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Investigation on viscoelastic properties of urea-formaldehyde microcapsules by using nanoindentation
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Material Properties
Investigation on viscoelastic properties of urea-formaldehyde microcapsules by using nanoindentation Rui Han a, Xianfeng Wang a, *, Guangming Zhu b, Ningxu Han a, Feng Xing a a
College of Civil and Transportation Engineering, Guangdong Provincial Key Laboratory of Durability for Marine Civil Engineering, Shenzhen University, Shenzhen, 518060, PR China b College of Materials Science and Engineering, Shenzhen University, Shenzhen, 518060, PR China
A R T I C L E I N F O
A B S T R A C T
Keywords: Microcapsule Viscoelastic Maxwell model Burgers model Finite element method (FEM) Nanoindentation
The viscoelastic properties of urea-formaldehyde (UF) microcapsules is very important for achieving self-healing process in the field of concrete. In this study, the load-displacement relationship of the microcapsule was ob tained by micromanipulation technique and the variation of force with time for compression and holding pro cess, from which the properties of elasticity and viscoelasticity were determined. Three viscoelastic models, including 3-parameter Maxwell model, 5-parameter Maxwell model and 4-parameter Burgers model based on shell theory, have been applied to study the viscoelastic properties of the microcapsules. This study has demonstrated that these models consistently predict viscoelastic properties for both ramping and stress relaxa tion periods. Utilization of the models can enrich the experimental nano-micro material mechanics in inter pretation of nanoindentation of microcapsules.
1. Introduction A microcapsule can be regarded as a core-wall composite structure. In which one material (core) is entirely microencapsulated by another (shell). In general, liquid, powder and solid can be used for core mate rial, while the shell is solid (polymeric or inorganic wall). The different utilizations of microcapsules mainly rely on the types of core materials and various trigger patterns [1]. The pioneering work started from Jong’s research in the late 1930s [2]. Afterwards, especially in recent years, inventing different kinds of functional microcapsules has attrac ted more and more researchers’ interest, and they have been extensively applied to improve the quality of foods [3,4], pharmaceutic [5], printing industry [6], cosmetics industry [7], as well as civil engineering etc. [8–11]. In this paper, urea-formaldehyde (UF) microcapsules were applied in healing microcracks in materials [12,13]. The repair process can be achieved by rupture of microcapsules’ shell as the generation and propagation of cracks. The application of microcapsules as vehicles for delivering agents in time at the right place [14–16]. Consequently, it is important how to trigger the burst of microcapsules. Mechanical trigger (External or enteral loads) is common among many triggering mechanisms, hence study of the mechanical properties of microcapsules is important for
determining whether or not triggering the rupture of microcapsules. In simple terms, the microcapsules must be mechanically stable and intact during the fabrication and storage and have optimal mechanical char acteristics if the successful release of the core materials need to be triggered under external/internal loads. Meanwhile, study of mechani cal characteristics of micro-nano particles becomes accessible with the development of precision instruments. There exist two methods, bulk and individual approaches, to study the mechanical characteristics of microcapsules. When it comes to the bulk methods, researchers only obtain the average mechanical parameters, such as Young’s modulus as well as shear modulus [17,18], therefore, these methods have not been used extensively now. In the following, individual micro-nano particle measurement equipment, such as atomic force microscope (AFM) and micro-nano indentation (micro-nano manipulation) will be presented. Whereas, among these advanced apparatus, electromagnetic tweezers and/or micropipette aspiration are generally used for soft samples [19]. Firstly, Cole studied the arbacia egg’s (diameter: 75 μm) mechanical behavior using a plate compression test in the 1930s [20]. After decades of development, precise compression of a single micro-nano particle was introduced by Zhang et al. [21]. The mechanical properties of a large number of samples, such as microcapsules, cells and beads, were
* Corresponding author. E-mail address: [email protected] (X. Wang). https://doi.org/10.1016/j.polymertesting.2019.106146 Received 18 July 2019; Received in revised form 14 September 2019; Accepted 8 October 2019 Available online 11 October 2019 0142-9418/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Rui Han, Polymer Testing, https://doi.org/10.1016/j.polymertesting.2019.106146
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determined by using micromanipulation. Specifically, the mechanical properties, including elasticity, plastic-elasticity and viscoelasticity of melamine formaldehyde (MF) microcapsules have been studied [22]. As far as we know, the research of mechanical properties of MF micro capsule have been comparatively comprehensive and systematic. The diameter, thickness, and composition of microcapsules directly result in the mechanical behavior. Even though, elastic-plastic properties of UF microcapsules has been estimated by Keller and Sottos [23] Ahangari et al. [24]. Ma et al. [25]. Mercade-Prieto et al. [26–28]. Meanwhile, numerical method combined with realistic experiment is an effective means in studying a variety of physical phenomena, in which the finite element method (FEM), as a numerical method for solving the mathe matical physics problems, has been widely used in various fields such as simulation of mechanical behavior of materials [29–31], fluid flow [32], optical communication [33,34], as well as sensor systems [35–37]. Nanoindentation techniques have been proven to be effective to char acterize the mechanical properties of nano-micro materials. However, most studies focused on the elastic-plastic properties of microcapsules, in other words, viscoelastic properties of a single UF microcapsule have not been comprehensively studied yet. In the present study, UF microcapsules are synthesized by an in-situ polymerization method and are characterized (diameter and thickness distribution) by scanning electron microscopy (SEM). The indentation on the microcapsules to a small deformation was completed, its elastic parameter (Young’s modulus) was determined by contact and shell theory. Moreover, according to the compression and holding tests, viscoelastic parameters were obtained for the Maxwell and Burgers models based on shell theory.
Table 1 Parameters and the specific settings in preparing UF microcapsules. Synthesis time (hour)
Rotational velocity (r/min)
Average diameter (μm)
Wall thickness (μm)
Core content (%)
2
200
198.4
8.1
78
it was placed on the test stand as shown in Fig. 4. G200 type Nano indenter (Agilent Inc., USA) with a flat end was applied in this study to obtain force and displacement curves. The creep and relaxation response of a material can be applied for determining its viscoelastic properties. In this study, the displacementcontrolled method was employed, and the advantage of using this approach is that the indenter displacement can be easily controlled and modified in experimental investigations. The indentation-relaxation tests were performed to the microcapsules at two different displace ment levels (dmax ¼ 5 μm and 10 μm) at a fixed rise time (tR) of 150 s for both displacement levels and with a 100s hold period at peak depth. The controlling parameters for the nanoindentation can be found in Table 2. A compression and holding procedure (displacement-controlled) is shown in Fig. 5. Firstly, a constant loading speed is performed in the indenter, and the reaction force increases gradually as the indenter displacement gets close to the sample growingly. And then, the indenter would be in position when the penetration depth reached pre-setting maximum value. Meanwhile, the reaction force measured would decrease slowly as time goes on due to the material’s viscosity.
2. Materials and methods
dðtÞ ¼ kt; 0 � t � tR
(1)
dðtÞ ¼ d0 ; tR � t � tτ
(2)
where d(t) is the penetration depth related to time t of the model, d0 is the penetration depth under holding procedure, k is the loading speed, tR is the staring time of holding procedure, tτ is the ending time of holding procedure.
2.1. UF microcapsules UF microcapsules with core materials being epoxy resin E 51 and butyl glycidyl ether were prepared using an in-situ polymerization method in this study, as shown in Fig. 1. The preparation details can be referred to Ref. [11]. Specific parameters of UF microcapsules can be found in Table 1. Fig. 2(a) and Fig. 2(b) show the microcapsules for calibrating the diameter and wall thickness, respectively. These geo metric parameters were characterized using SEM (Quanta TM 250 FEG) for counting the 300 microcapsules, and Fig. 3(a), (b) show the diameter distribution and the wall thickness distribution of the UF microcapsules, respectively. The mean diameter and wall thickness of the UF micro capsules are shown in Table 1.
2.3. Finite element model The nanoindentation test with flat-ended indenter was simulated using finite element method (FEM) by Abaqus 6.14, as shown in Fig. 6(a) and (b). A single UF microcapsule was modelled as a typical spherical shell structure (diameter ¼ 198.4 μm, wall thickness ¼ 8.1 μm) using 2D axisymmetric element, including CAX4R (4-node bilinear, reduced integration with hourglass control) and SAX1 (2-node axisymmetric shell element), and the comparison using these two elements was shown in appendix A. supplementary data. And the sample was simulated by an isotropic and homogenous linear elastic material due to its small deformation. In addition, the flat-ended indenter and the substrate were simulated by rigid elements. Friction was not considered in the tangential direction, and the hard contact model was applied in the normal direction. The spherical model was compressed using a displacement-control of the indenter to a depth of 5 μm. The bottom of the microcapsule and substrate was fully fixed. The variation of the material Poisson’s ratios from 0.1 to 0.5 and the adaptivity of the finite element mesh were considered to investigate their influence on the nanoindentation behavior of the microcapsule in Appendix A supple mentary data. The effect of the core liquid or air on the behavior of the microcapsule was not significant based on our previous research [29].
