Znd. Eng. Chem. Res. 1995,34, 4174-4184
4174
Computational Chemistry and the Design of New Materials for Xerographic Applications? Tom A. Kavassalis,*Peter M. Kazmaier, James P. Bareman, Pudupadi R. Sundararajan, and Joseph D. Wright* Xerox Research Centre of Canada, 2660 Speakman Drive, Mississauga, Ontario, Canada
L5K 2Ll
Modern xerographic systems involve the integration of mechanical, electronic, software, and materials subsystems. Designing materials for this technology presents a number of challenges in which computational chemistry methods have played an increasingly important role in recent years. In this paper we will outline several examples where molecular level simulations have helped to develop fundamental insights into material structurdproperty relations and to establish design rules based on computable molecular properties and their relationship to appropriate end use requirements. The paper presents very brief highlights of our accomplishments in the areas of photogenerator pigments and polymers. In addition, it provides a brief commentary on how we approached the integration of simulation methodology into what has traditionally been a n experimental environment. 1. Introduction
Xerographic or electrophotographic systems have been in use since the early 1960s. While the basic steps in the process have remained essentially the same, vast improvements in image quality have been achieved through process and materials refinement. Today, xerographic imaging is closing in on the resolution domain that is currently dominated by the lithographic printing process. The design of current and future xerographic products requires the integration of mechanical, electronic, software, and materials subsystems. Since the introduction of new xerographic technology is dependent on developments in so many fields, it is imperative that all opportunities for improved productivity be exercised. In each of the above-mentioned subsystems, substantial gains in design and developmentproductivity have been achieved in recent years through the use of advanced software tools. An example familiar to most is the semiconductor integrated circuit, where numerous simulations are performed routinely in order t o debug and assess the performance potential of a design before committing it to silicon. In this paper we describe the role that modeling and simulation plays in the development of new materials and processes for xerographic applications at the Xerox Research Centre of Canada. The aim of the paper is not to provide an exhaustive treatment of how to apply computational chemistry methodology to problems in the field of electrophotography. Instead, we attempt to provide a flavor for what some of the materials science issues are in this field and how we address them with our modeling and simulation research. Many of these issues are generic and should therefore be of interest to a larger community. In accordance with the stated themes of this symposium, we emphasize applications of computational chemistry only. Other computational approaches, that are more relevant to processes for
* Author to whom correspondence is addressed. E-mail: tkava.xrc&erox.com. Fax: (905) 822-7022. ' Presented a t the 1994 Annual Meeting of the American Institute of Chemical Engineers. ' Present address: Pulp and Paper Research Institute of Canada, 570 boul. St. Jean, Pointe-Claire, QuBbec, Canada H9R 359.
making materials, are equally important but will not be discussed here. The remainder of the paper is organized as follows. In section 2, we review the basic xerographic process in order t o introduce the main issues that arise in the development of new materials. This section spells out some of the important requirements of various materials subsystems. In subsequent sections, we present several examples where the application of computational chemistry to xerographic materials has led either to significant insights into the properties and/or behavior of the materials or to correlations between material structure and performance that form the basis of design rules for developing new materials. The materials systems are categorized as either small molecules or polymers, and these are discussed in sections 3 and 4,respectively. In each section we discuss a few sample problems and the basic approach used. The readers are directed to several references for more detailed information. In section 5 we conclude with a brief description of some of the challenges that remain.
2. Xerographic Process and Materials Issues The central component of the xerographic process is the photoreceptor, depicted schematically in Figure 1. This device typically consists of multiple layers, including an inert substrate or support layer, an electrode layer, a charge generation layer, and a charge transport layer. Typical layer thicknesses are shown in the figure. A very complete review of the physics of electrophotography was published recently by Pai and Springett (1993).A simplified version of the process is described below. The readers are referred to the review for a more in-depth discussion. The exposure and development of an image in xerography or electrophotography proceeds as follows. The top surface of the photoreceptor is first charged to a high potential, typically 1000 V relative to the electrode, by an ion cloud created by a corotron discharge. In order to create an image, laser light or light from a lamp and a combination of lenses is directed at the device's charge generation layer. The role of this layer is to create, with high efficiency, a population of excitations that can inject holes or electrons into the charge transport layers. The charge transport layer, which is more commonly a
0888-588519512634-4174$09.00/00 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34,No. 12, 1995 4175
}
charge Generation Layer
Substrate Layer
(> IOOOpmJ
Figure 1. Schematic crass section of a typical dual-layer xemgraphic photoreceptor. The function of each layer is described in section 2.
