J. Phys. Chem. B 2009, 113, 4549–4554
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ARTICLES Spherulitic Networks: From Structure to Rheological Property Jing Hua Shi,†,‡ Xiang Yang Liu,*,†,‡ Jing Liang Li,† Christina S. Strom,† and Hong Yao Xu§ Department of Physics, National UniVersity of Singapore, 2 Science DriVe 3, Singapore 117542, Institute of Crystal Materials, Shandong UniVersity, Shandong 250100, China, and Department of Polymer Science and Engineering, Donghua UniVersity, Shanghai 201620, China ReceiVed: April 19, 2008; ReVised Manuscript ReceiVed: February 04, 2009
A finite element method based on ABAQUS is employed to examine the correlation between the microstructure and the elastic response of planar Cayley treelike fiber networks. It is found that the elastic modulus of the fiber network decreases drastically with the fiber length, following the power law. The power law of elastic modulus G′ vs the correlation length ξ obtained from this simulation has an exponent of -1.71, which is close to the exponent of -1.5 for a single-domain network of agar gels. On the other hand, the experimental results from multidomain networks give rise to a power law index of -0.49. The difference between -1.5 and -0.49 can be attributed to the multidomain structure, which weakens the structure of the overall system and therefore suppresses the increase in G′. In addition, when the aspect ratio of the fiber is smaller than 20, the radius of the fiber cross-section has a great impact on the network elasticity, while, when the aspect ratio is larger than 20, it has almost no effect on the elastic property of the network. The stress distribution in the network is uniform due to the symmetrical network structure. This study provides a general understanding of the correlation between microscopic structure and the macroscopic properties of soft functional materials. Introduction Small-molecule-mass organogels (SMOGs) consisting of 3D interconnected fiber networks trapping organic liquid have attracted great attention in recent years due to their great potential applications in drug delivery, separation, catalyst supporters, scaffolds for tissue engineering, templating nanostructural materials, etc.1-13 In general, the applications of such materials are determined by their properties and microstructures. For instance, tissue engineering requires scaffolds with the appropriate mechanical properties and network structure to create and maintain conditions for tissue culture.14,15 Macroscopic properties, in particular the rheological properties of SMOGs, are determined by the microstructure of fiber networks. Fibrous networks with permanent interconnections will effectively entrap and immobilize liquid in the meshes and possess both the elastic properties of ideal solids and the viscosity properties of Newtonian liquids, leading to the formation of self-supporting supramolecular materials.1,3 In contrast, systems consisting of nonpermanent or transient interconnecting (or entangled) fibers or needles can only form weak and viscous paste at low concentrations.1 As found, any change in the topology and structure of fiber networks of SMOGs will exert a direct impact on the rheological properties.1-30 The stimuli giving rise to the impact range from chemicals to the cooling rate, seeding, sound, etc. In general, the smaller the mesh size of the fiber networks, the larger the gelling capacity and the higher the elastic modulus. We simply summarize the * To whom correspondence should be addressed. Phone: (65)-6516-2812. Fax: (65)-6777-6126. E-mail:
[email protected]. † National University of Singapore. ‡ Shandong University. § Donghua University.
knowledge already learned about SMOGs as follows: (i) the types of gelators and, in part, the relationship between the gelator structure and gelation ability;1,16-19 (ii) the morphology structures of the organogel which have been studied by a variety of different techniques including scanning electron microscopy (SEM), transmission electron microscopy (TEM), and so on20,21 (generally, the organogel networks can be grouped into two types: entangled continuous fibrous networks and Cayley treelike or spherulitic networks);22-27 (iii) the rheological and thermodynamic properties of SMOGs;28 (iv) the formation mechanism of SMOGs, which is widely accepted as nucleation-mediated branching or “crystallographic mismatch branching” (CMB).28-31 Nevertheless, as a part of the general study of soft functional material engineering, the relationship between the microstructure of an organogel network and its macroscopic properties, particularly, the mechanical properties, has not been established, due to the complexity of the organogel system. Since there is no sufficient understanding, significant efforts have been devoted to identify novel systems with threedimensional (3D) interconnecting self-organized network structurestoobtainfunctionalmaterialswiththedesirableproperties.1-6,32 This includes the screening of a large number of compounds and solvent molecules, as well as suitable solvents capable of forming these structures. In some cases, because of the limitation in the choice of materials, screening becomes impossible. On the other hand, if interconnecting 3D micro- or nanofiber networks with the required organization can be constructed, new functional materials with the required functionalities can be produced. Obviously, this is a completely new route in producing new functional materials. The correlation between microscopic structure and the macroscopic properties should be
10.1021/jp8035023 CCC: $40.75 2009 American Chemical Society Published on Web 03/17/2009
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Figure 1. Formation of a fiber network based on the CMB mechanism.