2.2. Preparation of specimen and nanoindentation Several UF microcapsules were dispersed on the top of 1 cm � 1 cm glass piece attached by epoxy resin and kept it 24 h to cure. Eventually,
2.4. Elastic models The Hertz model has been used to study the contact behavior of elastic spherical particles, especially when the strain is small, the sur faces are continuous and the contact is frictionless [38]. In this case, the classical Hertz contact model can be applied in determining the Young’s modulus [39]. The force-displacement (F-d) relation for ramping period
Fig. 1. Schematic diagram of preparing UF microcapsules. 2
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Fig. 2. (a) SEM picture of UF microcapsules for calibrating diameter, scale bar: 50 μm, (b) SEM picture of UF microcapsules for calibrating wall thickness, scale bar: 50 μm.
Fig. 3. (a) Distribution of the diameter of UF microcapsules, (b) Distribution of thickness of UF microcapsules.
Fig. 4. Schematic diagram of preparing samples for nanoindentation tests.
displacement of the loaded point on the microcapsule, ν and E denote the Poisson’s ratio and the Young’s modulus of the microcapsule, respec tively; R represents the radius of the microcapsule. A microcapsule can be considered as a core-wall structure, in which the wall material is thought to be thin and homogeneous. To this end, thin shell theory can be used to investigate the mechanical behavior of the microcapsule under a small deformation [29]. In such a case, the theoretical solution of the shallow spherical shell under the action of a concentrated load was proposed by Reissner [40,41],
Table 2 Specific settings during nanoindentation tests. Temperature (� C)
Relative humidity (RH) (%)
Loading velocity (μm/s)
Penetration depth (μm)
Holding time (s)
Thermal drift (nm/s)
25
65
0.5
5 &10
100
0.05
determined by the model is given by pffiffiffi 4E R 32 F¼ d 3ð1 v2 Þ
4h2 E F ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d 3ð1 v2 ÞR
(3)
(4)
where F indicates the load exerted on the microcapsule, d denotes the displacement of the loaded point on the microcapsule, ν and E denote the
=
where F indicates the load exerted on the microcapsule, d denotes the 3
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Fig. 5. (a) Schematic diagram of the nanoindentation for compression and holding process (displacement-controlled), (b) the upper curve represents displacementtime for compression and holding process, and the lower curve represents force-time for compression and holding process.
Poisson’s ratio and the Young’s modulus of the microcapsule, respec tively, R and h represents the radius and the wall thickness of the microcapsule, respectively. 2.5. Viscoelastic models Organic polymer materials usually exhibit viscoelastic behavior. Accordingly, the viscoelastic models consist of springs and dashpots, arranged in parallel or/and in series [42] as shown in Fig. 7. Viscoelastic solution was derived using the Reissner equation since Hertz contact model is often applied to spherical solid sample [43]. To determine the viscoelastic response for the UF microcapsule, Young’s modulus E in Eq. (4) is transformed into G(t) in Eq. (5), and Eq. (5) is transformed into the general Boltzmann integration Eq. (6). Numerical solutions of Eq. (7) and Eq. (8) are obtained using numerical simulation method due to the lack of Boltzmann integration’s analytical solution 4ð1 þ vÞh2 d0 F ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi GðtÞ 3ð1 v2 ÞR Z
t
Gðt
uÞd 2 ðuÞdu
0
n X
GðtÞ ¼ C0 þ
3
=
4ð1 þ vÞh2 d0 F ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ð1 v2 ÞR
(5) (6)
Ci expð
t = τi Þ
(7)
Bi expð
t = τi Þ
(8)
i¼1 n X
FðtÞ ¼ B0 þ i¼1
where F indicates the concentrated load exerted on the microcapsule, ν denotes the Poisson’s ratio the microcapsule, R and h represents the radius and the wall thickness of the microcapsule, respectively, F(t) is force related to time t of the model, G(t) denotes the relaxation modulus, d0 signifies the holding displacement, τi is the relaxation time, B0 and C0 are the fitting coefficients. Specifically, the schematics of 3-parameter and 5-parameter Maxwell models are shown in Fig. 8(a) and (b), respectively.
Fig. 6. (a) Finite element model for nanoindentation with flat-ended indenter on the UF microcapsule, (b) Von Mises stress contour in microcapsule at the end of compression. 4
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Fig. 7. Schematic of spring unit and dashpot unit, and Maxwell model, obtained with the two basic unit cascade; Voigt model, obtained with the two basic units parallel.
Eqs. (9) and (10) describe a single microcapsule’s viscoelastic response by �� � � 4ð1 þ νÞh2 d0 t FðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G∞ þ G1 exp (9) τ1 3ð1 ν2 ÞR � � 4ð1 þ νÞh2 d0 FðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G∞ þ G1 exp 3ð1 ν2 ÞR
t
τ1
�
� þ G2 exp
t
τ2
�� (10)
where ν denotes the Poisson’s ratio of the microcapsule, R and h rep resents the radius and the wall thickness of the microcapsule, respec tively, F(t) is the force related to time t of the model, d0 signifies the holding displacement, τ1,τ2 are the relaxation time constants, G∞ is equivalent shear modulus under long-time, Gi is the shear modulus of i unit. Additionally, Burgers model is customary in describing the visco elastic behavior of the material [44], which is obtained by the combi nation of spring and dashpot in parallel for the relaxation representation. Consequently, the Burgers model can be represented by two paralleled Maxwell elements as shown in Fig. 9. And the basic equation for 4-parameter Burgers model is given by � �� � � � 4ð1 þ νÞh2 d0 t t FðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G1 exp þ G2 exp (11) τ1 τ2 3ð1 ν2 ÞR
Fig. 9. Schematic of Burgers model for the relaxation representation.
3. Results and discussion 3.1. Elastic property of UF microcapsules
where ν denotes the Poisson’s ratio the microcapsule, R and h represents the radius and the wall thickness of the microcapsule, respectively, F(t) is the force related to time t of the model, d0 signifies the holding displacement, τ1 and τ2 are the relaxation time constants, Gi is the shear modulus of i unit.
3.1.1. Determination of Young’s modulus based on shell theory model (Reissner’s solution) & Hertz contact model As mentioned in section 2.4, Hertz model has been successfully applied to study the Young’s modulus of single microspheres and mi crocapsules as a whole solid identity [45], although they have a core-shell structure [46]. Hertz model was also applied here to the indentation data of UF microcapsules. It is assumed that the microcap sule material was incompressible (ν ¼ 0.5). The assumptions of Hertz model were applied to all microcapsules, and the data were fitted with Hertz model as shown in Fig. 10. According to the experimental data (Black dot) in Fig. 10, the indentation of a single microcapsule, of which the diameter and thickness are 198.4 μm and 8.1 μm, respectively, to a final deformation of 5% illustrates the deformation was small. The load imposed on the microcapsule increases gradually as the depth of pene tration increases. When the displacement reached a value of 5 μm, the force value was 5.84 mN. Since the deformation is small and lower than 10% strain, Eq. (1) of Hertz contact model can be applied to the loading data to study the elastic behavior and to determine Young’s modulus of the microcapsule (As a solid sphere like a cell), and the Hertz fitting is normally presented in a nonlinear form as illustrated red solid line in Fig. 10. The red solid line is the closest fitting result of Hertz model and the correlation co efficient (R-square) is 0.9613. The Young’s modulus of this UF micro capsule was extracted from the slope of the linear line and was found to be 43.42 MPa. Additionally, it can be seen that the Hertz contact model agrees well with the experimental data, especially at the linear stage
Fig. 8. (a) Schematic of Maxwell model of 3 parameters; and (b) schematic of Maxwell model of 5 parameters. 5
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The modelled results are plotted in Fig. 12 to establish a correlation between Ec and Ew. If there were a proven connection between them, the developed equation could be used to predict one of the parameters when the other one is known [47]. In Fig. 12, a roughly linear relationship was found between Ec and Ew. The slope of the trend line was then correlated to the wall thickness t and diameter of each microcapsule Dm. The relationship between Ec and Ew may be described by Eq. (12): Ec ¼ ð0:039 � 0:003Þ
hm Ew Dm
(12)
where hm is the thickness of the microcapsules, Dm is the mean diameter of the microcapsules, Ec and Ew determined from Hertz model and Reissner model, respectively. In shell theory, when the mechanical behavior of the microcapsule using Reissner model, no bending effects were considered for the wall of microcapsule. This was because the wall thickness of microcapsule was significantly smaller than the diameter. It should be noted that this approach can generally be applied to a micro-particle whose ratio of β (initial wall thickness to radius) is less than 0.133 [43]. This is to say that Eq. (11) becomes valid when β � 0.133. The β for UF microcapsules studied here was equal to 0.04 � 0.002, which is smaller than 0.133. Additionally, there is a small difference between computing results 0.039 0.003 and experimental results 0.04 � 0.002, this is because that each single microcapsule’s diameter is not completely similar even though the thickness of each microcapsule, which is mainly depend on the synthesis time (2 h), is the same [10].