hole-transporting material, will conduct the charge in the field created by the ions a t the surface of the device. Eventually, the surface potential in the areas above the exposed region will decay relative to that of the undeveloped areas. At this point, one has an image stored as variations in the surface charge that can be developed using oppositely charged powders called toners. The toner itself is another multicomponent material subsystem. The image, now recorded by the toner, can be transferred to a sheet of paper that has been charged opposite to the toner. Finally, the image is fused to the paper by a heat treatment. The materials requirements for the xerographic process are very complex. Consider, for example, the charge generation layer and the charge transport layer. These two components have very different requirements. The generation layer must strongly absorb light in the visible or near-IR portions of the spectrum. For laser printer and digital copier applications, the absorption maximum should coincide with the wavelength of the laser diode or LED array used. The transport layer, on the other hand, must be transparent in the same region of the spectrum in order to allow light to reach the generation layer. For most families of photogenerating materials, a high degree of crystallinity is required for the generation layer in order to provide a high quantum efficiency between the light absorbed and the charges injected into the transport layer. The optical absorption of many organic charge generation materials is significantly red shifted upon crystallization relative to their solution spectra. For many classes of organic photogenerators, such as the phthalocyanines and perylenes, the situation is somewhat complicated by the display of polymorphism, the ability to crystallize in multiple forms. Often, one polymorph displays properties that are significantly better than the other polymorphs. Understanding the role that crystal packing plays on the optical properties is one challenge. Developing processes that deliver the most desired polymorph is another. The transport layer, on the other hand, is ideally an amorphous molecular dispersion of a hole-transporting molecule and a binder polymer. As an amorphous dispersion above the charge-conducting percolation threshold, one will get the minimum of trapped excitations, which reduce the effectiveness of the device. Also, as an amorphous dispersion, the transparency in the visible portion of the spectrum can be guaranteed. The transport layer has a number of additional requirements as well. Since it is charged t o high voltages via a
corotron discharge, it must be resistant to chemical degradation by ions. I t must also be fairly resistant to mechanical abrasion, since excess toner is cleaned off using a blade or brush after each image transfer to paper. In photoreceptor belt applications, the charge transport layer should be flexible and should adhere well to the lower layers. The generation layer and the transport layer must also have radically different solubility properties if the multilayer device is to be produced using a low-cost coating process. The generation layer should be insoluble in organic solvents, while the transport layer should be soluble in a common and safe solvent. Xerographic toner has a number of special material requirements as well. Toner is a dispersion of pigment and other additives in a polymeric binder. The binder must melt and flow into the paper fibers, where it is expected to have good adhesion and mechanical strength. Other requirements of a good toner include proper triboelectrical (contact charging) properties and a long shelf life. Many of these requirements cannot be simultaneously met without compromises. For example, high molecular weights needed for mechanical strength will lead to poor flow during the fusing process. One can see that the materials requirements for xerography are quite complex and diverse. They involve optical properties, mechanical properties, miscibility properties, and other special processing requirements. Opportunities to impact the research and development process using computational chemistry approaches are vast. For example, quantum mechanical methods can be used to study the electronic structure of both isolated molecules and molecules in the solid state. Miscibility properties can be addressed with a combination of quantum mechanics and molecular mechanics level modeling. Very often, computational chemistry approaches only indirectly address the questions of interest. The performance of the overall system is usually dependent on a combination of both the material properties and end use conditions or applications. For example, the abrasive wear life of a photoreceptor is presumably related to various mechanical and surface properties of the material as well as the conditions under which the wear process is occurring. Those properties are in turn determined by the detailed chemical structure and morphology of the device. Unfortunately, the precise link between the wear performance of the device and the chemical microstructure and morphology is not known. In such situations, we rely largely on using computational chemistry approaches to establish empirical correlations between performance metrics and computable properties of the materials. While this is unsatisfactory from a pure structurdproperty point of view, it nonetheless allows us to make progress with very complex multicomponent systems. An example of this can be found in section 4. In the remainder of this paper we discuss some of our strategies for understanding the properties of these materials and for designing improved materials using the methods of computational chemistry. The information presented here is very cursory in the interest of presenting many examples.
3. Small Molecules In this section we describe our approach to modeling photogenerator pigment materials and their properties. We are principally concerned with three issues: the
4176 Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995
optical absorption, polymorphism, and surface structure of the pigment crystals. To address each issue, we use different approaches. However, there is an important interdependency on the results. For example, the manner in which pigment molecules pack together to form the crystalline state will affect the structure of the surface and also the optical properties of the aggregate. The structure of the surface is also expected to influence the efficiency with which the excitation can be transferred to the conducting matrix. For the remainder of this section, we restrict our discussion t o the matters of optical absorption and polymorphism and the interplay between the two. Since we are dealing with rather large molecules in the solid state, we are forced to make what are often drastic approximations. For example, our analysis of the optical properties of perylene pigments was performed using extended Huckel methods (Hoffmann, 1963). This simplification allows us to examine very many structures on workstation level computers, which would not be feasible if one adopted more rigorous approaches. Nonetheless, we have shown that several important insights and qualitative trends identified with our simplifications have a valid basis in the known experimental data. Likewise, the models we employ in our studies of polymorphism and the surfaces of crystals are of the molecular mechanics variety. These models, we feel, are appropriate level representations of the systems as far as several thermodynamic properties are concerned. 3.a. Band Structure of Organic Photogenerators: The Role of Crystal Packing. The effect of packing on the color of pigments is a general phenomenon seen in many classes of compounds. Perhaps its most marked manifestation is found in a remarkable series of compounds called the perylenes, which have been investigated by Graser and Hadicke (1980, 1984) and Hadicke and Graser (1986a,b) a t BASF. In Figure 2 we show 11perylene derivatives. Although the basic chromophore for each pigment is identical, very small substituent changes on the periphery of the pigment chromophore lead to enormous changes in the color of these compounds. For example, the change in oxygen position on going from 3-oxapentylperylene (structure 1 in Figure 2) t o 4-oxapentylperylene (structure 2) results in a transformation of the color from red to black (the absorption maximum changes from 564 to 613 nm and the band broadens considerably in the solid state, giving the pigment the black color). Since all of these materials have essentially the same absorption spectrum in solution, the differences in their solid-state color are the result of differences in solid-state packing. Our interest in this behavior stems from the technological need to develop organic photogenerators with optical properties tuned to overlap the output of lowcost solid-state lasers. Our approach to this problem involves interdependent experimental and modeling activities. The aim of the modeling activities is to synthesize design rules from a combination of computationally derived results and the properties of known materials. With these design rules, we then evolve our designs toward materials with optimal properties. In this application, computational chemistry methods provide a potential means by which to prototype several designs before committing to a synthetic and purification procedure. Here, we briefly describe our approach to understanding the role that crystal packing plays in determining the optical properties of perylene systems.