established to engineer the small-molecule-mass organogels at the nanoscale to acquire some specific functions. It is the purpose of this paper to establish in terms of modeling the relationship between the microscopic structure and macroscopic elastic coefficient of spherulitic networks of gel. We hope that the new knowledge acquired in this paper will serve as a guidance for the production of soft functional materials of new types. The challenge of this simulation work is how to acquire the elastic modulus G′ of the Cayley treelike network. In this regard, Head et al. have obtained the shear modulus of a semiflexible polymer network through modeling of the network using the finite element solver Nastran by MSC software.33-39 In this study, the effects of a series of microstructure parameters on the shear modulus are examined on the basis of a finite element solver, so-called ABAQUS, and the related model.37 The spatial stress distribution in the network after applied shear strain is also studied. Simulation Model and Method Network Model. According to the mechanism of CMB, the network formation can be described as follows: Initially gelators nucleate and subsequently grow mainly longitudinally. At a certain fiber length, the structural match between the parent fibers and the new fibers will be disturbed, which leads to the branching of the fibers. Thus, a small fiber network (structure unit), such as a Cayley tree, is formed as a result of the process of nucleation growth-CMB growth-CMB growth (Figure 1). The entire network is composed of a number of individual smaller fiber networks, each of which originated from a nucleation site. The fibers of one structure unit entangle with those of other structure units to form the composite network, as shown by Figure 2a.26 The structure units are characterized as spherulites.
Shi et al. The above results give rise to the following model: In a square area, initially fibers emanate from a central point, each fiber gives birth to two new-generation branches, and the angle between two branches, ϑ, is constant (Figure 1). It is assumed that once a growing fiber meets another growing fiber they stop growing and never branch again. When the boundaries are encountered, the growth of the network terminates. The fiber length (Figure 1) and the cross-section of the fiber segment are uniform in the whole network. Junctions are considered to be permanent linkages to mimic branching points. The model structure is given in Figure 2b. Note that there may be several spherulites coexisting in a gel system. Nevertheless, here we start with a system comprising only one spherulite. Simulation Method. All our simulations were carried out using the commercially available finite element solver ABAQUS/ Standard ver. 6.6 on the x86 HPC Linux Clusters. The fiber segment is a homogeneous, linear elastic material with a circular cross-section, and the radius is r. A static, general procedure is chosen to capture the response of the network after application of an external loading.40 There are two types of beam elements in ABAQUS that can be used to model the network,40 that is, the Euler-Bernoulli beam and Timoshenko beam. The classical Euler-Bernoulli beam takes only stretching and bending deformation into account. This type of beam element is suitable to model slender beams. The Timoshenko beam allows for transverse shear deformation (the cross-section of the fiber deforms along the longitudinal direction) besides stretching and bending.40 This type of beam element can be employed for thick beams as well as slender beams. Generally, the effect of transverse shear cannot be negligible for beams made of homogeneous materials, when the aspect ratio, l/r, is lower than 15. Here, l is the beam length and r is the radius of the beam cross-section. In our case, l is the length of the fiber segment (Figure 1) and r is the radius of the fiber cross-section. The experimental aspect ratio of a fiber segment could be smaller than 15, so in our modeling, the Timoshenko beam element is chosen to model the network. For an initially unstressed network, the strain energy is zero. After application of a shear strain γ, the stored strain energy εstrain is obtained through the simulation; it is written as41
1 εstrain ) G ′γ2LxLy(2r) 2
(1)
where G′ denotes the shear modulus of the network, γ is the applied shear strain, Lx and Ly are the length and the width of
Figure 2. (a) Optical micrograph of an organogel network. Refer to ref 26. (b) Simulation model: planar radial-growth network.
Spherulitic Networks
Figure 3. Effect of the junction density on G′. Black circles are the simulation data, and the red line is to guide the eye.
the area, and 2r is the diameter of the fiber cross-section. Consequently, the shear modulus of the network is obtained. Results and Discussion Influence on G′ of the Junction Density, Fiber Length l, and Radius r. The elastic response of a fiber segment between two neighboring branching points is characterized by lengthdependent force constants for stretching, µ (proportional to Er 2/ l), for bending, k (proportional to Er4/l3), and for transverse shear, g (proportional to Er2/[2(1 + ν)l]).36 E is Young’s modulus (the unit of E is N m-2) of the individual fiber, and ν is the Poisson ratio. The ratio of µ to k is proportional to (l/r) 2, the square of the aspect ratio. Thus, the aspect ratio can reflect the rigidity of the fiber segment. For radial network growth within a fixed area, a change in l causes both the fiber rigidity and the number of junctions to change, while a change in r causes only the fiber rigidity to change. We consider these effects on the macroscopic elasticity of the network. First, if l is changed, the junction density of the network, defined as the number of junctions per network area, varies accordingly. In this simulation, the area is 200 × 200 µm2, allowing the network to have more than 20 orbits. Figure 3 shows the relationship between the shear modulus, G′, and the junction density. It follows that G′ increases monotonically with the junction density. Figure 4a shows the dependence of G′ on l. Figure 4b is the log-log plot of the resulting G′ as a function of l. It gives log(G′) ) -1.71 log(l/r) + 8.31. We notice that the change in behavior noted near l/r ) 20 (decrease compared to leveling off) in Figure 4a is an artifact of the linear scale as no change in the l/r dependence of G′ is seen in Figure 4b at any aspect ratio.