Fig. 10. Force-displacement relationship based on Hertz model (Red solid line), Reissner model (Blue solid line) and the experimental data (Black dot).
where the deformation is less than 4 μm. Therefore, the good fitting of Hertz contact model to the elastic behavior of UF microcapsules dem onstrates that Hertz contact model can be used to describe the elastic behavior of microcapsules when the deformation is lower than a diameter of 5%. It should be noted that, herein, the microcapsule was treated as a solid ball in the Hertz contact model. The Young’s modulus obtained is then for a smeared ball instead of that for the wall material of microcapsule. The mechanical behavior of spherical shells subjected to a concen trated force under a small deformation can be predicted by elastic shell theory developed by Reissner for determining the Young’s modulus of the microcapsule’s wall material (As a core-wall structure) [38], and the Reissner model fitting is normally presented in a linear form as illus trated blue solid line in Fig. 10. The linear line is the best fit of Reissner model and the correlation coefficient (R-square) is 0.9603. Therefore, the entire modulus of this UF microcapsule can be extracted from the curve and was found to be 690.77 MPa. Furthermore, it can be seen that the Reissner model agrees well with the experimental data, especially at the linear stage where the deformation is less than 4.4 μm. Therefore, the good fitting of Reissner model to the elastic behavior of UF micro capsules demonstrates that Reissner model can also be used to describe the elastic behavior of microcapsules when the deformation is lower than a diameter of 5%.
3.1.3. Comparison of FEM models with various thicknesses For a core-shell structure, the structural deformation occurs using a flat-ended indenter. Therefore, elastic shell theory is often applied to predict its mechanical behavior. Whereas, for a spherical solid particle (thickness equals radius), its deformation under the same condition is the material’s deformation, hence, contact theory is generally employed to study its mechanical properties. However, there is no obvious limit to which theoretical model (shell theory & contact theory) is required for materials with various wall thickness. In this study, the microcapsules with different thicknesses, 8 μm, 20 μm–60 μm with an interval of 10 μm, and 80 μm and 100 μm, were simulated using FEM. The simu lation details have been described in section 2.3. The FEM models are shown in Fig. 13. The comparison of the force-displacement curves for the indentation of a microcapsule with different thicknesses between the FEM results and the results from the Hertz contact model and Reissner model as shown in Fig. 14, and it can be seen that Hertz contact model is appropriate for predicting the microcapsule’s mechanical behavior
3.1.2. Comparison of young’s modulus determined from Hertz model & shell theory model As discussed above, Ec (Young’s modulus of a sphere equivalent to the UF microcapsule) obtained from Hertz contact model, and Ew (Young’s modulus of UF microcapsule wall material) determined from the elastic shell theory as shown in Fig. 11. Ec and Ew determined from both models were based on the data of the same 25 UF microcapsules in a given sample.
Fig. 11. (a) Schematic of a single microcapsule (a core-shell structure) filled with liquid healing agent, (b) schematic of a spherical solid particle.
Fig. 12. Relationship between Young’s modulus Ec predicted from Hertz model, and Ew from Reissner model. Red solid line is a linear trend line. 6
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Fig. 13. Schematic representations of the different thickness of UF microcapsules. Table 4 Comparison of FEM results and Reissner model result. Thickness (μm)
FEM fitting function from Fd curve
Reissner’s solution
Rsquare
8
F1 ¼ 1.15 � d
F0 1 ¼ 1.55 � d
0.896
F3 ¼ 8.583 � d
F0 3 ¼ 21.78 � d
0.429
F0 5 ¼ 60.52 � d
0.333
F0 7 ¼ 154.93 � d
0.188
20 30 40 50 60 80 100
when the thickness is greater than 60 μm. While, the shell theory is better for predicting the microcapsule’s mechanical behavior when the thickness is less than 8 μm (thin shell structure). In order to quantify the fitting efficiency of both models, the parameter, R-square, is used as the evaluating indicator given by Eq. (13) square ¼
Norm½ðF
meanðFi ÞÞ�2
Norm½ðFi
meanðFÞÞ�2
Table 3 Comparison of FEM results and Hertz contact model result. Hertz’s solution
Rsquare
8
F1 ¼ 5.249 � d1.5
F ¼ 163.815 � d1.5
0.031
20 30 40 50 60 80 100
F2 ¼ 26.88 � d1.5 F3 ¼ 55.25 � d1.5 F4 ¼ 88.69 � d1.5 F5 ¼ 124.7 � d1.5 F6 ¼ 155.9 � d1.5 F7 ¼ 160.3 � d1.5 F8 ¼ 165.8 � d1.5
F6 ¼ 23.96 � d F7 ¼ 28.57 � d F8 ¼ 29.26 � d
F0 4 ¼ 38.73 � d
0.368
F0 6 ¼ 87.15 � d
0.276
F0 8 ¼ 242.09 � d
0.125
3.2.1. Compression and holding behaviors of UF microcapsules with various penetration depth A UF microcapsule with diameter and thickness of 203.4 μm and 8.1 μm respectively was compressed to a final deformation of 2.5% at point A1 and B1 and held for 100 s as shown Fig. 15(a), (b). Another single UF microcapsule with diameter and thickness of 198.7 μm and 8.1 μm respectively was compressed to a final deformation of 5% at point A2 and B2 and held for 100 s as shown Fig. 16(a), (b). In these curves, the origin O indicates that the load probe starts to contact the sample; the OA stage represents the compression process of the micro capsule; and AB segment depicts the holding process (100 s) of the compressed microcapsule. Overall, the force exerted on the microcap sules ascended when they were compressed. It can be seen that there is a force relaxation at the final deformation 2.5% and 5%, respectively, which illustrate that the UF microcapsules possessed viscoelastic behavior up to these deformations. Specifically, the force is 0.433 mN under the final deformation 2.5%, and the force is reduced to be 0.417 mN (Reduction of 0.016 mN). For the other sample, the force is 2 mN under the final deformation 5%, and the force is reduced to be 1.94 mN (Reduction of 0.06 mN). In other words, with the increase of the penetration depth, there is a significant viscoelastic characteristic (force relaxation). Furthermore, other researchers have studied the viscoelastic property of other types of microcapsules. For example, Hu et al. [22] and Liu has studied the viscoelastic behavior of MF micro capsules [47], and found that there was hardly force relaxation for MF microcapsule under a small deformation, and there was little force relaxation for them under a relatively large deformation. It was probably because holding time was too short (Holding time was 6 s in these
where: R-square is coefficient of determination,Fi is the reaction force obtained from FEM,F is computing results based on both models, Norm is the 2-dimensional Euclidean norm. Comparison of the FEM fitting functions from force-displacement curves with theoretical results from the Hertz contact model and Reissner model are displayed in Table 3 and Table 4, respectively. Ac cording to Table 3, the R-square value becomes larger and close to 1 with the increase in the wall thickness, which illustrates that for a
FEM fitting function from Fd curve
F5 ¼ 19.22 � d
0.534
3.2. Viscoelastic characteristics of UF microcapsules
(13)
Thickness (μm)
F4 ¼ 13.98 � d
F0 2 ¼ 9.68 � d
microcapsule with a thick wall, Hertz contact model is more appropriate to predict the mechanical behavior of the microcapsule. Especially, when the thickness is greater than 60 μm, the R-square value is higher than 0.95. While in Table 4, as the wall thickness decreases, the R-square value becomes larger, and close to 1, which illustrates that for a microcapsule with a thin wall, elastic shell model is more appropriate to predict the mechanical behavior of the microcapsule. Especially, when the thickness is lower than 8 μm, which illustrates the microcapsule is a thin shell structure. At this moment, the R-square value is close to 0.9.
Fig. 14. Comparison of the force-displacement curves for the microcapsule with different thicknesses between the FEM results and the results from Hertz contact model and Reissner model.
R
F2 ¼ 4.190 � d
0.166 0.337 0.546 0.761 0.951 0.964 0.981
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Fig. 15. (a) Plotting of force versus time curve for compression and holding process; (b) plotting of force versus time curve for holding process (Compression to 5 μm).
Fig. 16. (a) Plotting of force versus time curve for compression and holding process; (b) plotting of force versus time curve for holding process (Compression to 10 μm).
Fig. 17. Plotting of experimental nanoindentation load-time relaxation data (R ¼ 125 μm) at peak displacement of 5 μm and fitting curves. Black dot is experimental data, and red line is fitting curve; (a) for 3-parameter Maxwell model; (b) for 5-parameter Maxwell model; (c) for Burgers model. 8
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papers).
Table 5 Initial inputs and computing results for 3-parameter Maxwell viscoelastic model (Penetration depth: 5 μm).