R Group
Compound No.
Color of Crystals
1
CH",O"CH,
red
2
CH-,"CH,
black
3
C HZMC
4
C H 2 b o V C H 3
5
H3
C H CH3
2
m
red red
red
6
black
7
red
8
9
11
CH -'CH,
red-maroon
CH3
red
red-violet
Figure 2. Series of perylene pigments and their color in the solid state. Solution spectra of these materials are nearly identical. Differences in the solid-state color are due to specific differences in the solid-state packing geometry.
Figure 3. Definition of the longitudinal ( I ) and transverse ( t ) axis on a perylene imide structure.
Since we are seeking qualitative design rules, we have adopted the extended Huckel approach developed by Hoffmann (1963, 1987, 1988). A typical perylene structure is shown in Figure 3. In the solid state, these pancakelike molecules generally stack in long one-dimensional columns. For all planar perylene crystal structures known, the interplanar spacing, d , in these columns is approximately 3.5 A. Adjacent perylenes do not adopt a face-to-face orientation. Rather, they are staggered or offset by an amount that varies from one perylene derivative to another. In Figure 3 we define a longitudinal offset, 1, as a displacement along the long molecular axis (nitrogen-nitrogen axis) of the molecule and a transverse offset, t , as a displacement perpendicular to the longitudinal axis; d, I, and t describe the coordinates of a given atom in the next perylene down the stack relative to the preceding one. Suppose one were to eclipse an infinite stack of
Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995 4177 -loa
c
-11.80
-
-1210
I
-12.20
om
2.m
am
d.m
8m
I0.m
12m
,urn
16.m
~~~
t ,
Oth.t=OA
0 f h W r I . tA
Figure 4. Top and side views of perylene imide stacks. The figure on the left has an offset of 0.A,while the figure on the right has a longitudinal offset or displacement of 2.1 A. Extended Hiickel calculations have been performed on these stacks as a function of both the longitudinal and transverse displacement.
Figure 6. Variation of the valence (lower pair of curves) and conduction (upper pair of curves) bandwidths of perylene imide as a function of the longitudinal offset. Far bath the valence band and conduction band, the squares correspond to wave vector k = d a and the triangles correspond to k = 0.
-.
-10.10-
.to.a -10.80
-
.
-
.. . GO
20
40
60 80 IO0 OmnrrL7lWh*qnrm
120
110
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Figure 7. Variation of the valence (lower pair of curves) and conduction (upper pair of curves) bandwidth of perylene imide as a function of the transverse offset. Here, the squares refer to k = 0 and the triangles refer to k = da. Figure 5. Electron density distributions of the frontier orbitals for the perylene imide molecules,at the 90%probability level. The upper and lower diagrams are for the HOMO and LUMO, respectively. Light- and dark-shaded regions represent densities derived from positive and negative sign wave function, respectively.
perylene imides, as shown in Figure 4 a t left, and then gradually offset the two stacks in a longitudinal sense. What would happen to the band structure of the aggregate in the solid state? The bandwidth as a function of offset may be seen from a plot of the energy of the crystal orbitals with indices of k = 0 and k = n/u,the extrema of the crystal orbitals (Hoffmann, 1988; Kazmaier and Hoffmann, 1994). It has previously been shown that a transverse offset for a stack of ethylenes (the simplest n system) will in each case produce a simple diminution of the bandwidth of the bands derived from these MO’s (Kazmaier and Hoffmann, 1994). On the other hand, a longitudinal offset will lead to oscillations in the bandwidth, the number of maxima depending on the nodal structure: two nodes for the HOMO of the diene, three for the LUMO of the diene and HOMO of the triene, and four for the LUMO of the triene. The extrapolation to a flat n system having transverse extent, as a perylene does, is pretty clear. The frontier orbitals, depicted in Figure 5 for our perylene imide system, have “transverse” and “longitudinal” nodes, leading t o an oscillation in bandwidth for offset in either direction. In general, the number of minima (or zeros) in the bandwidth versus offset graph will equal the number of nodes in the given offset direction.
The calculated dependence of the bandwidth on the degree of longitudinal offset for our stack of perylene imides is shown in Figure 6, while Figure 7 shows the corresponding influence of transverse offset on the valence and conduction bandwidths. The behavior of the computed curves for the HOMO is consistent with our expectations; three longitudinal nodes and one transverse node lead to the same number of points of zero bandwidth. The three points of zero bandwidth in the transverse offset of the LUMO are similarly related to the three transverse nodes. The longitudinal offset behavior of the conduction band is more complicated. There are three points of maximum bandwidth seen in Figure 6, separated by two points of minimum, but not zero bandwidth. Some width always remains, as the location of the nodes of the LUMO does not perfectly follow the transversdongitudinal description. Klebe et al. (1989) correlated observed A,, values with 1 and t for a series of perylenes of known structure and found a linear, monotonic decrease of A,, with increasing 1. However, the property of crystallochromy is manifested not only in the shift of the absorption maximum but also in a general broadening of the absorption (Graser and Hadicke, 1980). Our molecular orbital analysis indicates that the nodal character of the HOMO wave function imposes conditions that can lead to a catastrophic collapse of bandwidth for certain, specific offsets with very large area overlaps. Thus, band broadening correlates with area of overlap only if one selects maxima in the oscillating bandwidth versus offset curve. Node crossing can lead to narrow bandwidths even for large area overlaps.
