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Hyperbranched Polymers and Aggregates: Distribution Kinetics of Dendrimer Growth
Journal of Colloid and Interface Science 216, 235–241 (1999) Article ID jcis.1999.6297, available online at http://www.idealibrary.com on
Hyperbranched Polymers and Aggregates: Distribution Kinetics of Dendrimer Growth Benjamin J. McCoy Department of Chemical Engineering and Materials Science, University of California, Davis, California 95616 E-mail: [email protected] Received October 5, 1998; accepted April 27, 1999
in volume and mass. An analytic method is needed to describe the dynamics of such cluster distributions and properties. When the branched structure is self-similar at different length scales, a fractal (power-law) relationship holds. Such dendrimers are extreme cases of branching, at the opposite pole from unbranched, linear-chain structures. Comprehending the relationship between kinetics and geometry for fractal dendrimers may help understand other hyperbranched molecules and aggregates, whether natural or synthetic. The present deterministic method differs from other iterative constructions that repeat branched groups from the preceding step (1, 2) (see A in Fig. 1) or grow branches along the length of all previous branches (7) (B in Fig. 1). Earlier work did not fully address the dynamics of growth governed by reaction and diffusion rates, or the distribution of clusters. The aim is to develop an analytic approach to growth dynamics of fractal aggregates based on kinetics of monomer addition to branched dendrimers. We assume that branching occurs at time intervals that provide average fractal dimensions for the distributed branch lengths (Fig. 2). Following the suggestion (8) that polymer concepts can be applied to colloidal growth processes, we require that branches in an aggregate grow by monomer addition. Cluster– cluster aggregation can form symmetric fractal structures only under special circumstances, such as illustrated in Fig. 1A. Growth by monomer addition occurs exclusively at the branch tips, thus resembling computer simulations that show diminished growth in the shielded interior of a diffusion-limited aggregate. The clusters grow from seeds (initial buds) by chain-end addition of monomer. To formulate cluster properties such as total cluster mass as a function of time, we specify the instants in time when iterative branching occurs (new buds form at the end of a branch). These points in time denote the sequential time intervals for hierarchical branch formation (Fig. 2). Exact expressions for moments of branch distributions, based on a continuous-distribution population balance, provide measurable properties, for example, number of branches, average cluster mass, volume, and mass-average molecular weight. These properties change with time for irreversible monomer addition.
We present an analytic solution for growth of branched aggregates or polymers with distributed cluster (dendrimer) size. Monomer addition to each branch follows first-order polymerization kinetics leading to a distribution of branch lengths. The rate constant for monomer addition is considered diffusion-dependent. Deterministic branching occurs so that at prescribed times t j ( j > 1), p branches emanate from the tip of a branch that began to grow at t j21. When the ratio of average branch lengths is constant, L j/L j21 5 a, the fractal dimension of branches is ln( p)/zln(a)z. Closed expressions for cluster mass moments show unbounded growth with time unless ap < 1. The number of clusters (zeroth moment) is constant during growth and equal to the number of initiating buds. Expressions for the number of branches and clusters, average cluster mass, volume, density, and viscosity in solution are functions of p and a. Density and molecular weight show features similar to observed behavior of dendrimers and hyperbranched polymers. © 1999 Academic Press Key Words: branched polymer; aggregate; fractal; dendrimer; distribution kinetics.
INTRODUCTION
Dendrimers and hyperbranched polymers occur in many natural phenomena and can also be synthesized for scientific, engineering, and medical applications (1–4). Symmetric dendrimers, produced by organic synthesis from initiator seeds (3, 4), can be tailored to have fractal branch properties by controlling the polymerization process. Monomer addition will form shells of branches around the initiating core. Such growth or aggregation is encountered in many processes, including polymerization and precipitation (1, 2). Rate studies for dendrimer synthesis by chainpropagation monomer addition, however, have been confined to diffusion-limited aggregation, which is simulated by monomers that undergo random walks and deposit on branches of a growing aggregate. Computer models of diffusion-limited aggregation yield structures that resemble electrolytic deposition, viscous fingering in porous media, and electrical discharge patterns (1, 2, 5). Computer simulation is restricted to monomer numbers that are far smaller than in realistic dendrimers (6). Moreover, multiple clusters formed by monomer addition will usually be distributed 235
0021-9797/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.
236
BENJAMIN J. MCCOY
during t j21 # t , t j , the zeroth moment (q 5 0) is the number of branches, the first moment (q 5 1) is the mass of branches, and the second moment (q 5 2) is related to the variance of the distribution. Average mass of branches formed during the jth interval is defined as ~0! c avg 5 c ~1! j j /c j .
[2]
Because each branch starts as a bud of zero mass, the initial condition for branch growth at each time interval is c j ( x, t 5 t j ) 5 c j(0) (t j ) d ( x), where at j 5 0, t j 5 0. For q $ 1 all moments of these initiating distributions vanish (buds have no mass, for example). For t 5 0 we write the number of cluster buds as c 0(0) (0) [ c o . We define the branching number (multiplicity) p as the number of buds started at the instant t 5 t j , ~0! ~t j ! 5 pc ~0! c j11 j ~t j !.
FIG. 1. Previous iterative methods to construct a fractal branched aggregate. (A) The branched aggregate from the preceding step is repeated and added to the aggregate (1, 2). (B) Branches are allowed to grow along the length of all previous branches (7).
For branches growing during the time interval t j21 # t , t j , we write the branch distribution such that c j ( x, t)dx is the number of branches with mass in the differential range ( x, x 1 dx). Mass moments are defined as c ~q! j ~t! 5
E
`
c j ~ x, t! x q dx.
For any branching instant t j ( j $ 0), it follows that the number of new branches is j c ~0! j ~t j ! 5 p c o.