3.2.2. Determination of viscoelastic parameters of UF microcapsules For the compression and holding test (Penetration depth is 5 μm), the experimental data and computing results are shown in Fig. 17(a)-(c) and Fig. 18(a)-(c) for the three viscoelastic models, including 3-parameter Maxwell model, 5-parameter Maxwell model and 4-parameter Burgers model. It can be seen that for 3-parameter Maxwell model the fitting curve is almost coincident with the experimental data (R-square: 0.994) in Figs. 17(a) and Fig. 18(a). In addition, for 5-parameter Maxwell model and 4-parameter Burgers model the fitting curves agrees well with the experimental data (R-square: 0.999) in Fig. 17(b), (c) and Fig. 18(b), (c), which indicate that the models with more parameters have better fitting efficiency to predict the viscoelastic behavior of the UF microcapsules. The fitting parameters for the three viscoelastic models, including 3parameter Maxwell model, 5-parameter Maxwell model and 4-param eter Burgers model, are presented in Tables 5–10, respectively. In order to eliminate the effect of a local optimum, several amendments of the values of the initial variables were performed in this study. After wards, according to Tables 5–10 the results of the optimization were stable, which indicated that the suggested approach is robust. During the compression and holding test (Depth: 5 μm), for the relaxation time constant, τ1, the highest value is 6.08 s for 5-parameter Maxwell model, and the lowest value is 0.66 s for 4-parameter Burgers model, addi tionally, the value is 3.63 s for 3-parameter Maxwell model. For the relaxation time constant, τ2, the values are 0.34 s and 191.95 s for 5parameter Maxwell model and 4-parameter Burgers model, respec tively. The equivalent shear modulus, G∞, are almost similar, 20.8 kPa and 20.6 kPa, for these 2 Mx models. For the shear modulus, G1, the highest value is 1.46 kPa for 5-parameter Maxwell model, and the lowest value is 0.27 kPa for 4-parameter Burgers model, and the value is 0.9 kPa for 3-parameter Maxwell model. For the last parameter, G2, the value is 0.16 kPa for 5-parameter Maxwell model, and it is 20.88 kPa for
Viscoelastic parameters
Initial inputs
Computing results
Average value
G∞/(kPa) G1/(kPa) τ1/(s)
10.0, 15.0, 20.0 0.1, 0.5, 1.0 10.0, 20.0, 30.0
19.70, 20.30, 22.40 0.84, 0.91, 0.96 3.20, 3.69, 4.11
20.80 0.90 3.63
Table 6 Initial inputs and computing results for 5-parameter Maxwell viscoelastic model (Penetration depth: 5 μm). Viscoelastic parameters
Initial inputs
Computing results
Average value
G∞/(kPa) G1/(kPa) G2/(kPa) τ1/(s) τ2/(s)
1.0, 5.0, 10.0 1.0, 1.5, 2.0 0.1, 0.5, 1.0 10.0, 20.0, 30.0 10.0, 20.0, 30.0
19.31, 20.71, 21.79 1.11, 1.45, 1.64 0.15, 0.17, 0.17 5.91, 6.02, 6.25 0.29, 0.41, 0.33
20.60 1.46 0.16 6.08 0.34
Table 7 Initial inputs and computing results for 4-parameter Burgers viscoelastic model (Penetration depth: 5 μm). Viscoelastic parameters
Initial inputs
Computing results
Average value
G1/(kPa) G2/(kPa) τ1/(s) τ2/(s)
0.1, 0.5, 1.0 0.1, 0.5, 1.0 1.0, 2.0, 3.0 10.0, 30.0, 50.0
0.24, 0.31, 0.26 20.51, 21.24, 20.78 0.63, 0.71, 0.69 193.32, 186.75, 195.61
0.27 20.88 0.66 191.95
Fig. 18. Plotting of experimental nanoindentation load-time relaxation data (R ¼ 120 μm) at peak displacement of 10 μm and fitting curves. Black dot is experimental data, and red line is fitting curve; (a) for 3-parameter Maxwell model; (b) for 5-parameter Maxwell model; (c) for Burgers model. 9
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4. Conclusions
Table 8 Initial inputs and computing results for 3-parameter Maxwell viscoelastic model (Penetration depth: 10 μm). Viscoelastic parameters
Initial inputs
Computing results
Average value
G∞/(kPa) G1/(kPa) τ1/(s)
10.0, 30.0, 50.0 1.0, 5.0, 10.0 1.0, 5.0, 10.0
91.96, 88.63, 93.22 3.11, 3.35, 3.06 3.20, 3.59, 2.91
91.27 3.17 3.23
The paper presented micro-scale compression and holding tests to study the viscoelastic characteristics of UF microcapsules, including determination of Young’s modulus and viscoelastic parameters of UF microcapsules. The following conclusions can be drawn: (1) The preparation and characterization of UF microcapsules were performed. Furthermore, they were indented with a flat-ended indenter for the compression and holding tests, and SEM obser vation was conducted as well. (2) Force-displacement curve under a small deformation was wellfitted by Hertz model and Reissner model for determining Young’s modulus of UF microcapsules. The comparison of Ec obtained from Hertz model (a whole UF microcapsule) and Ew determined from the elastic shell theory (a core-wall structure) shows a nearly linear correlation. (3) In the FEM analysis, the effect of element types (CAX4R and SAX1), i.e. solid element and shell element, as well as the sensi tivity of Poisson’s ratio, were analysed. It was obtained that the element type had no remarkable effect. Meanwhile, the me chanical response of UF microcapsules had a low sensitivity to Poisson’s ratio and element number. The finite-element models with various thicknesses were studied, and the authors found a range regarding the application of Hertz model or shell theory. (4) The tests show that the UF microcapsules have obvious visco elastic characteristics, especially under a larger deformation. The comparison of the experimental results with the computing re sults from 3-parameter Maxwell model, 5-parameter Maxwell model and 4-parameter Burgers model shows that each of them agrees well with the experimental data. The 3-parameter model can provide enough accuracy (R-square: 0.994) for both ramping and stress relaxation periods. The stability of the parameters was verified. Utilization of the models may enrich the experimental nano-micro material mechanics in interpretation of nano indentation of microcapsules.
Table 9 Initial inputs and computing results for 5-parameter Maxwell viscoelastic model (Penetration depth: 10 μm). Viscoelastic parameters
Initial inputs
Computing results
Average value
G∞/(kPa)
10.0, 50.0, 100.0 1.0, 1.5, 2.0 1.0, 5.0, 10.0 0.1, 0.5, 1.0 10.0, 20.0, 30.0
90.51, 89.60, 91.23 0.67, 0.66, 0.71 4.24, 4.16, 4.53 0.27, 0.29, 0.26 6.82, 7.17, 7.01
89.43
G1/(kPa) G2/(kPa) τ1/(s) τ2/(s)
0.68 4.31 0.27 10.51
Table 10 Initial inputs and computing results for 4-parameter Burgers viscoelastic model (Penetration depth: 10 μm). Viscoelastic parameters
Initial inputs
Computing results
Average value
G1/(kPa) G2/(kPa)
0.1, 0.5, 1.0 10.0, 50.0, 100.0 0.1, 0.5, 1.0 30.0, 50.0, 100.0
0.91, 0.88, 0.91 94.3, 91.5, 95.6
0.90 93.80
0.52, 0.48, 0.53 237.64, 246.49, 239.71
0.51 241.28
τ1/(s) τ2/(s)
4-parameter Burgers model. Since the UF microcapsules have a strong viscoelastic behavior under a large deformation, therefore, these viscoelastic parameters during the compression and holding test (Depth: 10 μm) are generally greater than the above mentioned. Specifically, for the relaxation time constant, τ1, the highest value is 3.23 s for 3-parameter Maxwell model, and the lowest value is 0.27 s for 3-parameter Maxwell model, additionally, the value is 0.51 s for 4-parameter Burgers model. For the relaxation time constant, τ2, the values are 10.51 s and 241.28 s for 5-parameter Maxwell model and 4-parameter Burgers model, respectively. The equivalent shear modulus, G∞, are almost similar, 91.27 kPa and 89.43 kPa, for these 2 Mx models. For the shear modulus, G1, the highest value is 3.17 kPa for 3-parameter Maxwell model, and the lowest value is 0.68 kPa for 5-parameter Maxwell model, and the value is 0.9 kPa for 4-parameter Burgers model. For the last parameter, G2, the value is 4.31 kPa for 5-parameter Maxwell model, and it is 93.8 kPa for 4-param eter Burgers model. The microcapsules composed by urea and formaldehyde under catalysis show an obvious viscoelastic behavior, as Shaw and MacKnight indicated when the organic polymer was subjected to various external forces, there was almost no exception to the viscoelastic response [48, 49]. In this work, UF microcapsules are applied for self-healing cementitious composites. Therefore, detailed study of the mechanical characteristics of the microcapsules, such as determination of elastic-plastic [39], and viscoelastic parameters, should be beneficial for uncovering the rupturing mechanism of the microcapsules embedded in the cementitious materials. Furthermore, the development would have wider application, including the determination of mechanical parame ters of a variety of microspheres, microcapsules and cells.
Data availability The raw/processed data required to reproduce these findings are available to download from https://doi.org/10.7910/DVN/WVESCH. Declaration of competing interest The authors declare that there is no conflict of interest. Acknowledgements The authors gratefully acknowledge the financial support provided by the General Program of the National Natural Science Foundation of China (No. 51978409), the International Cooperation and Exchange of the National Natural Science Foundation of China (51120185002), the Science and Technology Foundation for the Basic Research Plan of Shenzhen City (JCYJ20160422095146121). Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.polymertesting.2019.106146. References [1] A. Gray, S. Egan, S. Bakalis, Z. Zhang, Determination of microcapsule physicochemical, structural, and mechanical properties, Particuology 24 (2015) 32–43. [2] H. Jong, J. Bonner, Phosphatide auto-complex coacervates as ionic systems and their relation to the protoplasmic membrane, Protoplasma 24 (1935) 198–218.
