respectively, while the Li+ contribution is 5. For any of the nanocrystalline alkali halides, the individual planes will necessarily be square in cross section in the lowest energy structure because there is no mathematical distinction between the cations and anions in the present Born type model. The geometry is therefore defined by just the two parameters a and b defined in section 1. For a nanocrystal containing 1 mol of stoichiometric formula units and with geometry thus defined, the total crystal cohesive energy NAU(a, b), negative for a bound crystal and measured relative to the sum of the energies of the free isolated ions, is given by
(
NAU(a, b) ) NA Umad(a, b) +
)
2A CA + n bn a
(1)
Here, NA is Avogadros number so that NAUmad(a, b) is the total Madelung energy of 1 mol. The term 2A/bn describes the repulsion experienced by any one ion with its two neighboring ions in the same chain. The constant C is defined such that NACA/an is, for 1 mol, the sum of all the short-range repulsions between ions in neighboring chains. Numerical values of these constants are reported in Table 1. For systems, such as the 2 × 2 nanocrystal or a bulk cubic crystal, in which all the chains are located in the same environment, the constant C is the number of anions located at distance a from any one cation, any such anion being in a neighboring chain. Thus, for such systems, C equals the coordination number minus 2. The derivation of C for systems in which not all the chains are in the same environment can be illustrated by considering 1 mol of a 3 × 3 nanocrystal, this having NA/9 cations in each chain. Each cation at the center of one of the planes, having also cations at the four corners, experiences repulsion from four anions, so that all such cations contribute 4NA/9 repulsions to the molar energy. Although each cation at the corner experiences only two such repulsions, there are four such cations, so that all these cations contribute 8NA/9 repulsions to the molar energy. Planes of the neighboring type contain only cations at the middle of the edges, there being four such ions. Since each of these experiences three repulsions, all such ions contribute 12NA/9
5938 J. Phys. Chem. B, Vol. 110, No. 12, 2006
Bichoutskaia and Pyper
repulsions to the molar energy. The total number of repulsions from cations in three of the environments just considered is 24NA/9 which explains the value of 8/3 for C reported in Table 1. The C value for the 4 × 4 crystal can be similarly derived. The observation that every pair of ions experiencing such a short-range repulsion will also contribute -1/a to the Madelung energy of the crystal allows the C constants to be extracted from the calculation of Umad(a, b) to be presented in section 2.3. 2.2. Application to a One-Dimensional Crystal. For a onedimensional (1D) crystal consisting of a single alternating sequence of NA cations and NA anions, the form of expression 1 for its cohesive energy U1D(b), depending on just the single geometrical parameter b, reduces to
U1D(b) ) -
M1D 2A + n b b
(2)
The Madelung constant M1D is readily evaluated through ∞
(-1)m+1
m)1
m
M1D ) 2
∑
) 2 ln 2
(3)
The equilibrium separation, b1D e , is that minimizing U1D(b) and is found from 2 to be given by
b1D e )
( ) 2nA M1D
1/(n-1)
(4)
The standard expression for the cohesive energy of 1 mol of bulk three-dimensional crystal having the rock-salt structure with internuclear separation R differs from 2 only in that M1D is replaced by the Madelung constant M6:6() 1.74756456) and that the repulsive term becomes 6A/Rn. For this bulk crystal, the equilibrium separation, denoted R6:6 e , is given by
R6:6 e )
( ) 6nA M6:6
1/(n-1)
(5)
It follows from 4 and 5 that the ratio of the equilibrium separations in the 1D chain and the bulk is given by
b1D e
( )
1 M6:6 ) 3 M1D R6:6 e
1/(n-1)
(6)
Result 6 shows that the fractional change in the interionic distance on passing from the bulk to the 1D chain is independent of the strength, as defined by A, of the short-range repulsion. It is shown in the next section that this is a general result valid for all the nanocrystals. Thus, relation 6 shows that the fractional changes in the geometry are determined solely by the range of the repulsion defined by the exponent n. Since the ratio in brackets in 6 is less than unity, the ionic separation in the chain is predicted to be contracted relative to that of the bulk. It is 6:6 predicted from the numerical values for b1D e /Re , equal to 1/(n-1) 0.4202 and depicted in Figure 2 as a function of n, that the greatest fractional contraction of 0.841 will occur for LiF, for which n is 6. It is further predicted from this relation that the contraction decreases with increasing n until it vanishes in the hard sphere limit of n tending to infinity. In physical terms, it can be seen that these contractions arise because, on passing from the bulk to the 1D system, the factor of 3 reduction in the number of short-range repulsions is much greater than the fractional reduction in the Madelung energy by the ratio (2 ln 2)/1.747.