BRANCH PROPERTIES
[1]
0
For a discrete distribution with x 5 ix m , the integration is replaced by a summation (9) over i. For branches growing
[3]
[4]
In Fig. 2, we have p 5 3 and the number of branches at j 5 0, 1, 2, 3, and 4 is 0, 3, 9, 27, and 81, respectively. Unlike computer simulations, which have been limited to small degrees of branching due to the nature of the monomer attachment, our analytic approach allows arbitrarily large values of p, which would be appropriate for dense (compact) clusters. Growing branches are distributed in mass for the same reason that chain-end polymerization (9) yields a molecularweight distribution with a range of branch lengths (or mass). The length and number of branches have a fractal (power law)
FIG. 2. Growth of a cluster with p 5 3 and a ' 0.7. The growth interval, j, and the branching instant, t j , are illustrated. The number of branches, c j(0) (t j ), and average mass, c javg(t j ), follow Eqs. [4] and [6], respectively.
237
HYPERBRANCHED POLYMERS AND AGGREGATES
FIG. 3. Examples of clusters formed by monomer addition. The average length of each branch is L j , which is proportional to the average mass, c javg. The branching number is p, the number of iterations is n, and D 5 ln( p)/ln(1/a). A, B, and C are based on bifurcations ( p 5 2). C is formed by branching off six initial stems or by superposing three clusters with p 5 2.
relationship if the ratio of average mass (or length) at t j11 and t j is constant, avg c avg j /c j21 5 a.
[5]
Because branch length is proportional to branch mass, this can be written as L j /L j21 5 a, from which it follows that for j $ 1, L j 5 a j21 L 1 , or c avg 5 a j21 c a , j
[6]
in terms of c a [ c 1avg(t 1 ). Within the jth time interval for a given cluster the statistical self-similar relationship between number of branches and branch length can be expressed as the power law, p j 5 (L 1 /a j L 1 ) D . The fractal dimension for this geometrical structure is D 5 ln~ p!/uln~a!u.
Similar expressions (10) have been proposed for other regular branched structures. The dilational symmetry is determined by fractal geometry and not directly by mass or volume. If p 5 2 and a 5 1, the cluster is a Cayley tree (Bethe lattice) (11, 12). In the limit as both p and a approach unity, we have D 3 1, and the cluster is star shaped (Fig. 3E) with low density. If p is large and branches are short, the resulting compact cluster is very dense. Mandelbrot (12) illustrates treelike structures with D ' 1 to 2 when p 5 2 and a ' 0.5 to 0.7. In the present treatment, branch diameter, flexibility, sinuosity, and angle off a stem are unspecified, and branches may grow in any-dimensional space. Cluster properties are summations of all branches up to a branching instant j 5 n. The total number (13) of branches, by Eq. [4], is the finite sum of powers of p,
[7]
Oc n
Nn 5
j51
~0! j
~t j ! 5 pc o ~ p n 2 1!/~ p 2 1!,
[8]
238
BENJAMIN J. MCCOY
independent of branch length ratio, a (3, 4). The average number of branches per cluster is N an 5 N n /c o. The total cluster mass M n is the sum over all first moments, and can be expressed in terms of c a, the average cluster mass after the first growth of branches. We substitute c j(1) 5 c j(0) c javg, along with Eqs. [4] and [6]:
diffusion resistance (shielding effects in computer simulations) can cause inhomogeneous branch growth. The population balance equation for branch size distribution is (14)
c~ x, t!/t 5 2kc~ x, t!
Oc n
Mn 5
j51
O ~ pa! n
~1! j
~t j ! 5 c a c o p
E E
`
m~ x, t!dx
0
j21
j51
5 c a c o p~1 2 a n p n !/~1 2 ap!.
1k [9]
The average cluster mass after n time intervals is M n /c o , which is made dimensionless by dividing by c a ; thus M an 5 M n / (c o c a ). The number-average molecular weight is thus M n /c o 5 M anc a . For large n, the scaled cluster mass approaches a limit for ap , 1; thus M an 3 p/(1 2 ap), and an asymptote for ap $ 1; thus M an 3 p( pa) n /(ap 2 1). The variance of the cluster mass distribution is ~2! ~0! avg 2 c var j 5 c j /c j 2 ~c j ! .
[10]
Further computation of weight-average molecular weight or variance requires an expression for time dependence of the second moment, c j(2) . MONOMER ADDITION KINETICS
The time dependence of the cluster properties is determined by polymer growth kinetics (14, 15). Cluster growth occurs by monomer addition to the end of a branch (chain) in an existing cluster (or initiator). By adding one monomer at a time, a chain or branch of mass x, C( x), grows with a rate denoted by the coefficient k,
x
m~ x9, t!c~ x 2 x9, t!dx9.
[13]
0
For monomer of mass x m , one substitutes the distribution, m( x, t) 5 m (0) d ( x 2 x m ), into Eq. [13], dc/d t 5 2c~ x! 1 c~ x 2 x m !,
[14]
in terms of t [ km (0) t. If we define the discrete mass in units of i 5 x/x m , the discrete population balance equation is obtained. The solution to this difference-differential equation (9) is the Poisson distribution, t i exp(2 t )/i! for i 5 0, 1, 2, . . . , which has average and variance (for i) both equal to t. We assume that the monomer concentration, m (0) , is in great excess and therefore constant. This assumption is comparable to computer simulations for which new monomer enters the system at each addition step. If the monomer concentration were finite in a closed system, then aggregate growth would occur until monomer was depleted. The moment equations (14, 15) for growing branches can be derived by applying * 0` dxx q [ ] to Eq. [13]. In the last term, the order of integration for x and x9 is interchanged, y 5 x 2 x9 is defined to eliminate x, and the binomial expansion is applied. The result is
O ~ !c q
k
C~ x! 1 M~ x m ! ¡ C~ x 1 x m !.