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Material Properties
Investigation on viscoelastic properties of urea-formaldehyde microcapsules by using nanoindentation Rui Han a, Xianfeng Wang a, *, Guangming Zhu b, Ningxu Han a, Feng Xing a a
College of Civil and Transportation Engineering, Guangdong Provincial Key Laboratory of Durability for Marine Civil Engineering, Shenzhen University, Shenzhen, 518060, PR China b College of Materials Science and Engineering, Shenzhen University, Shenzhen, 518060, PR China
A R T I C L E I N F O
A B S T R A C T
Keywords: Microcapsule Viscoelastic Maxwell model Burgers model Finite element method (FEM) Nanoindentation
The viscoelastic properties of urea-formaldehyde (UF) microcapsules is very important for achieving self-healing process in the field of concrete. In this study, the load-displacement relationship of the microcapsule was ob tained by micromanipulation technique and the variation of force with time for compression and holding pro cess, from which the properties of elasticity and viscoelasticity were determined. Three viscoelastic models, including 3-parameter Maxwell model, 5-parameter Maxwell model and 4-parameter Burgers model based on shell theory, have been applied to study the viscoelastic properties of the microcapsules. This study has demonstrated that these models consistently predict viscoelastic properties for both ramping and stress relaxa tion periods. Utilization of the models can enrich the experimental nano-micro material mechanics in inter pretation of nanoindentation of microcapsules.
1. Introduction A microcapsule can be regarded as a core-wall composite structure. In which one material (core) is entirely microencapsulated by another (shell). In general, liquid, powder and solid can be used for core mate rial, while the shell is solid (polymeric or inorganic wall). The different utilizations of microcapsules mainly rely on the types of core materials and various trigger patterns [1]. The pioneering work started from Jong’s research in the late 1930s [2]. Afterwards, especially in recent years, inventing different kinds of functional microcapsules has attrac ted more and more researchers’ interest, and they have been extensively applied to improve the quality of foods [3,4], pharmaceutic [5], printing industry [6], cosmetics industry [7], as well as civil engineering etc. [8–11]. In this paper, urea-formaldehyde (UF) microcapsules were applied in healing microcracks in materials [12,13]. The repair process can be achieved by rupture of microcapsules’ shell as the generation and propagation of cracks. The application of microcapsules as vehicles for delivering agents in time at the right place [14–16]. Consequently, it is important how to trigger the burst of microcapsules. Mechanical trigger (External or enteral loads) is common among many triggering mechanisms, hence study of the mechanical properties of microcapsules is important for
determining whether or not triggering the rupture of microcapsules. In simple terms, the microcapsules must be mechanically stable and intact during the fabrication and storage and have optimal mechanical char acteristics if the successful release of the core materials need to be triggered under external/internal loads. Meanwhile, study of mechani cal characteristics of micro-nano particles becomes accessible with the development of precision instruments. There exist two methods, bulk and individual approaches, to study the mechanical characteristics of microcapsules. When it comes to the bulk methods, researchers only obtain the average mechanical parameters, such as Young’s modulus as well as shear modulus [17,18], therefore, these methods have not been used extensively now. In the following, individual micro-nano particle measurement equipment, such as atomic force microscope (AFM) and micro-nano indentation (micro-nano manipulation) will be presented. Whereas, among these advanced apparatus, electromagnetic tweezers and/or micropipette aspiration are generally used for soft samples [19]. Firstly, Cole studied the arbacia egg’s (diameter: 75 μm) mechanical behavior using a plate compression test in the 1930s [20]. After decades of development, precise compression of a single micro-nano particle was introduced by Zhang et al. [21]. The mechanical properties of a large number of samples, such as microcapsules, cells and beads, were
* Corresponding author. E-mail address: [email protected] (X. Wang). https://doi.org/10.1016/j.polymertesting.2019.106146 Received 18 July 2019; Received in revised form 14 September 2019; Accepted 8 October 2019 Available online 11 October 2019 0142-9418/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Rui Han, Polymer Testing, https://doi.org/10.1016/j.polymertesting.2019.106146
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determined by using micromanipulation. Specifically, the mechanical properties, including elasticity, plastic-elasticity and viscoelasticity of melamine formaldehyde (MF) microcapsules have been studied [22]. As far as we know, the research of mechanical properties of MF micro capsule have been comparatively comprehensive and systematic. The diameter, thickness, and composition of microcapsules directly result in the mechanical behavior. Even though, elastic-plastic properties of UF microcapsules has been estimated by Keller and Sottos [23] Ahangari et al. [24]. Ma et al. [25]. Mercade-Prieto et al. [26–28]. Meanwhile, numerical method combined with realistic experiment is an effective means in studying a variety of physical phenomena, in which the finite element method (FEM), as a numerical method for solving the mathe matical physics problems, has been widely used in various fields such as simulation of mechanical behavior of materials [29–31], fluid flow [32], optical communication [33,34], as well as sensor systems [35–37]. Nanoindentation techniques have been proven to be effective to char acterize the mechanical properties of nano-micro materials. However, most studies focused on the elastic-plastic properties of microcapsules, in other words, viscoelastic properties of a single UF microcapsule have not been comprehensively studied yet. In the present study, UF microcapsules are synthesized by an in-situ polymerization method and are characterized (diameter and thickness distribution) by scanning electron microscopy (SEM). The indentation on the microcapsules to a small deformation was completed, its elastic parameter (Young’s modulus) was determined by contact and shell theory. Moreover, according to the compression and holding tests, viscoelastic parameters were obtained for the Maxwell and Burgers models based on shell theory.
Table 1 Parameters and the specific settings in preparing UF microcapsules. Synthesis time (hour)
Rotational velocity (r/min)
Average diameter (μm)
Wall thickness (μm)
Core content (%)
2
200
198.4
8.1
78
it was placed on the test stand as shown in Fig. 4. G200 type Nano indenter (Agilent Inc., USA) with a flat end was applied in this study to obtain force and displacement curves. The creep and relaxation response of a material can be applied for determining its viscoelastic properties. In this study, the displacementcontrolled method was employed, and the advantage of using this approach is that the indenter displacement can be easily controlled and modified in experimental investigations. The indentation-relaxation tests were performed to the microcapsules at two different displace ment levels (dmax ¼ 5 μm and 10 μm) at a fixed rise time (tR) of 150 s for both displacement levels and with a 100s hold period at peak depth. The controlling parameters for the nanoindentation can be found in Table 2. A compression and holding procedure (displacement-controlled) is shown in Fig. 5. Firstly, a constant loading speed is performed in the indenter, and the reaction force increases gradually as the indenter displacement gets close to the sample growingly. And then, the indenter would be in position when the penetration depth reached pre-setting maximum value. Meanwhile, the reaction force measured would decrease slowly as time goes on due to the material’s viscosity.
2. Materials and methods
dðtÞ ¼ kt; 0 � t � tR
(1)
dðtÞ ¼ d0 ; tR � t � tτ
(2)
where d(t) is the penetration depth related to time t of the model, d0 is the penetration depth under holding procedure, k is the loading speed, tR is the staring time of holding procedure, tτ is the ending time of holding procedure.
2.1. UF microcapsules UF microcapsules with core materials being epoxy resin E 51 and butyl glycidyl ether were prepared using an in-situ polymerization method in this study, as shown in Fig. 1. The preparation details can be referred to Ref. [11]. Specific parameters of UF microcapsules can be found in Table 1. Fig. 2(a) and Fig. 2(b) show the microcapsules for calibrating the diameter and wall thickness, respectively. These geo metric parameters were characterized using SEM (Quanta TM 250 FEG) for counting the 300 microcapsules, and Fig. 3(a), (b) show the diameter distribution and the wall thickness distribution of the UF microcapsules, respectively. The mean diameter and wall thickness of the UF micro capsules are shown in Table 1.
2.3. Finite element model The nanoindentation test with flat-ended indenter was simulated using finite element method (FEM) by Abaqus 6.14, as shown in Fig. 6(a) and (b). A single UF microcapsule was modelled as a typical spherical shell structure (diameter ¼ 198.4 μm, wall thickness ¼ 8.1 μm) using 2D axisymmetric element, including CAX4R (4-node bilinear, reduced integration with hourglass control) and SAX1 (2-node axisymmetric shell element), and the comparison using these two elements was shown in appendix A. supplementary data. And the sample was simulated by an isotropic and homogenous linear elastic material due to its small deformation. In addition, the flat-ended indenter and the substrate were simulated by rigid elements. Friction was not considered in the tangential direction, and the hard contact model was applied in the normal direction. The spherical model was compressed using a displacement-control of the indenter to a depth of 5 μm. The bottom of the microcapsule and substrate was fully fixed. The variation of the material Poisson’s ratios from 0.1 to 0.5 and the adaptivity of the finite element mesh were considered to investigate their influence on the nanoindentation behavior of the microcapsule in Appendix A supple mentary data. The effect of the core liquid or air on the behavior of the microcapsule was not significant based on our previous research [29].