Figure 2. Predicted ratios of the optimum constrained (a ) b) separations in nanocrystals relative to the bulk. The lines are from Born model 54: solid lines are for the 1D system, and dotted lines are for the 2 × 1 crystal. A ) A6:6 for lines 1 and 2; lines 3 and 4 (dotted) have f ) 1.5; lines 5 (solid) and 6 have f ) 3. The crosses and circles are for the 1D crystals of indicated compositions. The crosses are derived from 54 and 53 using the f value taken from the ab initio computations, as described in section 3.2.1. The circles are derived by locating the R value minimizing the ab initio U1D(R) with inclusion of the dispersive attractions.
2.3. Difference between the Longitudinal and Transverse Separations. 2.3.1. Fundamental Equations and Deductions. For the nanocrystals containing more than a single chain, it is useful to distinguish between the intra- and interchain contributions to the total Madelung energy NAUmad(a, b). Although there are NA/nc cations in a nanocrystal composed of nc chains, the total intrachain Madelung energy in a crystal having nc chains is still given by -NA(2 ln 2)/b, unchanged from the single chain system. After defining the total interchain Madelung energy in 1 mol as NAUmad ic (a, b), the energy U(a, b) in expression 1 for the cohesion becomes
U(a, b) ) -
2A CA 2 ln 2 + Umad ic (a, b) + n + n b b a
(7)
For the purposes of calculating Umad ic (a, b), the relevant properties of any pairs of chains are specified by the integer q which yields the perpendicular distance between the two chains. This is the distance between two ions in the same plane, each of which lies in one of the two chains. If the ion in the second chain has x and y coordinates of ja and ka measured relative to the first ion, so that j and k are integers, then q is defined as (j2 + k2), so that distance a(j2 + k2)1/2 between the two ions is axq. Table 1 presents all the q values arising in each nanocrystal. Since the two ions will have the same charge if q is even but opposite charges when q is odd, the electrostatic interaction energy of an ion in the first chain with the ion in the second chain is (-1)q/(axq). The distance between the ion in the first chain and one in the second, if the latter resides in a plane having z coordinate of mb relative to the first plane, is (qa2 + m2b2)1/2, where m is a positive integer. The electrostatic energy of interaction between these two ions is therefore (-1)q+m/(qa2 + m2b2)1/2. The total electrostatic interaction energy of interaction between the one ion in the first chain with all ions in the second chain is therefore given by (-1)q/(axq) ∞ (-1)q+m/(qa2 + m2b2)1/2 because, for nonzero m, + 2∑m)1 there are two planes located at a distance mb from the first plane. Since any chain contains NA/nc cations, the electrostatic interaction energy between all these cations in the first chain with all
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J. Phys. Chem. B, Vol. 110, No. 12, 2006 5939
the ions in the second chain is given by multiplying this energy by NA/nc. If there are nqcp pairs of chains of the type defined by q, the total electrostatic interaction in all chain pairs of the type q between all cations in the first chain in any pair with all other ions is given by a further multiplication by nqcp. The required total interchain Madelung energy Umad ic (a, b) will be twice the energy already calculated in order to include the interaction of all anions, in the first chain in any of the pairs, with all other ions in the second chain in each pair. Hence, Umad ic (a, b) is given by
Umad ic (a,
b))
2
∑q
nc
(-1)qnqcp
(
∞
1
axq
-2
∑
)
(-1)m+1
m)1(qa
2
+ m2b2)1/2
(9)
which allows 8 to be written in the form
Umad ic (a, b) ) where
Mb(x) )
2 nc
∑q
(-1)q+1nqcp
(
Mb(x) b
∞
1
xxq
-2
∑
(10)
(-1)m+1
m)1(qx
2
)
+ m2)1/2
(11)