[11]
Here, M( x m ) is a monomer of mass x m . The addition rate is considered proportional to the number of chain-ends and is thus independent of the chain length. For diffusion-influenced reactions (16), the rate constant is k 5 k /@1 1 k /~4 p D m r!#,
[12]
in terms of the diffusion-independent reaction rate constant k, the monomer diffusivity D m , and the reaction radius r. For diffusion-controlled addition reaction, we have k /(4 p D m r) @ 1, and therefore k . 4 p D m r, the Smoluchowski representation of the rate constant for stationary clusters (9). For wellstirred mixtures of clusters and monomers, concentration gradients will not exist external to clusters, but intracluster
dc ~q! /dt 5 2kc ~q! m ~0! 1 k
q i
~q2i!
m ~i! ,
[15]
i50
where m (q) 5 m (0) x mq is the qth moment of the monomer distribution. The moment equations provide expressions applicable to chain-end polymerization (14, 15). The zeroth moment (q 5 0) of the branch distribution is determined by (Eq. [15]) dc ~0! /dt 5 0,
[16]
and the number of branches is constant during any growth interval, Dt j 5 t j 2 t j21 . The total number of clusters, c o, is thus also constant. When time is scaled by defining u 5 ( x m /c a )km (0) t, the first moment equation (q 5 1) is dc ~1! /d u 5 c a c ~0! ,
[17]
239
HYPERBRANCHED POLYMERS AND AGGREGATES
which in terms of the growth interval D u j 5 u j 2 u j21 integrates to ~0! j c ~1! j ~ u j! 5 c ac j D u j 5 c ac op D u j,
[18]
when Eq. [4] is substituted. Substituting Eqs. [2] and [6] shows how the time interval is related to branch parameters:
M aw 5 p@S 22 1 ~ x m /c a !S 12 #/S 11 .
[25]
When radius equals the maximum total length of branches that can form a stem, cluster volume in d-dimensional space is
O L! n
Vn 5 gd ~
j
d
5 g d L d1 @~1 2 a n !/~1 2 a!# d ,
[26]
j51
D u j 5 a j21 5 c avg j /c a .
[19]
Thus the average added shell thickness for the jth growth interval is proportional to Dt j . Total branching time during aggregation can be represented as the sum over time intervals for branch growth,
O Du 5 ~1 2 a !/~1 2 a!, n
un 5
n
j
[20]
j51
when a j21 in Eq. [19] is summed. For a , 1 and large n, we have a n ! 1, and elapsed time, 1/(1 2 a), is constant, an indication that added branches are so small that growth has effectively ceased. For a $ 1, cluster growth is unlimited as n increases. Other rate expressions in the population balance equation, which may be more appropriate to other branching phenomena, for example, biological systems, could be postulated. The present model thus provides a framework for generalized dendrimer growth dynamics. The second-moment differential equation according to Eq. [15] is dc ~2! /d u 5 c a @2c ~1! 1 x m c ~0! #,
[21]
which yields after substitution for the first and zeroth moment followed by integration c ~2! 5 x m c a c o p j D u j ~D u j c a /x m 1 1!.
[22]
When branch multiplicity is included by multiplying by p j (Eq. [4]) and the result is summed, we have
Oc n
~2! j
5 x m c a c o p 2 @~c a /x m !S 22 1 S 12 #
[23]
j51
in terms of S ij 5 @1 2 ~a i p j ! n #/~1 2 a i p i !#.
[24]
Weight-average molecular weight, M w avg, is given by Eq. [23] divided by M n from Eq. [9], and is furthermore made dimensionless by dividing by c a ; thus,
independent of branching number, p. For d 5 2 or 3, the coefficient for the circle or sphere is g d 5 p or 34p, respectively. The reduced average cluster volume is V an 5 @~1 2 a n !/~1 2 a!# d 5 u dn ,
[27]
when Eq. [20] is substituted. The reduced characteristic length for the cluster is obviously u n . Average cluster mass density can be represented as average mass divided by average volume of clusters. Scaled average cluster mass density is thus r an 5 M an/V an. RESULTS AND DISCUSSION
Some special cases of the general expressions are of interest. For p . 1 and n 5 1, the clusters are stars (Fig. 3E) composed of p branches that each grow as polymer chains. For p 5 1 and n 5 1 we have a distribution of single polymer chains growing at one end by monomer addition. As with the star, reduced average mass and variance are equal, as expected for chain-end polymerization (9). Evaluation of the closed-form equations for number, mass, volume, and density of clusters (N an, M an, V an, and r an) as functions of branching interval n is straightforward. Observation of branched clusters, whether generated by computer simulations (1, 2) or in the laboratory (3, 4), suggests that bifurcation ( p 5 2) is the common form of branching. Branch lengths are distributed by polymerization kinetics, leading to structures similar to Fig. 3C. Plotted versus the elapsed interval n for p 5 2, the mass (Eq. [9]) increases to limit values when a , 12 and increases without limit when a $ 21 (Fig. 4). The volume, however, even when defined to have its maximum possible value as in Eq. [26], approaches a finite value when a , 1. Except for a $ 1, volume versus n (Fig. 5) reaches limits that increase with a as indicated by Eq. [27]. Cluster density versus n (Fig. 6) shows realistically that a minimum occurs for a $ 21. This behavior is caused by the diminishing volume increase and stronger mass increase when a . 0.5, illustrated by Figs. 5 and 4, respectively. Such density minima were reported by Tomalia and Durst (3) for organically synthesized branched dendrimers and are in accord with the present theory. The regular branching patterns of dendrimers, such as those in Fig. 3, provide a basis for comparing and classifying
240
BENJAMIN J. MCCOY
FIG. 4. Average cluster mass, M an 5 p(1 2 a n p n )/(1 2 ap), versus branching interval, n, for p 5 2.