2.2. Preparation of specimen and nanoindentation Several UF microcapsules were dispersed on the top of 1 cm � 1 cm glass piece attached by epoxy resin and kept it 24 h to cure. Eventually,
2.4. Elastic models The Hertz model has been used to study the contact behavior of elastic spherical particles, especially when the strain is small, the sur faces are continuous and the contact is frictionless [38]. In this case, the classical Hertz contact model can be applied in determining the Young’s modulus [39]. The force-displacement (F-d) relation for ramping period
Fig. 1. Schematic diagram of preparing UF microcapsules. 2
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Fig. 2. (a) SEM picture of UF microcapsules for calibrating diameter, scale bar: 50 μm, (b) SEM picture of UF microcapsules for calibrating wall thickness, scale bar: 50 μm.
Fig. 3. (a) Distribution of the diameter of UF microcapsules, (b) Distribution of thickness of UF microcapsules.
Fig. 4. Schematic diagram of preparing samples for nanoindentation tests.
displacement of the loaded point on the microcapsule, ν and E denote the Poisson’s ratio and the Young’s modulus of the microcapsule, respec tively; R represents the radius of the microcapsule. A microcapsule can be considered as a core-wall structure, in which the wall material is thought to be thin and homogeneous. To this end, thin shell theory can be used to investigate the mechanical behavior of the microcapsule under a small deformation [29]. In such a case, the theoretical solution of the shallow spherical shell under the action of a concentrated load was proposed by Reissner [40,41],
Table 2 Specific settings during nanoindentation tests. Temperature (� C)
Relative humidity (RH) (%)
Loading velocity (μm/s)
Penetration depth (μm)
Holding time (s)
Thermal drift (nm/s)
25
65
0.5
5 &10
100
0.05
determined by the model is given by pffiffiffi 4E R 32 F¼ d 3ð1 v2 Þ
4h2 E F ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d 3ð1 v2 ÞR
(3)
(4)
where F indicates the load exerted on the microcapsule, d denotes the displacement of the loaded point on the microcapsule, ν and E denote the
=
where F indicates the load exerted on the microcapsule, d denotes the 3
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Fig. 5. (a) Schematic diagram of the nanoindentation for compression and holding process (displacement-controlled), (b) the upper curve represents displacementtime for compression and holding process, and the lower curve represents force-time for compression and holding process.
Poisson’s ratio and the Young’s modulus of the microcapsule, respec tively, R and h represents the radius and the wall thickness of the microcapsule, respectively. 2.5. Viscoelastic models Organic polymer materials usually exhibit viscoelastic behavior. Accordingly, the viscoelastic models consist of springs and dashpots, arranged in parallel or/and in series [42] as shown in Fig. 7. Viscoelastic solution was derived using the Reissner equation since Hertz contact model is often applied to spherical solid sample [43]. To determine the viscoelastic response for the UF microcapsule, Young’s modulus E in Eq. (4) is transformed into G(t) in Eq. (5), and Eq. (5) is transformed into the general Boltzmann integration Eq. (6). Numerical solutions of Eq. (7) and Eq. (8) are obtained using numerical simulation method due to the lack of Boltzmann integration’s analytical solution 4ð1 þ vÞh2 d0 F ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi GðtÞ 3ð1 v2 ÞR Z
t
Gðt
uÞd 2 ðuÞdu
0
n X
GðtÞ ¼ C0 þ
3
=
4ð1 þ vÞh2 d0 F ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ð1 v2 ÞR
(5) (6)
Ci expð
t = τi Þ
(7)
Bi expð
t = τi Þ
(8)
i¼1 n X
FðtÞ ¼ B0 þ i¼1
where F indicates the concentrated load exerted on the microcapsule, ν denotes the Poisson’s ratio the microcapsule, R and h represents the radius and the wall thickness of the microcapsule, respectively, F(t) is force related to time t of the model, G(t) denotes the relaxation modulus, d0 signifies the holding displacement, τi is the relaxation time, B0 and C0 are the fitting coefficients. Specifically, the schematics of 3-parameter and 5-parameter Maxwell models are shown in Fig. 8(a) and (b), respectively.
Fig. 6. (a) Finite element model for nanoindentation with flat-ended indenter on the UF microcapsule, (b) Von Mises stress contour in microcapsule at the end of compression. 4
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Fig. 7. Schematic of spring unit and dashpot unit, and Maxwell model, obtained with the two basic unit cascade; Voigt model, obtained with the two basic units parallel.
Eqs. (9) and (10) describe a single microcapsule’s viscoelastic response by �� � � 4ð1 þ νÞh2 d0 t FðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G∞ þ G1 exp (9) τ1 3ð1 ν2 ÞR � � 4ð1 þ νÞh2 d0 FðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G∞ þ G1 exp 3ð1 ν2 ÞR
t
τ1
�
� þ G2 exp
t
τ2
�� (10)
where ν denotes the Poisson’s ratio of the microcapsule, R and h rep resents the radius and the wall thickness of the microcapsule, respec tively, F(t) is the force related to time t of the model, d0 signifies the holding displacement, τ1,τ2 are the relaxation time constants, G∞ is equivalent shear modulus under long-time, Gi is the shear modulus of i unit. Additionally, Burgers model is customary in describing the visco elastic behavior of the material [44], which is obtained by the combi nation of spring and dashpot in parallel for the relaxation representation. Consequently, the Burgers model can be represented by two paralleled Maxwell elements as shown in Fig. 9. And the basic equation for 4-parameter Burgers model is given by � �� � � � 4ð1 þ νÞh2 d0 t t FðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G1 exp þ G2 exp (11) τ1 τ2 3ð1 ν2 ÞR
Fig. 9. Schematic of Burgers model for the relaxation representation.
3. Results and discussion 3.1. Elastic property of UF microcapsules
where ν denotes the Poisson’s ratio the microcapsule, R and h represents the radius and the wall thickness of the microcapsule, respectively, F(t) is the force related to time t of the model, d0 signifies the holding displacement, τ1 and τ2 are the relaxation time constants, Gi is the shear modulus of i unit.
3.1.1. Determination of Young’s modulus based on shell theory model (Reissner’s solution) & Hertz contact model As mentioned in section 2.4, Hertz model has been successfully applied to study the Young’s modulus of single microspheres and mi crocapsules as a whole solid identity [45], although they have a core-shell structure [46]. Hertz model was also applied here to the indentation data of UF microcapsules. It is assumed that the microcap sule material was incompressible (ν ¼ 0.5). The assumptions of Hertz model were applied to all microcapsules, and the data were fitted with Hertz model as shown in Fig. 10. According to the experimental data (Black dot) in Fig. 10, the indentation of a single microcapsule, of which the diameter and thickness are 198.4 μm and 8.1 μm, respectively, to a final deformation of 5% illustrates the deformation was small. The load imposed on the microcapsule increases gradually as the depth of pene tration increases. When the displacement reached a value of 5 μm, the force value was 5.84 mN. Since the deformation is small and lower than 10% strain, Eq. (1) of Hertz contact model can be applied to the loading data to study the elastic behavior and to determine Young’s modulus of the microcapsule (As a solid sphere like a cell), and the Hertz fitting is normally presented in a nonlinear form as illustrated red solid line in Fig. 10. The red solid line is the closest fitting result of Hertz model and the correlation co efficient (R-square) is 0.9613. The Young’s modulus of this UF micro capsule was extracted from the slope of the linear line and was found to be 43.42 MPa. Additionally, it can be seen that the Hertz contact model agrees well with the experimental data, especially at the linear stage
Fig. 8. (a) Schematic of Maxwell model of 3 parameters; and (b) schematic of Maxwell model of 5 parameters. 5
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The modelled results are plotted in Fig. 12 to establish a correlation between Ec and Ew. If there were a proven connection between them, the developed equation could be used to predict one of the parameters when the other one is known [47]. In Fig. 12, a roughly linear relationship was found between Ec and Ew. The slope of the trend line was then correlated to the wall thickness t and diameter of each microcapsule Dm. The relationship between Ec and Ew may be described by Eq. (12): Ec ¼ ð0:039 � 0:003Þ
hm Ew Dm
(12)
where hm is the thickness of the microcapsules, Dm is the mean diameter of the microcapsules, Ec and Ew determined from Hertz model and Reissner model, respectively. In shell theory, when the mechanical behavior of the microcapsule using Reissner model, no bending effects were considered for the wall of microcapsule. This was because the wall thickness of microcapsule was significantly smaller than the diameter. It should be noted that this approach can generally be applied to a micro-particle whose ratio of β (initial wall thickness to radius) is less than 0.133 [43]. This is to say that Eq. (11) becomes valid when β � 0.133. The β for UF microcapsules studied here was equal to 0.04 � 0.002, which is smaller than 0.133. Additionally, there is a small difference between computing results 0.039 0.003 and experimental results 0.04 � 0.002, this is because that each single microcapsule’s diameter is not completely similar even though the thickness of each microcapsule, which is mainly depend on the synthesis time (2 h), is the same [10].