This allows U(a, b) to be expressed as
U(a, b) ) -
2 ln 2 + Mb(x) 2A CA + n+ n b b a
(12)
Since Mb(x) is dimensionless being, for any given structural type, a function of solely the variable x, it can be regarded as a generalization of the Madelung constant that can be called the Madelung function. The structures of the nanocrystals are best understood by first considering optimizing the geometry under the constraint that a ) b, followed by elucidating any further changes that may result from removing this constraint. Any such constrained value of b() a) will be denoted R with the R minimizing U(b, b) () U(R)) denoted Re. Noting that, for such constrained geometries, Mb(x) becomes the constant Mb(1) shows that
dU(R) 2 ln 2 + Mb(1) nA(2 + C) ) dR R2 Rn+1
(13)
Application of the condition dU(R)/dR|R)Re ) 0 yields the relation
2 ln 2 + Mb(1) -
nA(2 + C) n-1
R
)0
(
nA(2 + C) 2 ln 2 + Mb(1)
(14)
)
1/(n-1)
(
)
nA(2 + C) MR
)
1/(n-1)
(15)
The quantity MR, defined through the second equality (15) as 2 ln 2 + Mb(1) is just the Madelung constant for the constrained crystal. Division of eq 15 by result 5 for the separation R6:6 e in the bulk shows that
R6:6 e
(8)
The numbers of pairs of chains of the type q are presented in Table 1. The factor NA(2/nc)n1cp gives, for 1 mol, the number of electrostatic interactions between closest ion pairs, so that this equals the number of short-range repulsions. This provides an alternative derivation of the coefficient C in eq 7, showing it to take the value (2/nc)n1cp. It proves useful to introduce the definition of the ratio x of a to b by
a b
Re )
Re
nqcp
x)
which shows Re to be given by
)
(
)
2 + C M6:6 6 MR
1/(n-1)
(16)
The values of Mb(1) evaluated from eq 47, as described in section 3.1.1, are presented in Table 1. The value for the 2 × 1 crystal shows, as expected, that two chains attract, their interaction energy being -0.116741/R. However, this interchain attraction is much less than that of - (2 ln 2)/R within a single chain. Examination of the top part of Figure 1 reveals the physical origin of this difference. Within any chain, each ion has two closest neighbors at a distance R and of opposite charge with the two closest neighbors of the same charge being located at the large distance of 2R. However, not only does each ion have only one oppositely charged neighbor at a distance R in the adjacent chain, but also there are two neighbors of the same charge at a distance of x2R in the other chain. For 1 mol of 2 × 2 crystal, although there are twice as many closest interchain attractions of energy -0.116741/R, the total interchain interaction energy is less than twice the value for the 2 × 1 crystal because each pair of chains separated by the distance x2R repel rather than attract, these contributing +0.02857/R to the Madelung energy. Examination of the bottom part of Figure 1 reveals the origin of this repulsion, the closest interaction at a distance x2R being between two ions of the same charge. The values of Mb(1) in Table 1 show that the ratio M6:6/(2 ln 2 + Mb(1)), although greater than one, is always closer to unity than is the fraction (2 + C)/6. It then follows from 16 that the constrained (Re) equilibrium separation is reduced compared to that of the bulk. Furthermore, on descending in Table 1, that is, on passing to nanocrystals with progressively larger cross sections, the Madelung energy ratio decreases less rapidly than the increase of the ratio (2 + C)/6. This shows that the fractional contraction of the constrained Re from the bulk separation R6:6 e decreases with increasing nanocrystal cross section. For the 2 × 1 crystal, the dependence on n of ratio 16, equal to 0.58141/(n-1), is shown in Figure 2. Comparison of this result with that for the 1D chain shows that the latter exhibits the largest fractional contraction at fixed n. This therefore continues, to the limit, the trend of increasing fractional contraction with decreasing nanocrystal cross section. The direction of the individual changes in a and b which lower the energy on relaxing the structure from the optimum constrained a ) b, x ) 1 geometry is determined by the signs of the two derivatives dU(a, b)/da|b and dU(a, b)/db|a when evaluated at x ) 1. These derivatives are readily derived from 12 for all values of x by using the chain rule and noting that dx/da|b ) 1/b and dx/db|a ) -a/b2. The results are