FIG. 6. Average cluster density, r an 5 M an/V an, versus branching interval, n, for p 5 2.
branched polymers (17, 18). Ideal dendrimers are synthesized with a single monomeric unit making up each branch, so that the polydispersity index, P D, is essentially unity (17). This uniformity is consistent with Eqs. [9] and [25] for a 5 1, x m /c a 5 2, and large n in the definition
preted with parameter values that reflect structural differences in the two polysaccharides. Such polymers, less regular than ideal dendrimers, display a minimum in the Mark–Houwink (intrinsic viscosity versus mass) curve (17, 18). Because it accounts for the polydispersity of branches, the present theory is suitable for describing the general behavior of intrinsic viscosity, which is proportional to a power of the weight-average molecular weight,
P D 5 M aw/M an. Hyperbranched polysaccharides, however, show substantial polydispersity (17). To determine if the present theory can represent such polymers, values of x m /c a and a were calculated at n 5 10 given P D, M n avg, and x m 5 162 (for the saccharide repeat unit (17)). Results in Table 1 show that the range of polydispersities of dextrans and maltodextrins can be inter-
@ h # 5 K~M w avg! b ,
[28]
Here, b 5 0.6 for irregular hyperbranched polymers and b 5 20.2 for regular dendrimers (17). For a ' 0.2–0.3, dimensionless molecular weight, M aw, displays a minimum when plotted versus n (Fig. 7). The minimum occurs at about n 5 4, which is the generation number (17, 18) for the experimentally observed minimum. It has been postulated that an extremum in the Mark–Houwink curve is a result of a transformation from a planar to a spherical structure (17), but the present theory
TABLE 1 Interpretation of Polysaccharide Polydispersity Index (17)
Dextrans D7.2 D11.7 D20 D50.8 Maltrins M040 M100 M180 FIG. 5. Average cluster volume, V an 5 [(1 2 a n )/(1 2 a)] 3 , versus branching interval, n, for p 5 2.
PD
M n avg
x m /c a
a
2.76 1.81 1.69 1.75
2686 6400 13216 28768
0.39 0.20 0.15 0.20
0.35 0.38 0.44 0.56
32.7 21.5 26.3
2779 1974 957
20 2.3 2.5
Note. x m /c a and a were calculated given P D, M n avg, and x m 5 162
0.79 0.54 0.46
241
HYPERBRANCHED POLYMERS AND AGGREGATES
The present theory, although it does not specify the orientation, diameter, or flexibility of the branches, simulates some observed features of computer- and laboratory-generated dendrimers. Further detailed branch morphology could be imposed upon the basic construction, but we have focused here on effects of branching number p and length ratio a. For certain values of these parameters, the distribution resembles randomly distributed branch lengths for diffusion-limited aggregates. The mass distribution of clusters is characterized by its moments, which provide average properties for cluster dynamics as functions of time. The results allow cluster mass, volume, density, and molecular weight, which depend on fractal geometry through p and a, to be compared to measured values from laboratory experiments or computer simulations of aggregation processes. ACKNOWLEDGMENT FIG. 7. Weight-average molecular weight, M w avg/c a , versus branching interval, n, for p 5 2 and c a /x m 5 10.
This work was supported in part by NSF Grant CTS-9810194.
REFERENCES
indicates that fractal geometry can also cause this behavior. For small a (,0.2) the curve decreases to a constant value, as expected for dendrimers with strongly decreasing branch lengths. The model thus appears to simulate properties of hyperbranched dendrimers that occur naturally or are synthesized in the laboratory. CONCLUSION
Building structures molecule by molecule is a form of nanotechnology with appealing prospects. Existing methods are limited for mathematically constructing branched macromolecules or dendrimers and investigating their properties. The present theory, based on an iterative fractal construction, yields relatively straightforward relations for the characteristic moments of the dendritic clusters. Stochastic computer models of aggregation are replaced by an analytic, statistical model based on polymerization kinetics by monomer addition. In essence, this is a kinetic theory of branched aggregate growth for the distribution of branch lengths (or mass). Exact solutions show how the properties of fractal clusters grow with time. Branching is assumed to occur deterministically at defined time intervals. Possibly the model can be generalized to random time intervals, but the inherently random process of chain-end monomer addition already leads to a distribution of branch lengths during each time interval.
1. Bunde, A., and Havlin, S., (Eds.), “Fractals and Disordered Systems,” 2nd ed. Springer-Verlag, Berlin, 1996. 2. Vicsek, T., “Fractal Growth Phenomena.” World Scientific, Singapore, 1992; L. M. Sander, Sci. Am. 256, 94 (1987). 3. Tomalia, D. A., and Durst, H. D., Top. Curr. Chem. 165, 193 (1993). 4. Newkome, G. R., Moorefield, C. N., and Vogtle, F., “Dendritic Molecules.” VCH, Weinheim, 1996. 5. Stanley, H. E., and Meakin, P., Nature 335, 405 (1988). 6. Stanley, H. E., in “Fractals and Disordered Systems” (A. Bunde and S. Havlin, Eds.), 2nd ed. Springer-Verlag, Berlin, 1996. 7. Hinrichsen, E., Maloy, K. J., Feder, J., and Jossang, J., J. Phys. A 22, L271 (1989). 8. Schaefer, D. W., Science 243, 1023 (1989). 9. Dotson, N. A., Galvan, R., Laurence, R. L., and Tirrell, M., “Polymerization Process Modeling.” VCH, New York, 1996. 10. Pastorasatorras, R., and Wagensberg, J., Physica A 224, 463 (1996). 11. Mandelbrot, B., “The Fractal Geometry of Nature.” Freeman, San Francisco, 1982. 12. Schroeder, M. R., “Fractals, Chaos, Power Laws.” Freeman, New York, 1991. 13. Abramowitz, M., and Stegun, I. A., “Handbook of Mathematical Functions.” National Bureau of Standards, 1968. 14. McCoy, B. J., and Madras, G., J. Coll. Interface Sci. 201, 200 (1998). 15. Kodera, Y., and McCoy, B. J., AIChE J. 43, 3205 (1997). 16. Boudart, M., “Kinetics of Chemical Processes.” Prentice-Hall, Englewood Cliffs, NJ, 1968. 17. Striegel, A. M., Plattner, R. D., and Willett, J. L., Anal. Chem. 71, 978 (1999). 18. Frechet, J. M. J., Hawker, C. J., Gitsov, I., and Leon, J. W., Pure Appl. Chem. A33, 1399 (1996).