-
4178 Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995
1.4
-
2
4
1.2
-
1.0
-
0.8
-
I
F
,” ._ e ._
.; E
-
b-
E -ma T .
u
1.7
1.9
2.1
2.3
2.5
O b i e m d Transition Energy st Lambda Max(eV)
Figure 8. Observed and calculated transitions for perylene pigments. The open circles represent the band-gap transition (the minimum energy difference between the valence band and the conduction band), while the solid circles represent the transition fmm the bottom of the valence band to the top of the conduction band.
In order to determine whether or not the molecular orbital results are consistent with the observed ahsorption maxima reported for perylenes, extended Hiickel calculations were performed at the values of d , I, and t reported by Klehe et al. (1989). The calculated band gap and the maximum energy valence-conduction hand transitions are plotted as a function of the experimental absorption maxima in Figure 8. The calculated results correlate reasonably linearly with the experimental results. Not only does the hand gap correlate with the energy of absorption, but the broadness of the ahsorption qualitatively correlates with the hand broadening. It is not surprising that the extended Hiickel method underestimates the overall energy of the transition since the propensity for the extended Hiickel method t o give too small band gaps is well-known (Whangbo et al., 1979; Hong et al., 1992). The amount hy which the present calculations underestimate the hand gap in perylenes, approximately 50% of the experimental values, is similar in magnitude to that reported for the conducting polymer benzimidazobenzophenanthroline by Hong et al. (1992). In summary then, tight-binding extended Hiickel calculations on infinite one-dimensional stacks of perylene pigments have shown that the bandwidth of the valence and conduction bands is determined not only by the degree of overlap between adjacent perylene moieties in the stack but also hy the nodal characteristics of the HOMO and LUMO orbitals of the individual perylenes. Thus,some intermolecular orientations with very large area overlaps between adjacent perylenes may have very small valence and/or conduction handwidths if the primary intermolecular overlap vanishes. This is a quantum mechanical interference effect. Comparison of the calculated band gaps to the energy of the absorption maximum for a series of perylenes of known crystal structure indicates that the calculations can simulate both the trend in the wavelength maximum and the broadening of the absorption as a fnnction
of the one-dimensional geometry, even though threedimensional interactions have been omitted. 3.b. Polymorphism and Computational Chemistry. The dramatic effects of changes in crystal packing are further complicated by the recognition that most organic-based molecules can he made t o pack in a number of distinct crystal structures, or polymorphs, some ofwhich may he metastable (Kitaigorodsky, 1973). From the preceding discussion, it is clear that such polymorphism has important consequences for materials used in electrophotography. A well-known example is the case of the oxotitanium phthalocyanine (TiOPc) photogenerator pigment, for which at least four distinct polymorphs are known (Enokida et al., 1990; Saito et al., 1993, 1994). The photoconductivitiesand absorption spectra of the various TiOPc polymorphs differ significantly. The dependence of photoconductivity on crystalline packing makes the design of new photogenerator materials somewhat of a hit-or-miss process, as traditional molecular design techniques are geared toward the properties of individual molecules, rather than aggregates. (Recently,however, works have appeared that have as a goal the engineering of specific crystal packiugs using H-bonding interactions. See, for example, Zerkowski et al. (1994)J Ideally, one would like to incorporate knowledge of molecular packing behavior (and its relationship to the photogeneration process) into the a priori design of photogenerator materials. Failing that, one would a t least like to be able t o determine which polymorphs are possible, given a molecular structure. The relationship between photogeneration and crystal packing is not yet well understood (see previous section), partially because it is difficult to obtain single-crystalX-ray structures for many pigment polymorphs. Recently, however, significant advances have been made in the area of pigment structure “determination” from experimental powder diffraction data, through the application of molecular modeling techniques in a modified Rietveld refinement process (Oka et al., 1992; Harris et al., 1994). Structures determined in this way have relatively high uncertainties but can yield significant new insights into desirable packing properties when used with care (Okada et al., 1993).
Two scenarios one can imagine for the generation of possible polymorphic states using molecular modeling techniques are outright prediction based on molecular structure alone and application of heat and/or stress t o known crystal structures. Both of these techniques depend critically on an accurate description of the interactions between the molecules. Substantial progress has recently been made toward the goal of predicting crystal structures based on molecular structure information alone (Gavezzotti, 1991; Karfunkel and Gdanitz, 1992; Perlstein, 1994). All of these methods employ classical force fields to describe intermolecular interactions and are therefore unlikely to be able to reproduce effects caused by changes in the molecular electron density distribution on going to the solid state. Whether or not crystal structures are truly predictable remains open to debate. The interested reader is referred to a recent paper hy Gavezzotti (1994) for further discussion of this point. Our efforts in this area have focused on attempting to apply computational chemistry techniques, in particular molecular dynamics (MD) simulations, t o derive possible polymorphs from existing known crystal struc-
Ind. Eng. Chem. Res., Vol. 34,No. 12,1995 4179 Table 1. Unit Cell Data for Several Phases of Acridine" phase I1 I11
IV
alA
b/A
CIA
/3
16.34 18.90 6.08 95'5' 11.41 5.99 13.69 98'48' 15.75 29.43 6.20
space Zb group
density/ (g.rno1-l)
8 P21Ia 4 P21/n 12 P212121
1.27 1.29 1.24
a Data taken from Kitaigorodsky (1973). Number of molecules in the unit cell.
