高分子纳米纤维及其结构支架的力学性能研究与计算建模
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摘要
高分子纳米纤维支架具有与天然细胞外基质相似的形态结构,能够为细胞的种植和生长提供合适的微环境,在组织工程中得到了广泛的应用。支架是由单根的高分子纳米纤维组成,为了能够在纳米尺度下对支架进行设计以达到特定细胞生长所需要的力学强度,需要研究单根高分子纳米纤维的力学性能。另外,支架的孔隙率和纤维角度取向程度等结构参数也是支架设计过程中的重要参考因素,因此也需要研究支架本身的力学性能与支架结构参数之间的关系。本论文主要针对单根高分子纳米纤维的力学性能进行了实验和理论研究,并建立了支架力学性能的理论计算模型。
实验部分以聚己内酯(PCL)纳米纤维为研究对象,分别利用纳米拉伸设备和原子力显微镜测试了单根PCL纳米纤维在轴向拉伸和三点弯曲实验中的力学性能。拉伸实验的统计结果显示,PCL纳米纤维的弹性模量与其直径之间没有明显的尺度效应关系,而根据文献中的另一组实验数据,PCL纳米纤维的弹性模量具有尺度效应,这似乎与我们的实验结果存在矛盾。三点弯曲的实验结果显示纤维的弹性模量与纤维直径之间存在尺度效应关系,与文献中对高分子纳米纤维的弯曲实验得到的尺度效应一致,即随着纤维直径的减小,纤维的弹性模量呈显著增大趋势。
在单根纳米纤维力学性能的建模部分,分别针对纤维的拉伸和弯曲变形建立了不同的理论模型。考虑到高分子材料内部的微结构对纤维力学性能具有影响,采用应变梯度弹性(SGE)理论研究了纤维力学性能的尺度效应,并根据高阶应变梯度弹性(HSGE)理论建立了纤维在均匀拉伸变形中的尺度效应模型。
根据理论模型预测,纤维的等效弹性模量存在两种尺度效应:与纤维长度相关的尺度效应(L-SD)和与纤维直径相关的尺度效应(D-SD)。在纤维的拉伸变形建模中,根据SGE可以得到L-SD尺度效应,根据HSGE可以得到D-SD尺度效应,而根据表面效应理论(SE)同样可以得到D-SD尺度效应,但与HSGE模型得到的D-SD不同。由于针对纳米纤维的拉伸实验只有D-SD的数据,L-SD的实验数据尚未见到报道,分别利用HSGE模型和SE模型拟合拉伸实验中的D-SD数据。比较两种模型的拟合结果可以发现,HSGE模型的拟合结果与实验数据比较吻合,而SE模型的拟合结果与实验数据之间存在一定的偏差。
在纤维的弯曲变形建模中,根据SGE和SE建立了纤维在三种弯曲边界条件(悬臂梁、两端固支梁和简支梁)下等效弹性模量的尺度效应模型。根据SGE可以得到L-SD和D-SD两种尺度效应,而根据SE可以得到D-SD尺度效应。分别利用SGE模型和SE模型拟合弯曲实验中的D-SD数据,比较两种模型的拟合结果可以发现,两种模型均能解释纤维弯曲变形的尺度效应,但对于高分子纳米纤维材料,SGE的拟合结果与实验数据更加吻合。
基于对单根纳米纤维力学性能的理论分析,我们进一步研究纤维支架的力学性能。将纳米纤维等效处理为具有特定长度的线弹性杆状结构,假设纤维与纤维之间的交点符合泊松分布,以纤维交点之间的纤维片段作为研究对象,并综合考虑纤维密度逾渗阈值的影响,得到了角度随机分布和角度取向分布两种纤维支架的力学性能计算模型。角度随机分布的纤维支架是各向同性,而角度取向分布的纤维支架是正交各向异性,需要分别研究其平行于纤维取向方向和垂直于纤维取向方向的力学性能。对于角度随机分布的纤维支架,模型的计算结果与有限元模拟情况吻合;而对于角度取向分布的纤维支架,模型的计算结果与文献中的实验数据基本吻合,但存在一定的偏差。
为了研究单根纳米纤维尺度效应对支架力学性能的影响,在支架力学性能计算模型中分别用HSGE和SGE表示纤维在拉伸和弯曲变形中的D-SD尺度效应,得到了包含尺度效应的支架力学性能计算模型。利用该模型分别比较在不考虑尺度效应以及考虑尺度效应两种情况下,支架的弹性模量与其结构参数之间的关系:对于角度随机分布的纤维支架,得到了支架的弹性模量与纤维密度和直径之间的关系;对于角度取向分布的纤维支架,得到了支架两个方向的弹性模量与纤维密度、直径以及角度取向程度之间的关系。
综上所述,高分子纳米纤维的尺度效应比较复杂。根据对单根纳米纤维的力学建模,在不同的实验条件下,纳米纤维的力学性能可能表现出D-SD或者L-SD的尺度效应。根据对纳米纤维支架的力学建模,支架的力学性能不仅与其结构参数(纤维密度、角度取向程度等)有关,而且会受到单根纤维尺度效应D-SD的影响,因此在纤维结构支架的设计与制备中需要综合考虑以上各种因素的影响。
With the rapid development of tissue engineering,nanofibrous scaffolds have been gaining popularity for use as an extra-cellular matrix for cell seeding.Since the fibers used in these scaffolds are nano-scaled,an accurate quantification of their strength characteristics becomes necessary in order to design a scaffold that meets the specifc strength requirements for handling different cell types.Also,it is necessary to quantify the strength characteristics of the scaffold itself as the nanofiber arrangement in terms of fiber orientation and porosity can be a significant factor.Thus,this thesis is concerned with an investigation of the single-strand nanofiber,as well as,the nanofibrous scaffold.For the modeling of the single-strand nanofiber,we have provided experimental data to verify the predictions.As for the scaffold,we presented limited experimental verifications since there are very few reported experiments on the mechanical testing of the scaffold in the literature.