Hyperbranched Polymers and Aggregates: Distribution Kinetics of Dendrimer Growth Benjamin J. McCoy Department of Chemical Engineering and Materials Science, University of California, Davis, California 95616 E-mail: [email protected] Received October 5, 1998; accepted April 27, 1999
in volume and mass. An analytic method is needed to describe the dynamics of such cluster distributions and properties. When the branched structure is self-similar at different length scales, a fractal (power-law) relationship holds. Such dendrimers are extreme cases of branching, at the opposite pole from unbranched, linear-chain structures. Comprehending the relationship between kinetics and geometry for fractal dendrimers may help understand other hyperbranched molecules and aggregates, whether natural or synthetic. The present deterministic method differs from other iterative constructions that repeat branched groups from the preceding step (1, 2) (see A in Fig. 1) or grow branches along the length of all previous branches (7) (B in Fig. 1). Earlier work did not fully address the dynamics of growth governed by reaction and diffusion rates, or the distribution of clusters. The aim is to develop an analytic approach to growth dynamics of fractal aggregates based on kinetics of monomer addition to branched dendrimers. We assume that branching occurs at time intervals that provide average fractal dimensions for the distributed branch lengths (Fig. 2). Following the suggestion (8) that polymer concepts can be applied to colloidal growth processes, we require that branches in an aggregate grow by monomer addition. Cluster– cluster aggregation can form symmetric fractal structures only under special circumstances, such as illustrated in Fig. 1A. Growth by monomer addition occurs exclusively at the branch tips, thus resembling computer simulations that show diminished growth in the shielded interior of a diffusion-limited aggregate. The clusters grow from seeds (initial buds) by chain-end addition of monomer. To formulate cluster properties such as total cluster mass as a function of time, we specify the instants in time when iterative branching occurs (new buds form at the end of a branch). These points in time denote the sequential time intervals for hierarchical branch formation (Fig. 2). Exact expressions for moments of branch distributions, based on a continuous-distribution population balance, provide measurable properties, for example, number of branches, average cluster mass, volume, and mass-average molecular weight. These properties change with time for irreversible monomer addition.
We present an analytic solution for growth of branched aggregates or polymers with distributed cluster (dendrimer) size. Monomer addition to each branch follows first-order polymerization kinetics leading to a distribution of branch lengths. The rate constant for monomer addition is considered diffusion-dependent. Deterministic branching occurs so that at prescribed times t j ( j > 1), p branches emanate from the tip of a branch that began to grow at t j21. When the ratio of average branch lengths is constant, L j/L j21 5 a, the fractal dimension of branches is ln( p)/zln(a)z. Closed expressions for cluster mass moments show unbounded growth with time unless ap < 1. The number of clusters (zeroth moment) is constant during growth and equal to the number of initiating buds. Expressions for the number of branches and clusters, average cluster mass, volume, density, and viscosity in solution are functions of p and a. Density and molecular weight show features similar to observed behavior of dendrimers and hyperbranched polymers. © 1999 Academic Press Key Words: branched polymer; aggregate; fractal; dendrimer; distribution kinetics.
INTRODUCTION
Dendrimers and hyperbranched polymers occur in many natural phenomena and can also be synthesized for scientific, engineering, and medical applications (1–4). Symmetric dendrimers, produced by organic synthesis from initiator seeds (3, 4), can be tailored to have fractal branch properties by controlling the polymerization process. Monomer addition will form shells of branches around the initiating core. Such growth or aggregation is encountered in many processes, including polymerization and precipitation (1, 2). Rate studies for dendrimer synthesis by chainpropagation monomer addition, however, have been confined to diffusion-limited aggregation, which is simulated by monomers that undergo random walks and deposit on branches of a growing aggregate. Computer models of diffusion-limited aggregation yield structures that resemble electrolytic deposition, viscous fingering in porous media, and electrical discharge patterns (1, 2, 5). Computer simulation is restricted to monomer numbers that are far smaller than in realistic dendrimers (6). Moreover, multiple clusters formed by monomer addition will usually be distributed 235
0021-9797/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.
236
BENJAMIN J. MCCOY
during t j21 # t , t j , the zeroth moment (q 5 0) is the number of branches, the first moment (q 5 1) is the mass of branches, and the second moment (q 5 2) is related to the variance of the distribution. Average mass of branches formed during the jth interval is defined as ~0! c avg 5 c ~1! j j /c j .
[2]
Because each branch starts as a bud of zero mass, the initial condition for branch growth at each time interval is c j ( x, t 5 t j ) 5 c j(0) (t j ) d ( x), where at j 5 0, t j 5 0. For q $ 1 all moments of these initiating distributions vanish (buds have no mass, for example). For t 5 0 we write the number of cluster buds as c 0(0) (0) [ c o . We define the branching number (multiplicity) p as the number of buds started at the instant t 5 t j , ~0! ~t j ! 5 pc ~0! c j11 j ~t j !.
FIG. 1. Previous iterative methods to construct a fractal branched aggregate. (A) The branched aggregate from the preceding step is repeated and added to the aggregate (1, 2). (B) Branches are allowed to grow along the length of all previous branches (7).
For branches growing during the time interval t j21 # t , t j , we write the branch distribution such that c j ( x, t)dx is the number of branches with mass in the differential range ( x, x 1 dx). Mass moments are defined as c ~q! j ~t! 5
E
`
c j ~ x, t! x q dx.
For any branching instant t j ( j $ 0), it follows that the number of new branches is j c ~0! j ~t j ! 5 p c o.