tures. Key to the application of this technique is the ability to carry out the simulations under conditions of constant pressure and temperature, with variable cell lengths and angles (Allen and Tildesley, 1987). The inclusion of kinetic energy in MD methods allows the simulated system to pass over barriers, thus reducing the magnitude of the multiple minimum problem found with straight minimization (molecular mechanics). Advantages of MD over Monte Carlo methods include a more efficient sampling of collective modes, which may be involved in phase transitions. Again, these methods employ classical force fields, with the accompanying limitations. In order to explore the MD approach, we have applied constant-pressure and -temperature MD methods t o study solid acridine. Acridine (2,3,5,6-dibenzopyridine) is known to crystallize in at least four distinct polymorphs, one of which is a hydrate (Lowde et al., 1953; Phillips, 1954,1956;Phillips et al., 1960). Unit cell data for three of the polymorphs are' shown in Table 1. Experiments show that both phases I1 and I11 are reasonably stable at room temperature but that acridine I11 is transformed to acridine I1 with increasing rapidity on heating above 45 "C. Similarly, acridine IV transforms t o acridine I1 at 70 "C. Constant NPT (number of particles, pressure, and temperature) MD simulations were used to explore the potential energy surface around the minimum energy acridine I1 and acridine I11 crystal structures as a function of T with P = 0. In order to allow the possibility of reproducing the experimentally observed acridine I11 acridine I1 phase transition, it was necessary for the number of molecules in the MD box to be commensurate with the number of molecules in the unit cell of either,phase. Therefore, the simulations were carried out with 64 molecules, or a total of 1472 atoms, starting from both the acridine I1 and acridine I11 crystal structures. Parameters for the intermolecular interactions were taken from the Dreiding force field (Mayo et al., 1990). During the course of the simulations, T was gradually increased from an initially low value, allowing a relatively long run at each value of T in an effort to properly sample some of the longer wavelength collective fluctuations. As T was increased, phase transitions accompanied by discontinuous changes in the average volume, (V), were observed for the acridine I1 run at (T) = 480 K and (T) = 505 K and for the acridine I11 run at (T) = 520 K. The final average volumes and potential energies were essentially the same for both runs. The final structures also appeared to be very similar. Rather than reproducing the acridine I11 acridine I1 transition, the simulations found that both acridine I11 and acridine I1 transform suddenly to a common phase at high temperature. It is interesting to note that the common final structure observed in the simulations is a plausible intermediate for the acridine I11 acridine I1 transition. Failure of the simulations to reproduce the acridine I11 acridine I1 transition could be due to a number of
-
factors, some of the more obvious ones being an inadequate force field or not simulating long enough times to observe relatively rare collective phenomena. More basic is the question of whether or not it is possible to observe such a transition via MD simulations, even given a perfect force field and extremely long runs. Solid-solid phase transitions can be broadly classified as being either reconstructive, displacive, orientationswitching, or martensitic, in terms of the relationship of a molecule to its neighbors (Megaw, 1973). Realistically, MD methods are capable of reproducing only the last three forms. Of the observable transitions, the martensitic variety would probably be the most difficult to simulate since they often involve a large shearing of the unit cell. It is possible that the acridine I11 acridine I1 transition involves such a large shearing motion.
-
4. Polymers In this section we describe our approach to modeling the properties of binder polymers used in xerographic applications. We also describe our approach to modeling a novel pseudo living free-radical polymerization process developed by this laboratory. In both our toner and photoreceptor applications there are requirements imposed on the material's mechanical properties and glass transition temperatures. These properties are difficult to predict a priori. Our approach here is to establish correlations between computable molecular parameters of our systems and measured properties. Once established, these correlations then form the basis of design rules for further improvements in material properties. 4.a. Glass Transition Temperature. One of the important functional properties of polymers is the glass transition temperature, Tg.It has long been recognized that the local conformational flexibility of the chain influences the Tg(Gibbs and Di Marzio, 19581,although other factors, such as intermolecular interactions and their resultant effect on the cooperative motion of the chain segments, also contribute to the glass transition. Attempts to calculate the Tg using group additivity concepts have been reported in the literature (Van Krevelen, 1976). However, such an approach fails when one needs to predict the Tgof systems such as nonrandom copolymers, stereospecific polymers, etc. We have attempted to correlate the Tgof a homologous series of polycarbonates, with different types of substituents, to the calculated flexibility of the polymer segments. The conformational entropy is a direct measure of the degree of flexibility of a polymer system. A very flexible polymer will have a larger conformational entropy, when compared with a stiff polymer. Such comparisons are particularly useful when one is evaluating the functional suitability of polymers within a homologous series with systematic changes to the structure or when one is comparingdifferent copolymer compositions using a set of monomers. The conformational entropy is defined as
-
-
-
where
Pi= exp(-E,IRT)lZ
(2)
The summation in eq 1 includes each of the rotational states, i , of all the bonds which impart flexibility to the
4180 Ind. Eng. Chem. Res., Vol. 34,No. 12, 1995
--P
200-
cp
1
'
160:
120
\
\
.PPI
\
-
;\PPo
8 0 2 " " 1 "3 " 1 " " " 4" '
-e Figure 9. Isoenergy contours illustrating the flexibility of a polycarbonate chain segment arising from rotations 4 and p. The energy minima are marked by x. The contours are drawn at intervals of 1 kcavmol, relative to the minimum. All the conformational states outside of the 3 kcavmol contour will be of rare occurrence.