|
dU(a, b) 1 dMb(x) nCA - n+1 )- 2 da b dx b a
|
(17)
2 ln 2 + Mb(x) a dMb(x) 2nA dU(a, b) ) + 3 - n+1 (18) db a dx b2 b b
5940 J. Phys. Chem. B, Vol. 110, No. 12, 2006
Bichoutskaia and Pyper
Evaluation of these derivatives at the optimal constrained (x ) 1) geometry for which a ) b ) Re and addition of the results show that
|
dU(a, b) da
+
a)b)Re
|
dU(a, b) ) db a)b)Re 2 ln 2 + Mb(1) 2
-
nA(2 + C)
Re
Ren+1
) 0 (19)
The last step, showing that the entire quantity 19 vanishes, follows because the expression on the right of the first equality sign is nothing but constrained equilibrium condition 14. Result 19 shows that one of the distances a or b must decrease on relaxing the structure from the constrained equilibrium geometry while the other increases. This shows that at least one of these distances must be less than the separation (R6:6 e ) in the bulk material. 2.3.2. Ratio between the Longitudinal and TransVerse Separations. Comparison of ratio x () a/b) with unity will determine which of the distances contracts on relaxing from the optimum constrained geometry. An equation for x is derived by writing the term CA/an in 12 as (1/bn)CA/xn, thus converting U(a, b) into a function U(x, b) of just the variables x and b. The two partial derivatives dU(x,b)/db|x and dU(x, b)/dx|b, which are both zero at the optimal unconstrained geometry for which b ) be and x ) xe, are then found from 12 to be
|
2 ln 2 + Mb(x) nA(2 + Cx-n) dU(x, b) ) db x b2 bn+1
|
dU(x, b) nCA 1 dMb(x) - n n+1 )dx b b dx bx
(20)
(21)
The vanishing of 20 at the relaxed equilibrium geometry shows that
2 ln 2 + Mb(xe) nA ) n-1 be 2 + Cxe-n
(22)
Substitution of this relation into 21 shows, after noting that this also vanishes at the relaxed equilibrium geometry, that
-
2 ln 2 + Mb(xe)
|
dMb(x) dx
) xe(1 + 2C-1xen)
(23)
x)xe
Four interesting results plus a further useful inequality can be deduced from this relation. The useful inequality is that, since the three quantities (2 ln 2 + Mb(xe)), xe, and C are necessarily positive, it follows that dMb(x)/dx|x)xe must be negative. The first main deduction from 23 is that this shows that, given the structural type of the nanocrystal, the ratio of a/b is independent of the strength A of the short-range cation-anion repulsion with xe being determined solely by the range of the potential as defined by the Born exponent n. This range decreases with increasing n. The second main deduction from 23 is, as would be expected, that xe is not unity for the nanocrystals. This follows after noting that for the bulk material, which is just the limit of the sequence of nanocrystals considered in Table 1 for which the planes consist of m1 rows and columns as m1 tends to infinity, 23 is satisfied for xe ) 1 with C ) 4. Considering still the case of x ) 1, the ratio on the left-hand side of 23 will be different for