Fig. 10. Force-displacement relationship based on Hertz model (Red solid line), Reissner model (Blue solid line) and the experimental data (Black dot).
where the deformation is less than 4 μm. Therefore, the good fitting of Hertz contact model to the elastic behavior of UF microcapsules dem onstrates that Hertz contact model can be used to describe the elastic behavior of microcapsules when the deformation is lower than a diameter of 5%. It should be noted that, herein, the microcapsule was treated as a solid ball in the Hertz contact model. The Young’s modulus obtained is then for a smeared ball instead of that for the wall material of microcapsule. The mechanical behavior of spherical shells subjected to a concen trated force under a small deformation can be predicted by elastic shell theory developed by Reissner for determining the Young’s modulus of the microcapsule’s wall material (As a core-wall structure) [38], and the Reissner model fitting is normally presented in a linear form as illus trated blue solid line in Fig. 10. The linear line is the best fit of Reissner model and the correlation coefficient (R-square) is 0.9603. Therefore, the entire modulus of this UF microcapsule can be extracted from the curve and was found to be 690.77 MPa. Furthermore, it can be seen that the Reissner model agrees well with the experimental data, especially at the linear stage where the deformation is less than 4.4 μm. Therefore, the good fitting of Reissner model to the elastic behavior of UF micro capsules demonstrates that Reissner model can also be used to describe the elastic behavior of microcapsules when the deformation is lower than a diameter of 5%.
3.1.3. Comparison of FEM models with various thicknesses For a core-shell structure, the structural deformation occurs using a flat-ended indenter. Therefore, elastic shell theory is often applied to predict its mechanical behavior. Whereas, for a spherical solid particle (thickness equals radius), its deformation under the same condition is the material’s deformation, hence, contact theory is generally employed to study its mechanical properties. However, there is no obvious limit to which theoretical model (shell theory & contact theory) is required for materials with various wall thickness. In this study, the microcapsules with different thicknesses, 8 μm, 20 μm–60 μm with an interval of 10 μm, and 80 μm and 100 μm, were simulated using FEM. The simu lation details have been described in section 2.3. The FEM models are shown in Fig. 13. The comparison of the force-displacement curves for the indentation of a microcapsule with different thicknesses between the FEM results and the results from the Hertz contact model and Reissner model as shown in Fig. 14, and it can be seen that Hertz contact model is appropriate for predicting the microcapsule’s mechanical behavior
3.1.2. Comparison of young’s modulus determined from Hertz model & shell theory model As discussed above, Ec (Young’s modulus of a sphere equivalent to the UF microcapsule) obtained from Hertz contact model, and Ew (Young’s modulus of UF microcapsule wall material) determined from the elastic shell theory as shown in Fig. 11. Ec and Ew determined from both models were based on the data of the same 25 UF microcapsules in a given sample.
Fig. 11. (a) Schematic of a single microcapsule (a core-shell structure) filled with liquid healing agent, (b) schematic of a spherical solid particle.
Fig. 12. Relationship between Young’s modulus Ec predicted from Hertz model, and Ew from Reissner model. Red solid line is a linear trend line. 6
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Fig. 13. Schematic representations of the different thickness of UF microcapsules. Table 4 Comparison of FEM results and Reissner model result. Thickness (μm)
FEM fitting function from Fd curve
Reissner’s solution
Rsquare
8
F1 ¼ 1.15 � d
F0 1 ¼ 1.55 � d
0.896
F3 ¼ 8.583 � d
F0 3 ¼ 21.78 � d
0.429
F0 5 ¼ 60.52 � d
0.333
F0 7 ¼ 154.93 � d
0.188
20 30 40 50 60 80 100
when the thickness is greater than 60 μm. While, the shell theory is better for predicting the microcapsule’s mechanical behavior when the thickness is less than 8 μm (thin shell structure). In order to quantify the fitting efficiency of both models, the parameter, R-square, is used as the evaluating indicator given by Eq. (13) square ¼
Norm½ðF
meanðFi ÞÞ�2
Norm½ðFi
meanðFÞÞ�2
Table 3 Comparison of FEM results and Hertz contact model result. Hertz’s solution
Rsquare
8
F1 ¼ 5.249 � d1.5
F ¼ 163.815 � d1.5
0.031
20 30 40 50 60 80 100
F2 ¼ 26.88 � d1.5 F3 ¼ 55.25 � d1.5 F4 ¼ 88.69 � d1.5 F5 ¼ 124.7 � d1.5 F6 ¼ 155.9 � d1.5 F7 ¼ 160.3 � d1.5 F8 ¼ 165.8 � d1.5
F6 ¼ 23.96 � d F7 ¼ 28.57 � d F8 ¼ 29.26 � d
F0 4 ¼ 38.73 � d
0.368
F0 6 ¼ 87.15 � d
0.276
F0 8 ¼ 242.09 � d
0.125
3.2.1. Compression and holding behaviors of UF microcapsules with various penetration depth A UF microcapsule with diameter and thickness of 203.4 μm and 8.1 μm respectively was compressed to a final deformation of 2.5% at point A1 and B1 and held for 100 s as shown Fig. 15(a), (b). Another single UF microcapsule with diameter and thickness of 198.7 μm and 8.1 μm respectively was compressed to a final deformation of 5% at point A2 and B2 and held for 100 s as shown Fig. 16(a), (b). In these curves, the origin O indicates that the load probe starts to contact the sample; the OA stage represents the compression process of the micro capsule; and AB segment depicts the holding process (100 s) of the compressed microcapsule. Overall, the force exerted on the microcap sules ascended when they were compressed. It can be seen that there is a force relaxation at the final deformation 2.5% and 5%, respectively, which illustrate that the UF microcapsules possessed viscoelastic behavior up to these deformations. Specifically, the force is 0.433 mN under the final deformation 2.5%, and the force is reduced to be 0.417 mN (Reduction of 0.016 mN). For the other sample, the force is 2 mN under the final deformation 5%, and the force is reduced to be 1.94 mN (Reduction of 0.06 mN). In other words, with the increase of the penetration depth, there is a significant viscoelastic characteristic (force relaxation). Furthermore, other researchers have studied the viscoelastic property of other types of microcapsules. For example, Hu et al. [22] and Liu has studied the viscoelastic behavior of MF micro capsules [47], and found that there was hardly force relaxation for MF microcapsule under a small deformation, and there was little force relaxation for them under a relatively large deformation. It was probably because holding time was too short (Holding time was 6 s in these
where: R-square is coefficient of determination,Fi is the reaction force obtained from FEM,F is computing results based on both models, Norm is the 2-dimensional Euclidean norm. Comparison of the FEM fitting functions from force-displacement curves with theoretical results from the Hertz contact model and Reissner model are displayed in Table 3 and Table 4, respectively. Ac cording to Table 3, the R-square value becomes larger and close to 1 with the increase in the wall thickness, which illustrates that for a
FEM fitting function from Fd curve
F5 ¼ 19.22 � d
0.534
3.2. Viscoelastic characteristics of UF microcapsules
(13)
Thickness (μm)
F4 ¼ 13.98 � d
F0 2 ¼ 9.68 � d
microcapsule with a thick wall, Hertz contact model is more appropriate to predict the mechanical behavior of the microcapsule. Especially, when the thickness is greater than 60 μm, the R-square value is higher than 0.95. While in Table 4, as the wall thickness decreases, the R-square value becomes larger, and close to 1, which illustrates that for a microcapsule with a thin wall, elastic shell model is more appropriate to predict the mechanical behavior of the microcapsule. Especially, when the thickness is lower than 8 μm, which illustrates the microcapsule is a thin shell structure. At this moment, the R-square value is close to 0.9.
Fig. 14. Comparison of the force-displacement curves for the microcapsule with different thicknesses between the FEM results and the results from Hertz contact model and Reissner model.
R
F2 ¼ 4.190 � d
0.166 0.337 0.546 0.761 0.951 0.964 0.981
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Fig. 15. (a) Plotting of force versus time curve for compression and holding process; (b) plotting of force versus time curve for holding process (Compression to 5 μm).
Fig. 16. (a) Plotting of force versus time curve for compression and holding process; (b) plotting of force versus time curve for holding process (Compression to 10 μm).
Fig. 17. Plotting of experimental nanoindentation load-time relaxation data (R ¼ 125 μm) at peak displacement of 5 μm and fitting curves. Black dot is experimental data, and red line is fitting curve; (a) for 3-parameter Maxwell model; (b) for 5-parameter Maxwell model; (c) for Burgers model. 8
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papers).
Table 5 Initial inputs and computing results for 3-parameter Maxwell viscoelastic model (Penetration depth: 5 μm).