polymer. Ei is the energy of the ith state, and R and T are the gas constant and the absolute temperature, respectively. The conformational partition function is given by
In order to compute the conformational entropy, one needs to determine a conformational energy map for two neighboring, interdependent torsion angles. Molecular mechanics type models are most commonly used for this purpose, but semiempirical or ab initio quantum methods can also be used. In our studies, we have used molecular mechanics approaches with either commercially available (we used POLYGRAF, Version 3.0from Molecular Simulations Inc.) or in-house developed software. An example is shown in Figure 9, which depicts the flexibility of a segment of the bisphenol A polycarbonate chain (Sundararajan, 1989). The torsional angles, 4 and q, of the contiguous phenyl rings are varied about the backbone bonds of the isopropylidene moiety (see label in Figure 91, and the energies of the conformations are plotted as isoenergy contours. This figure shows that a number of conformations have energies within 3 kcal/mol of the minimum energy, indicating this polymer will be flexible at relatively low temperatures. The conformational entropy is evaluated with the energies derived from this map, using eqs 1-3. The effect of substituents on the flexibility can be evaluated by comparing similarly constructed maps. Figure 10 shows a plot of the calculated conformational entropy versus the experimental Tgvalues taken from various sources in the literature (Sundararajan, 1993). A linear correlation is evident between the calculated conformational entropy and Tgfor the series of polycarbonates. Using this, one can predict the Tg of a hypothetical polycarbonate of this type. The Tgof a new polycarbonate with a bulky substituent was
5
8
Sin (cal/mol/drg)
Figure 10. Plot of the calculated conformational entropy versus the experimental Tgvalues for the polycarbonates with different types of substituents at the a-carbon atom. Also shown are the results corresponding to the following poly(ppheny1enes): PPO, poly(pheny1ene oxide); PPS, poly(pheny1ene sulfide); PPO(CH3,CH3), poly(dimethylpheny1ene oxide); PPO(C&,C&), poly(diphenylphenylene oxide).
confirmed by experiments t o be within 10" of the predicted value. Other poly(l,4-p-phenylene)s, such as poly(pheny1ene oxide), poly(pheny1ene sulfide), poly(2,6-dimethylphenylene oxide), and poly(2,6-diphenylphenylene oxide), also follow the same trend, as illustrated in Figure 10. Another feature seen in this plot is that chains with articulated side groups follow a different correlation. For example, although the entropy of the polycarbonate segment with the CH2CH3,CH2CH3 articulated side group is much smaller than that of the Bisphenol A polycarbonate (with CH3,CH3 side group), the Tp's of the two polymers are the same. From the conformational maps of the type shown in Figure 9, it was found that the articulated side groups restrict the flexibility of the main-chain bonds severely. While this might be expected to increase the Tg, the large "volume" of the articulated side group decreases the intermolecular cohesion and leads to a lower Tg. Thus, in addition to providing useful correlations for a homologous series of polymers, these calculations also enable understanding of the various other factors that govern the functional properties. 4.b. Fracture Mechanics and Particle Attrition Processes. Fine powders of materials with a polymeric base are often prepared by grinding and jetting processes. The manufacture of toner is one example where a very fine powder is produced by a jet milling process. In a jet mill, high-velocity air streams move the particles in a fluidized bed, inducing collisions between particles and with walls. These collisions break down the particles into smaller particles. The degree to which a material "jets" will depend on its mechanical properties. There are, however, numerous mechanical properties that reflect a material's resistance to failure in processes that subject the material to stress or deformation. Which properties are important depends on the nature of the deformation and whether the material is prone to fail by either a brittle or ductile mechanism, which in turn can depend on the temperature and the rate of strain. One of our goals is to use molecular simulation methods to develop design rules that will identify highly jettable polymers, usually defined as a jet mill output per unit of time. One can take several approaches here.
-
Ind. Eng. Chem. Res., Vol. 34,No. 12, 1995 4181 X1-X,
2x1’
Remion
1
H H ) A H
t I
7.5
e
1
I
8.5
I
1
,
9.5
-I
I
10.5
Conformatlonsl Entropy
Figure 11. Plot of the calculated conformational entropy versus the jetting rate of polyesters with various comonomer compositions.
MD simulations can be used to predict the stress-strain behavior of models of either crystalline or amorphous polymers. (For semicrystalline polymers the picture is less clear since there is no unique state for a semicrystalline polymer.) This approach may yield some insight into the mechanisms of deformation and, therefore, how polymer structure influences mechanical properties. For example, the deformation of the polymer may be largely accomplished by distortion of the backbone torsion angles. Changes in substituents that directly increase the barrier heights of the backbone torsional potentials will then be expected to “toughen” the material. While we have been able t o pursue the approach mentioned above, a more successful route for us has been to establish a direct correlation between our performancemetric, the jetting rate, and a quantity that describes a polymer’s overall flexibility, the conformational entropy as described in the previous section. In Figure 11 we show a plot of a measured jetting rate versus computed conformational entropy. The points in this figure correspond t o various polyester copolymer compositions. The trend here is consistent with our physical intuition in that the highest jetting rate is observed for polymers with the lowest conformational entropy. The development of Figure 11is interesting in itself. The points drawn as circles were known from previous internal studies. The point drawn as a triangle corresponds to a material for which calculations had predicted a high jetting rate, but the material had not been synthesized. This material was selected from a list of over a dozen for which the same calculation of the conformational entropy had been performed. The material was then synthesized, its properties were measured, and it was confirmed to have the best jetting rate. In this study, a simple design rule was derived by comparing a performance metric with a property easily computed from the molecular structure of a material. The correlation that was established formed the basis for evolving the material design and eventually led to the design of a new material with improved properties. 4.c. Polymer Reactions. The development of a living free-radical polymerization process that enables the preparation of narrow polydispersity resins and controlled block length polymers remains a key challenge in polymer chemistry (Endo et al., 1992). Living polymer systems, as defined by Webster (1991), have three primary characteristics. First of all, assuming that the initiator is 100% efficient, the ultimate molecular weight is given by
Figure 12. Initiation-Transfer-Termination or “iniferter” reaction scheme of Otsu and Yoshida (1982) demonstrated for the dithiocarbamate system (XI = XZ= Me2N-CSz). The living nature of this reaction scheme has been shown in polymer and block copolymer synthesis.