different nanocrystals, all these ratios being different from that in the bulk, and moreover, this ratio will not vary with the nanocrystal as 1 + 2C-1. The right-hand side (RHS) of 23 depends only on the number of nearest neighbors and involves terms of short range while the left-hand side originates from the electrostatic interactions which, being of long range, involve the interactions of all the ions. These observations show that, for the nanocrystals, 23 will be satisfied by values of xe that are not unity. The third deduction follows from 23 by noting that while its left-hand side is independent of n, the right-hand side contains xen with the exponent n being a number no less than 6. Since it has already been shown that xe will not be unity for a nanocrystal, the right-hand side of 23 will vary much more rapidly with x for fixed n than will the left-hand side. It therefore follows that, as n increases, a smaller deviation of xe from unity is required to satisfy 23. This means that the relaxation of any nanocrystal from its optimal constrained geometry defined by 15 will decrease with increasing n. The fourth deduction from 23 is the determination of the size of x when compared to unity. This determines whether, on relaxation from the optimal constrained geometry, a increases with b decreasing or vice-versa. On passing from the bulk upward from the bottom through the nanocrystal sequence in Table 1, the right-hand side of 23 has the values, 3/2, 5/3, 7/4, 2, and 3 for x ) 1. The magnitudes of dMb(x)/dx should be expected to follow Mb(x) in decreasing with decreasing nanocrystal cross section and, moreover, to be roughly comparable with the Mb(x). This shows that -dMb(x)/dx will be small compared with 2 ln 2 because the data in Table 1 show that the Mb(x) values have this property. The values of the left-hand side of 23 evaluated for x ) 1, therefore, increase more rapidly with decreasing nanocrystal cross section than does the factor 1 + 2/C already listed. This shows that since, for the bulk, 23 is satisfied with x ) 1 with both sides equal to 3/2, the lefthand side will become progressively larger than the right-hand side on traversing this sequence if x is kept at unity. For example, for the 2 × 1 crystal, evaluation of dMb(x)/dx|x)1 from 11 yields -0.426 which taken in conjunction with the Mb(1) value in Table 1 leads to a value of 3.53 for the left side of 23. Since the right side of 23 contains x raised to the high power n, increasing x from unity will cause the right side to increase much more rapidly than the left side of 23 which contains only inverse powers of geometric parameters. Hence, 23 requires values of x, increasingly greater than unity, to be satisfied as one passes up through the sequence of nanocrystals in Table 1. It has therefore been shown that relaxation of any nanocrystal from its constrained equilibrium geometry will cause b to decrease while a increases. This result is readily understood physically as being a consequence of the Coulombic binding within each chain being much greater than that between different chains. 2.3.3. Interchain Coulomb Energy and Nanocrystal Relaxation. Result 18, when taken in conjunction with both the corresponding derivatives of relation 2 for energy of a 1D chain, yields expressions which show how the interchain coulomb energy varies as the crystal structure is relaxed toward its equilibrium structure. Division of eq 4 by eq 15 yields
(
MR b1D e 2 ) Re 2 (2 + C) ln 2
)
1/(n-1)
(24)
Since the ratio MR/(2 ln 2) is closer to unity than is 2/(2 + C), result 24 shows that, in any nanocrystal having a cross section
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J. Phys. Chem. B, Vol. 110, No. 12, 2006 5941
of two or more chains, its constrained equilibrium separation Re is greater than that b1D e in the 1D system. The observation that, for all distances b greater than b1D e in the 1D system, the derivative dU1D(b)/db of 2 will be positive shows that
2 ln 2 2An - n+1 > 0, for all b > b1D e b2 b
(25)
which on a simple rearrangement yields
C (1 + )n+1 ) [(G - 1) + (H - 1)] 2
It is useful to introduce the two further constants defined by
k1 )
This shows from 18 that, since dU(a, b)/db|a)ae,b)be is zero,
Mb(xe) +
|
ae dMb(x) be dx
x)xe