J. Phys. Chem. B, Vol. 113, No. 14, 2009 4551 A similar relationship has been obtained experimentally for a gel system formed by dissolving 8 wt % dibutyllauroylglutamic acid (GP-1) (>98%, from Aijnomoto) in isostearyl alcohol, where different G′ and correlation lengths were obtained from the gels by fixing the gelator concentration while changing the gelation temperature.23 It can be seen (Figure 4a and inset) that both the simulations and the experiments show a consistent relationship between G′ and l or ξ as G′max ≈ ξ-p (ξ is the correlation length, defined as the distance between two adjacent branching points along one fiber, and is proportional to l in our simulations). Nevertheless, the power of ξ, p, is different: the experiment gives G′max ≈ ξ-p (where p ) 0.49),23 while our simulations give G′ ≈ l-p (where p ) 1.71). In another study,42 the network formation of agarose gels based on both theoretical analysis and experiments gives rise to G′max ≈ ξ-p (where p ) 1.5), which is reasonably close to our simulation p ) 1.71. The difference may be attributed to the fact that, in our modeling, a single spherulitic network is treated, while, in the GP-1 experiments, several spherulitic networks occur simultaneously in the system (Figure 5). As the interactions between adjacent spherulites are weak and transient, there will be two consequences: (1) The enhancement of G′ due to the shortening of the fiber length is weakened with the increase of the organogel spherulitic network domains. This is because the weak interactions between adjacent spherulites cancel the enhancement effect of G′ due to the shortening of the fiber length. (2) The power p in the expression G′ ≈ ξ-p is determined by the topology and the structure of the fiber networks. When the single-domain spherulitic fiber network (in our simulations) is replaced by the multidomain (or spherulitic) fiber networks, both the topology and the overall structure of the networks are changed. The two consequences eventually give rise to the drop of the experimental exponent, compared to the simulations. When the networks are interpenetrated, the clear boundary between adjacent fiber networks vanishes. The multidomain networks will then behave like a single network. p is close to our simulation value, which was observed in the case of agarose gels.42 We notice that the dimension difference between the experimental and our simulation systems might exert some impact on the exponent. This is subject to further investigation. The above results also indicate that if in the real gel system the total number of spherulites is approximately constant, the rheological properties, mainly G′, will be substantially enhanced by reducing the mesh size or the average length of the fiber segments of the fiber network. In other words, if the mesh size of the network is approximately constant, the elastic modulus can also be significantly enhanced by decreasing the number of spherulites due to the reduction of the weak interaction in
Figure 4. (a) Effect of the fiber length l on G′, where r is fixed. The inset is the experimental relationship between G′ and ξ. (b) log-log plot of G′ vs l. Black circles are the simulation data, and the red line is the linear fit of the data. Refer to ref 23 for the details of the experiments.
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Figure 5. A spherulite fiber network formed by 3 wt % GP-1 in propylene glycol. The gelation temperature is 40 °C. Refer to ref 23.
Figure 6. G′ vs l/r where r is varied and l is fixed. ϑ is the branching angle. The number 0.99 (0.96) means that ϑ of a new generation equals 0.99 (0.96) times ϑ of the previous generation. The inset shows the linear relationship in the log-log plot of G′ vs l/r, indicating the power law dependence of G′ on l/r.