(4) where D, is the degree of polymerization, [MI0 is the initial concentration of monomer, and [I1 is the concentration of growing polymer (chains generated by the initiator). Second, the polydispersity is narrow and is given by (1 l/D& in the ideal case. Finally, the molecular weight increases linearly as a function of the conversion, in contrast to conventional free-radical polymerization reactions, where high molecular weight material is formed early in the reaction and the molecular weight remains relatively constant as a function of conversion. In practice, all living polymerizations deviate to some degree from this ideal case (Webster, 1991; Ivan, 1993). The deviation of the polydispersity from the ideal value may be regarded as a metric for measuring the degree t o which a quasiliving polymerization reaction approaches the ideal living case. An important milestone in the development of living free-radical polymerization reactions was the discovery of iniferter reactions. The term “iniferter”, coined by Otsu and Yoshida (19821, is derived from InitiationTransfer-Termination and is illustrated by the dithiocarbamate system shown in Figure 12 (XI= Xz = MezN-CSZ). In this scheme, which relies primarily on photochemical initiation, the dithiocarbamate radical serves as both an initiator of new chains (Figure 12, reactions 3 and 4) and a reversible trapping agent (Figure 12, reaction 2). The living nature of this reaction has been demonstrated by block copolymer formation (Turner and Blevins, 1988). Although the iniferter reaction demonstrates some of the characteristics of a living polymerization, such as a linear increase of molecular weight with time, there are also a number of deficiencies. For example, there is a significant loss of active end groups from the living polymer and reequilibration to the dithiuram initiator during the course of the reaction (Lambrinos et al., 1990). Furthermore, by the very nature of the iniferter mechanism, the dithiocarbamate radicals can initiate new chains throughout the course of the reaction, albeit very slowly (Turner and Blevins, 1988; Lambrinos et al., 1990). As a consequence, the reported polydispersities are relatively large and a substantial amount of homopolymer is formed along with block copolymer.
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4182 Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995
The development of a living free-radical reaction that meets the challenging test of providing resins with narrow polydispersities requires the design of reversible termination agents that balance the competing requirements of rapid propagation with no new initiation once all of the living chains have been generated. Semiempirical molecular orbital calculations, coupled with experimental investigations, have provided substantial direction in the selection of these reversible terminator compounds. Our approach t o this problem is described briefly below. In order to design a living free-radical polymerization reaction that produces a polymer with a narrow molecular weight distribution, it is essential that at least three conditions are satisfied. First, all propagating chains must be initiated within a very short time period. Second, the chains must be captured in a reversible manner, as demonstrated in the iniferter system, so that the process does not terminate prematurely. This preserves the living nature of the reaction. Finally, during the course of stepwise chain growth, it is essential that no new chains are initiated by any of the intermediate radical species. The iniferter reaction has a n inherent limitation for achieving all the features of a living reaction. By design, iniferter reagents are required to both initiate the polymerization and reversibly capture the propagating radical to ensure that the propagating species does not terminate prematurely. An attractive solution would be to separate the initiation and propagation processes altogether (Georges et al., 1993a,b;Veregin et al., 1993). The living characteristic of the reaction can be preserved if a reversible terminator, XZ,is chosen that can rapidly capture the propagating carbon radical as an a-carbonXZ adduct, provided the resulting bond is weak enough for thermal equilibration between the free radical and the capped radical (reaction 4). This equilibrium must allow a sufficient concentration of free radical for propagation but not enough for irreversible chain termination. A trade-off is still required, but one that is much more manageable than the one required of an iniferter reagent. In addition, the radical XZ must be unreactive with respect to initiation of new chains. Initiation is achieved with another radical (XI) which forms a strong XI-carbon bond with styrene and irreversibly binds to the polymer. The process of selecting a reversibly terminating agent could be greatly simplified if one could reliably predict the bond dissociation enthalpy, m b . d . , of reaction 4,Figure 12, for a variety of possible structures. We have found molecular orbital calculations to be of significant benefit in this selection process. The application of molecular orbital calculations to this problem is facilitated by the experimental observation that radical reactions are relatively insensitive to solvent conditions (Karelson et al., 1991). To a first approximation, therefore, solvent effects can be ignored. Still, it is necessary to determine what kind of calculations should be used and whether or not these calculations provide a sufficiently accurate estimate of M b . d . to allow reliable predictions. Both ab initio and semiempirical molecular orbital calculations have been successfully applied to radical processes (Dannenberg, 1990). However, the size of the model systems required for this study initially compelled the use of semiempirical rather than ab initio methods. Ideally, one would like to investigate the transition state associated with the free radical-carbon bond dissocia-
HYH3 Figure 13. Reaction scheme developed as a small-molecule analogue of the reversible termination step for a pseudo living freeradical polymerization process. Table 2. Calculated Bond Dissociation Enthalpies for the Reaction Shown in Figure 13“ radical fragment X PhCOz CH3S mercaptothiazole dithiocarbamate TEMPO
bond dissociation enthalpy (kcavmol) AM1 PM3 b 56 53 38 22
63 52 57 39 26
The systems with smaller values are better candidates for use as reversible terminators in pseudo living free-radical polymerization. * SCF convergence problems with both RHF-half-electron and UHF methods were experienced with the benzoyloxy radical system in MOPAC. Hyperchem UHF calculations gave a bond dissociation enthalpy of 58.5 kcal/mol.