3.2.2. Determination of viscoelastic parameters of UF microcapsules For the compression and holding test (Penetration depth is 5 μm), the experimental data and computing results are shown in Fig. 17(a)-(c) and Fig. 18(a)-(c) for the three viscoelastic models, including 3-parameter Maxwell model, 5-parameter Maxwell model and 4-parameter Burgers model. It can be seen that for 3-parameter Maxwell model the fitting curve is almost coincident with the experimental data (R-square: 0.994) in Figs. 17(a) and Fig. 18(a). In addition, for 5-parameter Maxwell model and 4-parameter Burgers model the fitting curves agrees well with the experimental data (R-square: 0.999) in Fig. 17(b), (c) and Fig. 18(b), (c), which indicate that the models with more parameters have better fitting efficiency to predict the viscoelastic behavior of the UF microcapsules. The fitting parameters for the three viscoelastic models, including 3parameter Maxwell model, 5-parameter Maxwell model and 4-param eter Burgers model, are presented in Tables 5–10, respectively. In order to eliminate the effect of a local optimum, several amendments of the values of the initial variables were performed in this study. After wards, according to Tables 5–10 the results of the optimization were stable, which indicated that the suggested approach is robust. During the compression and holding test (Depth: 5 μm), for the relaxation time constant, τ1, the highest value is 6.08 s for 5-parameter Maxwell model, and the lowest value is 0.66 s for 4-parameter Burgers model, addi tionally, the value is 3.63 s for 3-parameter Maxwell model. For the relaxation time constant, τ2, the values are 0.34 s and 191.95 s for 5parameter Maxwell model and 4-parameter Burgers model, respec tively. The equivalent shear modulus, G∞, are almost similar, 20.8 kPa and 20.6 kPa, for these 2 Mx models. For the shear modulus, G1, the highest value is 1.46 kPa for 5-parameter Maxwell model, and the lowest value is 0.27 kPa for 4-parameter Burgers model, and the value is 0.9 kPa for 3-parameter Maxwell model. For the last parameter, G2, the value is 0.16 kPa for 5-parameter Maxwell model, and it is 20.88 kPa for
Viscoelastic parameters
Initial inputs
Computing results
Average value
G∞/(kPa) G1/(kPa) τ1/(s)
10.0, 15.0, 20.0 0.1, 0.5, 1.0 10.0, 20.0, 30.0
19.70, 20.30, 22.40 0.84, 0.91, 0.96 3.20, 3.69, 4.11
20.80 0.90 3.63
Table 6 Initial inputs and computing results for 5-parameter Maxwell viscoelastic model (Penetration depth: 5 μm). Viscoelastic parameters
Initial inputs
Computing results
Average value
G∞/(kPa) G1/(kPa) G2/(kPa) τ1/(s) τ2/(s)
1.0, 5.0, 10.0 1.0, 1.5, 2.0 0.1, 0.5, 1.0 10.0, 20.0, 30.0 10.0, 20.0, 30.0
19.31, 20.71, 21.79 1.11, 1.45, 1.64 0.15, 0.17, 0.17 5.91, 6.02, 6.25 0.29, 0.41, 0.33
20.60 1.46 0.16 6.08 0.34
Table 7 Initial inputs and computing results for 4-parameter Burgers viscoelastic model (Penetration depth: 5 μm). Viscoelastic parameters
Initial inputs
Computing results
Average value
G1/(kPa) G2/(kPa) τ1/(s) τ2/(s)
0.1, 0.5, 1.0 0.1, 0.5, 1.0 1.0, 2.0, 3.0 10.0, 30.0, 50.0
0.24, 0.31, 0.26 20.51, 21.24, 20.78 0.63, 0.71, 0.69 193.32, 186.75, 195.61
0.27 20.88 0.66 191.95
Fig. 18. Plotting of experimental nanoindentation load-time relaxation data (R ¼ 120 μm) at peak displacement of 10 μm and fitting curves. Black dot is experimental data, and red line is fitting curve; (a) for 3-parameter Maxwell model; (b) for 5-parameter Maxwell model; (c) for Burgers model. 9
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4. Conclusions
Table 8 Initial inputs and computing results for 3-parameter Maxwell viscoelastic model (Penetration depth: 10 μm). Viscoelastic parameters
Initial inputs
Computing results
Average value
G∞/(kPa) G1/(kPa) τ1/(s)
10.0, 30.0, 50.0 1.0, 5.0, 10.0 1.0, 5.0, 10.0
91.96, 88.63, 93.22 3.11, 3.35, 3.06 3.20, 3.59, 2.91
91.27 3.17 3.23
The paper presented micro-scale compression and holding tests to study the viscoelastic characteristics of UF microcapsules, including determination of Young’s modulus and viscoelastic parameters of UF microcapsules. The following conclusions can be drawn: (1) The preparation and characterization of UF microcapsules were performed. Furthermore, they were indented with a flat-ended indenter for the compression and holding tests, and SEM obser vation was conducted as well. (2) Force-displacement curve under a small deformation was wellfitted by Hertz model and Reissner model for determining Young’s modulus of UF microcapsules. The comparison of Ec obtained from Hertz model (a whole UF microcapsule) and Ew determined from the elastic shell theory (a core-wall structure) shows a nearly linear correlation. (3) In the FEM analysis, the effect of element types (CAX4R and SAX1), i.e. solid element and shell element, as well as the sensi tivity of Poisson’s ratio, were analysed. It was obtained that the element type had no remarkable effect. Meanwhile, the me chanical response of UF microcapsules had a low sensitivity to Poisson’s ratio and element number. The finite-element models with various thicknesses were studied, and the authors found a range regarding the application of Hertz model or shell theory. (4) The tests show that the UF microcapsules have obvious visco elastic characteristics, especially under a larger deformation. The comparison of the experimental results with the computing re sults from 3-parameter Maxwell model, 5-parameter Maxwell model and 4-parameter Burgers model shows that each of them agrees well with the experimental data. The 3-parameter model can provide enough accuracy (R-square: 0.994) for both ramping and stress relaxation periods. The stability of the parameters was verified. Utilization of the models may enrich the experimental nano-micro material mechanics in interpretation of nano indentation of microcapsules.
Table 9 Initial inputs and computing results for 5-parameter Maxwell viscoelastic model (Penetration depth: 10 μm). Viscoelastic parameters
Initial inputs
Computing results
Average value
G∞/(kPa)
10.0, 50.0, 100.0 1.0, 1.5, 2.0 1.0, 5.0, 10.0 0.1, 0.5, 1.0 10.0, 20.0, 30.0
90.51, 89.60, 91.23 0.67, 0.66, 0.71 4.24, 4.16, 4.53 0.27, 0.29, 0.26 6.82, 7.17, 7.01
89.43
G1/(kPa) G2/(kPa) τ1/(s) τ2/(s)
0.68 4.31 0.27 10.51
Table 10 Initial inputs and computing results for 4-parameter Burgers viscoelastic model (Penetration depth: 10 μm). Viscoelastic parameters
Initial inputs
Computing results
Average value
G1/(kPa) G2/(kPa)
0.1, 0.5, 1.0 10.0, 50.0, 100.0 0.1, 0.5, 1.0 30.0, 50.0, 100.0
0.91, 0.88, 0.91 94.3, 91.5, 95.6
0.90 93.80
0.52, 0.48, 0.53 237.64, 246.49, 239.71
0.51 241.28
τ1/(s) τ2/(s)
4-parameter Burgers model. Since the UF microcapsules have a strong viscoelastic behavior under a large deformation, therefore, these viscoelastic parameters during the compression and holding test (Depth: 10 μm) are generally greater than the above mentioned. Specifically, for the relaxation time constant, τ1, the highest value is 3.23 s for 3-parameter Maxwell model, and the lowest value is 0.27 s for 3-parameter Maxwell model, additionally, the value is 0.51 s for 4-parameter Burgers model. For the relaxation time constant, τ2, the values are 10.51 s and 241.28 s for 5-parameter Maxwell model and 4-parameter Burgers model, respectively. The equivalent shear modulus, G∞, are almost similar, 91.27 kPa and 89.43 kPa, for these 2 Mx models. For the shear modulus, G1, the highest value is 3.17 kPa for 3-parameter Maxwell model, and the lowest value is 0.68 kPa for 5-parameter Maxwell model, and the value is 0.9 kPa for 4-parameter Burgers model. For the last parameter, G2, the value is 4.31 kPa for 5-parameter Maxwell model, and it is 93.8 kPa for 4-param eter Burgers model. The microcapsules composed by urea and formaldehyde under catalysis show an obvious viscoelastic behavior, as Shaw and MacKnight indicated when the organic polymer was subjected to various external forces, there was almost no exception to the viscoelastic response [48, 49]. In this work, UF microcapsules are applied for self-healing cementitious composites. Therefore, detailed study of the mechanical characteristics of the microcapsules, such as determination of elastic-plastic [39], and viscoelastic parameters, should be beneficial for uncovering the rupturing mechanism of the microcapsules embedded in the cementitious materials. Furthermore, the development would have wider application, including the determination of mechanical parame ters of a variety of microspheres, microcapsules and cells.
Data availability The raw/processed data required to reproduce these findings are available to download from https://doi.org/10.7910/DVN/WVESCH. Declaration of competing interest The authors declare that there is no conflict of interest. Acknowledgements The authors gratefully acknowledge the financial support provided by the General Program of the National Natural Science Foundation of China (No. 51978409), the International Cooperation and Exchange of the National Natural Science Foundation of China (51120185002), the Science and Technology Foundation for the Basic Research Plan of Shenzhen City (JCYJ20160422095146121). Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.polymertesting.2019.106146. References [1] A. Gray, S. Egan, S. Bakalis, Z. Zhang, Determination of microcapsule physicochemical, structural, and mechanical properties, Particuology 24 (2015) 32–43. [2] H. Jong, J. Bonner, Phosphatide auto-complex coacervates as ionic systems and their relation to the protoplasmic membrane, Protoplasma 24 (1935) 198–218.
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