BRANCH PROPERTIES
[1]
0
For a discrete distribution with x 5 ix m , the integration is replaced by a summation (9) over i. For branches growing
[3]
[4]
In Fig. 2, we have p 5 3 and the number of branches at j 5 0, 1, 2, 3, and 4 is 0, 3, 9, 27, and 81, respectively. Unlike computer simulations, which have been limited to small degrees of branching due to the nature of the monomer attachment, our analytic approach allows arbitrarily large values of p, which would be appropriate for dense (compact) clusters. Growing branches are distributed in mass for the same reason that chain-end polymerization (9) yields a molecularweight distribution with a range of branch lengths (or mass). The length and number of branches have a fractal (power law)
FIG. 2. Growth of a cluster with p 5 3 and a ' 0.7. The growth interval, j, and the branching instant, t j , are illustrated. The number of branches, c j(0) (t j ), and average mass, c javg(t j ), follow Eqs. [4] and [6], respectively.
237
HYPERBRANCHED POLYMERS AND AGGREGATES
FIG. 3. Examples of clusters formed by monomer addition. The average length of each branch is L j , which is proportional to the average mass, c javg. The branching number is p, the number of iterations is n, and D 5 ln( p)/ln(1/a). A, B, and C are based on bifurcations ( p 5 2). C is formed by branching off six initial stems or by superposing three clusters with p 5 2.
relationship if the ratio of average mass (or length) at t j11 and t j is constant, avg c avg j /c j21 5 a.
[5]
Because branch length is proportional to branch mass, this can be written as L j /L j21 5 a, from which it follows that for j $ 1, L j 5 a j21 L 1 , or c avg 5 a j21 c a , j
[6]
in terms of c a [ c 1avg(t 1 ). Within the jth time interval for a given cluster the statistical self-similar relationship between number of branches and branch length can be expressed as the power law, p j 5 (L 1 /a j L 1 ) D . The fractal dimension for this geometrical structure is D 5 ln~ p!/uln~a!u.
Similar expressions (10) have been proposed for other regular branched structures. The dilational symmetry is determined by fractal geometry and not directly by mass or volume. If p 5 2 and a 5 1, the cluster is a Cayley tree (Bethe lattice) (11, 12). In the limit as both p and a approach unity, we have D 3 1, and the cluster is star shaped (Fig. 3E) with low density. If p is large and branches are short, the resulting compact cluster is very dense. Mandelbrot (12) illustrates treelike structures with D ' 1 to 2 when p 5 2 and a ' 0.5 to 0.7. In the present treatment, branch diameter, flexibility, sinuosity, and angle off a stem are unspecified, and branches may grow in any-dimensional space. Cluster properties are summations of all branches up to a branching instant j 5 n. The total number (13) of branches, by Eq. [4], is the finite sum of powers of p,
[7]
Oc n
Nn 5
j51
~0! j
~t j ! 5 pc o ~ p n 2 1!/~ p 2 1!,
[8]
238
BENJAMIN J. MCCOY
independent of branch length ratio, a (3, 4). The average number of branches per cluster is N an 5 N n /c o. The total cluster mass M n is the sum over all first moments, and can be expressed in terms of c a, the average cluster mass after the first growth of branches. We substitute c j(1) 5 c j(0) c javg, along with Eqs. [4] and [6]:
diffusion resistance (shielding effects in computer simulations) can cause inhomogeneous branch growth. The population balance equation for branch size distribution is (14)
c~ x, t!/t 5 2kc~ x, t!
Oc n
Mn 5
j51
O ~ pa! n
~1! j
~t j ! 5 c a c o p
E E
`
m~ x, t!dx
0
j21
j51
5 c a c o p~1 2 a n p n !/~1 2 ap!.
1k [9]
The average cluster mass after n time intervals is M n /c o , which is made dimensionless by dividing by c a ; thus M an 5 M n / (c o c a ). The number-average molecular weight is thus M n /c o 5 M anc a . For large n, the scaled cluster mass approaches a limit for ap , 1; thus M an 3 p/(1 2 ap), and an asymptote for ap $ 1; thus M an 3 p( pa) n /(ap 2 1). The variance of the cluster mass distribution is ~2! ~0! avg 2 c var j 5 c j /c j 2 ~c j ! .
[10]
Further computation of weight-average molecular weight or variance requires an expression for time dependence of the second moment, c j(2) . MONOMER ADDITION KINETICS
The time dependence of the cluster properties is determined by polymer growth kinetics (14, 15). Cluster growth occurs by monomer addition to the end of a branch (chain) in an existing cluster (or initiator). By adding one monomer at a time, a chain or branch of mass x, C( x), grows with a rate denoted by the coefficient k,
x
m~ x9, t!c~ x 2 x9, t!dx9.
[13]
0
For monomer of mass x m , one substitutes the distribution, m( x, t) 5 m (0) d ( x 2 x m ), into Eq. [13], dc/d t 5 2c~ x! 1 c~ x 2 x m !,
[14]
in terms of t [ km (0) t. If we define the discrete mass in units of i 5 x/x m , the discrete population balance equation is obtained. The solution to this difference-differential equation (9) is the Poisson distribution, t i exp(2 t )/i! for i 5 0, 1, 2, . . . , which has average and variance (for i) both equal to t. We assume that the monomer concentration, m (0) , is in great excess and therefore constant. This assumption is comparable to computer simulations for which new monomer enters the system at each addition step. If the monomer concentration were finite in a closed system, then aggregate growth would occur until monomer was depleted. The moment equations (14, 15) for growing branches can be derived by applying * 0` dxx q [ ] to Eq. [13]. In the last term, the order of integration for x and x9 is interchanged, y 5 x 2 x9 is defined to eliminate x, and the binomial expansion is applied. The result is
O ~ !c q
k
C~ x! 1 M~ x m ! ¡ C~ x 1 x m !.