tion. However, locating a transition state is still a formidable challenge, particularly when the starting material is a closed-shell species while the products are open-shell entities. Rather than challenging the complexities of the transition state calculation, it was decided to only calculate m b . d . . The radical derived from ethylbenzene (Figure 13) was chosen as the model representing the polymer. Initial investigations using the propylbenzene radical indicated that the longer chain length necessitated the evaluation of many conformational minima involving the a-carbon bond without improving the reliability of the results (Georges et al., 1993a,b). The choice of ethylbenzene as the model system removed this complication. For the TEMPO radical, the nitroxide moiety was oriented in such a way that the N-0 bond was equatorial with respect t o the cyclic nitroxide. Conformational searches using molecular mechanics and semiempirical calculations indicated that the energy of the adduct was not significantly affected by this nitroxide conformational constraint. Estimated values of m b . d . were obtained by calculating the heats of formation of the species shown in the reaction in Figure 13. Subtraction of the heats of formation of the reactants from those of the products provided m b . d . for the bond scission reaction. Table 2 summarizes the results of AM1 and PM3 calculations on the bond dissociation enthalpies for a variety of systems. Initiators such as benzoyloxy radical (Webster, 1991) show large (63 kcal/mol) m b . d . ’ S , consistent with the perception that these intermediates lead t o irreversible termination. Chain-transfer reagents, represented by the methyl thiyl radical, show substantially smaller m b . d . ’ S , while iniferters (Ivan, 1993)have weaker bonds still. We therefore expect that the use of these different radical terminators will give rise to very different equilibrium polymer radical concentrations, which will influence directly the polymerization rate and the final polydispersity. In summary then, semiempirical molecular orbital calculations provide a way for designing new, reversible terminators which have the property of reversibly regenerating propagating radicals for stepwise polymerization.
Ind. Eng. Chem. Res., Vol. 34,No. 12, 1995 4183
5. Challenges In our experience, the methods of computational chemistry have an important role to play in the future of specialty materials development. The benefits from investing in this technology include improved understanding of the materials and new routes to developing design rules. As with any field, there are a number of outstanding challenges. For polymer systems, the challenges can be split into two groups-those imposed by the limitation of our computational hardware and software and those imposed by our limited understanding of the underlying science. Let us begin with the former, which is perhaps easiest to discuss. Many polymer properties, including some of the ones mentioned above, are known to be molecular weight dependent. Often this molecular weight dependence occurs over a limited range and the properties reach plateau values at high molecular weight. The molecular weight at which this occurs depends on the type of polymer but can be anywhere from a few thousand to several hundred thousand. Due t o their high molecular weight, the dynamics of high polymers can span the regime from microseconds up to seconds. Today, even with the use of the leading supercomputer systems, simulations of polymer systems with very high molecular weight out to suffciently long times to exhibit true rheological behavior are not possible. In order to use molecular simulations as a basis for studying such properties, one must develop hybrid approaches that, for example, describe long distance and long-time behavior in terms of properties that can be derived from short-time simulations on small systems. The need for, these hybrid approaches will persist through several generations of new computer architectures. In predicting the optical properties of organic photogenerator pigments, we face a similar problem. The sizes of the systems we deal with are simply too large to apply rigorous quantum mechanical methods. The empirical approaches, such as the extended Huckel method we applied in section 3, only provide qualitative guidance. Force fields for molecular simulations, and methods for deriving them, are highly evolved today compared with just a few years ago. However, while most popular force fields do a credible job predicting molecular structures, the accuracy of nonbonded interactions still makes quantitative predictions for condensed phases difficult. This is particularly important to our studies of polymorphism if we are to use molecular mechanics level models to derive plausible structures for use with structural refinement techniques and X-ray data. Another class of challenges we face with polymer and pigment simulations is due to our limited understanding of the underlying microphysical processes that occur. For example, polymer fracture can occur via several different mechanisms. In one picture, it occurs first by nucleating small voids which, under tensile deformation, grow to initiate a crack. The crack then propagates until the material fails. To what degree is bond breaking occurring during the nucleation phase, to what extent is there small-scale plastic deformation occurring, etc.? Similar questions can be asked about polymorphic conversion in pigments. The answers to these questions, which are largely unknown, can perhaps be addressed with molecular simulations. However, without clearer guidance it is very difficult to construct
appropriate models. Quantitative predictions on fracture, based on molecular simulations, have eluded us to date.
Acknowledgment The authors thank Prof. M. Ferrario (Messina) for the use of his molecular dynamics code and Prof. R. Hoffmann (Cornell) for a fruitful collaboration on the application of extended Huckel methods to pigment systems.
Nomenclature a, b, c = unit cell dimensions (for one dimensional stacks, a is the repeat distance in the stacking direction) AM1 = Austin model 1,a semiemperical parametrization d = distance between planes containing two planar mol-
ecules in a stack
D, = degree of polymerization
Ei = potential energy of state i [I1 = concentration of growing polymer chains IR = infrared k = wave number or band index 1 = longitudinal offset distance of one perylene molecule relative to another LED = light-emitting diode [MI0 = initial monomer concentration MD = molecular dynamics Me = methyl group N = number of particles P = external pressure Pi = probability of a given state, i PM3 = parametric method 3, a semiemperical parameterization R = ideal gas constant S = conformational entropy T = absolute temperature TEMPO = 2,2,6,6-tetramethyl-l-piperidinyloxy Tg= glass transition temperature t = transverse offset distance of one perylene molecule relative to another TiOPc = oxotitanium phthalocyanine V = volume X, XI, Xz = molecular radical species 2 = conformational partition function 2 = number of molecules in the unit cell /3 = unique unit cell angle for monoclinic space groups m b . d . = bond dissociation enthalpy ,A = wavelength of the absorption maximum 4, w = torsional angles ( ) = indicates a statistical mechanical ensemble average of a given property
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Abstract published i n Advance A C S Abstracts, November 15, 1995. @