the system. According to our experiments, this can be achieved by introducing additives or changing the gelation conditions.23-26 It is worth mentioning that the range of the aspect ratio in Figure 4a is relatively small, from 8 to 30, which corresponds to the number of orbits (or generations) of the network ranging from 21 to 6. There are two reasons for this. First, according to the program, if we want to generate a network with R orbits,
Shi et al. we need to store a number of 2(R+1) junctions. (Note that the actual number of junctions in the model is much less than 2(R+1) because the fibers stop growing when they meet any other fibers.) Due to such a large number of junctions, the computer can only generate a network with less than 20 orbits, i.e., R < 20, to function within its capacity. Hence, in a fixed area the range of fiber lengths is limited. Second, the aspect ratios of the fiber network, as observed from experiments, can be as small as 8. Thus, the range of aspect ratios in our simulation was chosen from 8 to 30, and the largest number of orbits is reached when the aspect ratio equals 8. The optical images given by Figures 2 and 5 show that the number of orbits of one small spherulite is below 10, whereas for large spherulites the number of orbits varies along different directions. In some cases it is below 10, and in others it is much larger than 30. On average, the range of orbits in our simulation, which is from 6 to 21, is reasonable in replicating the nature of the network. During the simulation, we notice that not only the fiber length but also the radius of the fiber cross-section, r, can greatly influence the network elasticity property. Because changing r does not affect either the planar topology of the network or the junction density, its value is not restricted. Figure 6 shows the dependence of G′ on 1/r and the log-log plot of G′ vs 1/r (the inset). It can be seen that when r is large and the aspect ratio, l/r, is smaller than about 20, G′ increases very quickly with r. Once l/r becomes larger than 20, the relation tends to level off. This implies that when the fiber is short and thick, the radius of the fiber cross-section has a great impact on the network elastic property; once the fiber becomes long, the effect of r on G′ can be neglected. In this investigation, the impact of the branching angle, ϑ, of the fibers is also examined. As seen in Figure 6, besides considering a constant ϑ, varying branching angles are also considered. The number 0.99 (0.96) means that ϑ of a new generation equals 0.99 (0.96) times ϑ of its previous generation. The fact that three straight lines simulated by Mathematica share nearly the same slope (see the inset of Figure 6) indicates that the powers of the three power law curves are almost the same. These results mean that ϑ exerts a certain influence on the shear modulus, but not affecting the behavior of G′ with respect to l/r. Influence of Young’s Modulus and the Poisson Ratio on G′. The effects of the material parameters on G′, such as Young’s modulus and the Poisson ratio of an individual fiber, are examined as well. The results are given in Figure 7. In Figure 7a, two networks with different fiber lengths (l ) 16 and 8) are
Figure 7. (a) Effect of Young’s modulus of an individual fiber on the shear modulus. Two kinds of networks with l/r ) 10 and 20 are investigated. For each kind of network three networks with different branching angles are tested. (b) effect of the Poisson ratio of an individual fiber on the shear modulus.
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J. Phys. Chem. B, Vol. 113, No. 14, 2009 4553 obtained are in reasonably good agreement with the experiments. Evidently, the simulation is a first basic step toward research of more complicated systems, such as networks consisting of many spherulites, three-dimensional networks, which will provide a guide to the design and fabrication of soft functional materials in practice. In this regard, the following conclusions can be obtained from our simulations. (1) The network elastic modulus decreases dramatically with the fiber length, which has a power law relationship. By exploring the difference between the computational and the experimental results, we suggest two ways to engineer the microstructure of the organogel network for the desired rheological properties: under the condition of the total number of spherulites being approximately constant, the network elastic modulus can be substantially enhanced by decreasing the mesh size or the average length of the fiber segments of the fiber networks, or if the mesh size of the network is approximately constant, the elastic modulus can also be significantly enhanced by decreasing the number of spherulites due to the resulting reduction of the weak interaction between different network domains. (2) When the aspect ratio of the fiber is smaller than 20, the radius of the fiber cross-section has a great impact on the network elasticity; while, when the aspect ratio is larger than 20, it exerts little effect on the elasticity property of the network due to the power law dependence of G′ on l/r. (3) The elasticity property of the network will increase with the junction density and fiber stiffness of the fiber network. Acknowledgment. This research is supported by FOS, the NUS Fund, and the Tier 1 Academic Research Fund (Grants R-144-000-169-101/112 and R-144-000-199-112). We thank Dr. Jun Ying Xiong for his kind assistance and advice at the beginning of this research. References and Notes
Figure 8. (a) Stress distribution in networks with a fiber aspect ratio of 16. (b) Stress distribution in networks with a fiber aspect ratio of 8. Stress distribution sequence: from red to blue the stress varies correspondingly from maximum to minimum.
taken into account. For a given fiber length, three networks with different branching angles (ϑ ) constant, 0.99, and 0.96; refer to Figure 6 for the meaning of 0.99 and 0.96). For all these networks, the investigated G′ increases linearly with Young’s modulus. Figure 7b shows the relation between G′ and the Poisson ratio. Increasing the Poisson ratio has a negative effect on G′, but the impact is trivial. Spatial Stress Distribution. Figure 8 shows the stress distribution in the networks with different fiber lengths. When the fiber length is long, the stress is distributed over a large area (Figure 8a), while, when l is short, the stress distribution is mainly along several radii (Figure 8b). In any case, the area around the central point undergoes a high level of stress. Summary and Conclusions In this work, we studied the Cayley treelike network in detail. Although the simulated network is two-dimensional, the results
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