[11]
Here, M( x m ) is a monomer of mass x m . The addition rate is considered proportional to the number of chain-ends and is thus independent of the chain length. For diffusion-influenced reactions (16), the rate constant is k 5 k /@1 1 k /~4 p D m r!#,
[12]
in terms of the diffusion-independent reaction rate constant k, the monomer diffusivity D m , and the reaction radius r. For diffusion-controlled addition reaction, we have k /(4 p D m r) @ 1, and therefore k . 4 p D m r, the Smoluchowski representation of the rate constant for stationary clusters (9). For wellstirred mixtures of clusters and monomers, concentration gradients will not exist external to clusters, but intracluster
dc ~q! /dt 5 2kc ~q! m ~0! 1 k
q i
~q2i!
m ~i! ,
[15]
i50
where m (q) 5 m (0) x mq is the qth moment of the monomer distribution. The moment equations provide expressions applicable to chain-end polymerization (14, 15). The zeroth moment (q 5 0) of the branch distribution is determined by (Eq. [15]) dc ~0! /dt 5 0,
[16]
and the number of branches is constant during any growth interval, Dt j 5 t j 2 t j21 . The total number of clusters, c o, is thus also constant. When time is scaled by defining u 5 ( x m /c a )km (0) t, the first moment equation (q 5 1) is dc ~1! /d u 5 c a c ~0! ,
[17]
239
HYPERBRANCHED POLYMERS AND AGGREGATES
which in terms of the growth interval D u j 5 u j 2 u j21 integrates to ~0! j c ~1! j ~ u j! 5 c ac j D u j 5 c ac op D u j,
[18]
when Eq. [4] is substituted. Substituting Eqs. [2] and [6] shows how the time interval is related to branch parameters:
M aw 5 p@S 22 1 ~ x m /c a !S 12 #/S 11 .
[25]
When radius equals the maximum total length of branches that can form a stem, cluster volume in d-dimensional space is
O L! n
Vn 5 gd ~
j
d
5 g d L d1 @~1 2 a n !/~1 2 a!# d ,
[26]
j51
D u j 5 a j21 5 c avg j /c a .
[19]
Thus the average added shell thickness for the jth growth interval is proportional to Dt j . Total branching time during aggregation can be represented as the sum over time intervals for branch growth,
O Du 5 ~1 2 a !/~1 2 a!, n
un 5
n
j
[20]
j51
when a j21 in Eq. [19] is summed. For a , 1 and large n, we have a n ! 1, and elapsed time, 1/(1 2 a), is constant, an indication that added branches are so small that growth has effectively ceased. For a $ 1, cluster growth is unlimited as n increases. Other rate expressions in the population balance equation, which may be more appropriate to other branching phenomena, for example, biological systems, could be postulated. The present model thus provides a framework for generalized dendrimer growth dynamics. The second-moment differential equation according to Eq. [15] is dc ~2! /d u 5 c a @2c ~1! 1 x m c ~0! #,
[21]
which yields after substitution for the first and zeroth moment followed by integration c ~2! 5 x m c a c o p j D u j ~D u j c a /x m 1 1!.
[22]
When branch multiplicity is included by multiplying by p j (Eq. [4]) and the result is summed, we have
Oc n
~2! j
5 x m c a c o p 2 @~c a /x m !S 22 1 S 12 #
[23]
j51
in terms of S ij 5 @1 2 ~a i p j ! n #/~1 2 a i p i !#.
[24]
Weight-average molecular weight, M w avg, is given by Eq. [23] divided by M n from Eq. [9], and is furthermore made dimensionless by dividing by c a ; thus,
independent of branching number, p. For d 5 2 or 3, the coefficient for the circle or sphere is g d 5 p or 34p, respectively. The reduced average cluster volume is V an 5 @~1 2 a n !/~1 2 a!# d 5 u dn ,
[27]
when Eq. [20] is substituted. The reduced characteristic length for the cluster is obviously u n . Average cluster mass density can be represented as average mass divided by average volume of clusters. Scaled average cluster mass density is thus r an 5 M an/V an. RESULTS AND DISCUSSION
Some special cases of the general expressions are of interest. For p . 1 and n 5 1, the clusters are stars (Fig. 3E) composed of p branches that each grow as polymer chains. For p 5 1 and n 5 1 we have a distribution of single polymer chains growing at one end by monomer addition. As with the star, reduced average mass and variance are equal, as expected for chain-end polymerization (9). Evaluation of the closed-form equations for number, mass, volume, and density of clusters (N an, M an, V an, and r an) as functions of branching interval n is straightforward. Observation of branched clusters, whether generated by computer simulations (1, 2) or in the laboratory (3, 4), suggests that bifurcation ( p 5 2) is the common form of branching. Branch lengths are distributed by polymerization kinetics, leading to structures similar to Fig. 3C. Plotted versus the elapsed interval n for p 5 2, the mass (Eq. [9]) increases to limit values when a , 12 and increases without limit when a $ 21 (Fig. 4). The volume, however, even when defined to have its maximum possible value as in Eq. [26], approaches a finite value when a , 1. Except for a $ 1, volume versus n (Fig. 5) reaches limits that increase with a as indicated by Eq. [27]. Cluster density versus n (Fig. 6) shows realistically that a minimum occurs for a $ 21. This behavior is caused by the diminishing volume increase and stronger mass increase when a . 0.5, illustrated by Figs. 5 and 4, respectively. Such density minima were reported by Tomalia and Durst (3) for organically synthesized branched dendrimers and are in accord with the present theory. The regular branching patterns of dendrimers, such as those in Fig. 3, provide a basis for comparing and classifying
240
BENJAMIN J. MCCOY
FIG. 4. Average cluster mass, M an 5 p(1 2 a n p n )/(1 2 ap), versus branching interval, n, for p 5 2.
FIG. 6. Average cluster density, r an 5 M an/V an, versus branching interval, n, for p 5 2.
branched polymers (17, 18). Ideal dendrimers are synthesized with a single monomeric unit making up each branch, so that the polydispersity index, P D, is essentially unity (17). This uniformity is consistent with Eqs. [9] and [25] for a 5 1, x m /c a 5 2, and large n in the definition
preted with parameter values that reflect structural differences in the two polysaccharides. Such polymers, less regular than ideal dendrimers, display a minimum in the Mark–Houwink (intrinsic viscosity versus mass) curve (17, 18). Because it accounts for the polydispersity of branches, the present theory is suitable for describing the general behavior of intrinsic viscosity, which is proportional to a power of the weight-average molecular weight,
P D 5 M aw/M an. Hyperbranched polysaccharides, however, show substantial polydispersity (17). To determine if the present theory can represent such polymers, values of x m /c a and a were calculated at n 5 10 given P D, M n avg, and x m 5 162 (for the saccharide repeat unit (17)). Results in Table 1 show that the range of polydispersities of dextrans and maltodextrins can be inter-
@ h # 5 K~M w avg! b ,
[28]
Here, b 5 0.6 for irregular hyperbranched polymers and b 5 20.2 for regular dendrimers (17). For a ' 0.2–0.3, dimensionless molecular weight, M aw, displays a minimum when plotted versus n (Fig. 7). The minimum occurs at about n 5 4, which is the generation number (17, 18) for the experimentally observed minimum. It has been postulated that an extremum in the Mark–Houwink curve is a result of a transformation from a planar to a spherical structure (17), but the present theory
TABLE 1 Interpretation of Polysaccharide Polydispersity Index (17)
Dextrans D7.2 D11.7 D20 D50.8 Maltrins M040 M100 M180 FIG. 5. Average cluster volume, V an 5 [(1 2 a n )/(1 2 a)] 3 , versus branching interval, n, for p 5 2.
PD
M n avg
x m /c a
a
2.76 1.81 1.69 1.75
2686 6400 13216 28768
0.39 0.20 0.15 0.20
0.35 0.38 0.44 0.56
32.7 21.5 26.3
2779 1974 957
20 2.3 2.5
Note. x m /c a and a were calculated given P D, M n avg, and x m 5 162
0.79 0.54 0.46
241
HYPERBRANCHED POLYMERS AND AGGREGATES
The present theory, although it does not specify the orientation, diameter, or flexibility of the branches, simulates some observed features of computer- and laboratory-generated dendrimers. Further detailed branch morphology could be imposed upon the basic construction, but we have focused here on effects of branching number p and length ratio a. For certain values of these parameters, the distribution resembles randomly distributed branch lengths for diffusion-limited aggregates. The mass distribution of clusters is characterized by its moments, which provide average properties for cluster dynamics as functions of time. The results allow cluster mass, volume, density, and molecular weight, which depend on fractal geometry through p and a, to be compared to measured values from laboratory experiments or computer simulations of aggregation processes. ACKNOWLEDGMENT FIG. 7. Weight-average molecular weight, M w avg/c a , versus branching interval, n, for p 5 2 and c a /x m 5 10.
This work was supported in part by NSF Grant CTS-9810194.
REFERENCES
indicates that fractal geometry can also cause this behavior. For small a (,0.2) the curve decreases to a constant value, as expected for dendrimers with strongly decreasing branch lengths. The model thus appears to simulate properties of hyperbranched dendrimers that occur naturally or are synthesized in the laboratory. CONCLUSION
Building structures molecule by molecule is a form of nanotechnology with appealing prospects. Existing methods are limited for mathematically constructing branched macromolecules or dendrimers and investigating their properties. The present theory, based on an iterative fractal construction, yields relatively straightforward relations for the characteristic moments of the dendritic clusters. Stochastic computer models of aggregation are replaced by an analytic, statistical model based on polymerization kinetics by monomer addition. In essence, this is a kinetic theory of branched aggregate growth for the distribution of branch lengths (or mass). Exact solutions show how the properties of fractal clusters grow with time. Branching is assumed to occur deterministically at defined time intervals. Possibly the model can be generalized to random time intervals, but the inherently random process of chain-end monomer addition already leads to a distribution of branch lengths during each time interval.
1. Bunde, A., and Havlin, S., (Eds.), “Fractals and Disordered Systems,” 2nd ed. Springer-Verlag, Berlin, 1996. 2. Vicsek, T., “Fractal Growth Phenomena.” World Scientific, Singapore, 1992; L. M. Sander, Sci. Am. 256, 94 (1987). 3. Tomalia, D. A., and Durst, H. D., Top. Curr. Chem. 165, 193 (1993). 4. Newkome, G. R., Moorefield, C. N., and Vogtle, F., “Dendritic Molecules.” VCH, Weinheim, 1996. 5. Stanley, H. E., and Meakin, P., Nature 335, 405 (1988). 6. Stanley, H. E., in “Fractals and Disordered Systems” (A. Bunde and S. Havlin, Eds.), 2nd ed. Springer-Verlag, Berlin, 1996. 7. Hinrichsen, E., Maloy, K. J., Feder, J., and Jossang, J., J. Phys. A 22, L271 (1989). 8. Schaefer, D. W., Science 243, 1023 (1989). 9. Dotson, N. A., Galvan, R., Laurence, R. L., and Tirrell, M., “Polymerization Process Modeling.” VCH, New York, 1996. 10. Pastorasatorras, R., and Wagensberg, J., Physica A 224, 463 (1996). 11. Mandelbrot, B., “The Fractal Geometry of Nature.” Freeman, San Francisco, 1982. 12. Schroeder, M. R., “Fractals, Chaos, Power Laws.” Freeman, New York, 1991. 13. Abramowitz, M., and Stegun, I. A., “Handbook of Mathematical Functions.” National Bureau of Standards, 1968. 14. McCoy, B. J., and Madras, G., J. Coll. Interface Sci. 201, 200 (1998). 15. Kodera, Y., and McCoy, B. J., AIChE J. 43, 3205 (1997). 16. Boudart, M., “Kinetics of Chemical Processes.” Prentice-Hall, Englewood Cliffs, NJ, 1968. 17. Striegel, A. M., Plattner, R. D., and Willett, J. L., Anal. Chem. 71, 978 (1999). 18. Frechet, J. M. J., Hawker, C. J., Gitsov, I., and Leon, J. W., Pure Appl. Chem. A33, 1399 (1996).