In our experimental measurements with single-strand polycaprolactone(PCL) nanofibers,we conducted uniaxial tensile and three-point bending tests using the Nano Universal Tensile Machine and diametral measurements using the atomic force microscopy.The tensile test data shows that the elastic modulus of PCL nanofibers is statistically invariant with changing fiber diameters.Our results appear to contradict one set of reported test data for the same fiber type,where it depicts a size dependent trend in the elastic modulus-diameter plot.For the three-point bending tests both our data and reported experimental results are consistent in that the elastic modulus clearly exhibits a significant inverse size-dependent trend with the fiber diameter.
Our modeling effort also begins with the single-strand nanofiber as without a prior knowledge of this basic building block,it is not possible to quantify the mechanical properties of the scaffold.We have developed several computational models for studying a single-strand nanofiber under the 2 types of loading;tensile and bending.Recognizing the non-local interactions between discrete elements of the polymeric material,we employed the strain gradient elasticity(SGE) theory to model the size dependent behavior.To handle the reported tensile size dependency,we expanded on the traditional SGE theory by incorporating 2nd-order strain gradients and termed the resulting approach as higher-order strain gradient elasticity(HSGE).
Our theoretical work indicates that there are 2 kinds of size dependent response; one that pertains to a decreasing fiber length(while still maintaining the fiber aspect ratio) and the other to a decreasing fiber diameter.We referred the former as L-SD (length size-dependency) and the latter as D-SD(diameter size-dependency).The L-SD is observed in our SGE tensile predictions while the D-SD is seen in our HSGE predictions.Since all the reported experiment data(including ours) are designed to measure only D-SD,the agreement between the model predictions and experimental results is very good.In this thesis,we have also presented a modified surface effect (SE) approach to study size-dependency in polymeric nanofibers and showed that its predictions also matched the measured data but not as closely as our HSGE model.
In the bending size dependency work,we developed computational models using both SGE and SE theories for 3 types of boundary conditions:cantilever,fixed-fixed and simply supported.Application of SGE to the bending deformation of nanofibers produces both L-SD and D-SD,while the application of SE leads to D-SD only.Both D-SD models are able to track the bending size-dependent behavior;however the SGE model appears to produce a better fit with the measured data for polymeric nanofibers.
With the knowledge gained from the modeling of single strand nanofibers,we now consider the micromechanical modeling of a nanofibrous scaffold.The nanofibers are assumed to be linear-elastic straight rods of constant length.However, in the scaffold the nanofibers are not viewed as distinct elements;instead,it is the individual fiber segment between any 2 fiber crossings that is considered.Further,the length of a fiber segment is assumed to obey a Poisson distribution of fiber-to-fiber contacts.In our 2-D scaffold model we studied 2 kinds of fiber distribution - random and aligned,and obtained its elastic modulus that included an exponential decay term to capture the percolation effect for both these 2 cases.For the randomly distributed nanofibers,the elastic modulus predictions agree with results from a finite element analysis to a reasonably good degree of accuracy.For the aligned nanofibers the scaffold is orthotropic in nature and thus,the elastic modulus has 2 components-parallel and perpendicular to the fiber orientation.A comparison of the 2 moduli with published experimental data shows that although the model is able to capture the data trend the agreement still has minor errors.
We also incorporated size dependency into the micromechanical model of the scaffold.Specifically,we introduced HSGE to capture tensile D-SD and SGE for bending D-SD.We presented several plots of the scaffold's elastic modulus versus the fiber density and fiber diameter for the randomly distributed fibers with and without size-dependency.Likewise,we did the same for the aligned fibers but with an additional parameter - the fiber orientation.
To summarize,we note that size dependency in polymeric nanofibers is a complex phenomenon.Our research on single strand nanofibers indicates that it can manifest either as a diameter-based and/or a length-based size dependency.The phenomenon in a scaffold is even more complex as it can be affected by additional parameters such as fiber density and fiber orientation.
实验部分以聚己内酯(PCL)纳米纤维为研究对象,分别利用纳米拉伸设备和原子力显微镜测试了单根PCL纳米纤维在轴向拉伸和三点弯曲实验中的力学性能。拉伸实验的统计结果显示,PCL纳米纤维的弹性模量与其直径之间没有明显的尺度效应关系,而根据文献中的另一组实验数据,PCL纳米纤维的弹性模量具有尺度效应,这似乎与我们的实验结果存在矛盾。三点弯曲的实验结果显示纤维的弹性模量与纤维直径之间存在尺度效应关系,与文献中对高分子纳米纤维的弯曲实验得到的尺度效应一致,即随着纤维直径的减小,纤维的弹性模量呈显著增大趋势。
在单根纳米纤维力学性能的建模部分,分别针对纤维的拉伸和弯曲变形建立了不同的理论模型。考虑到高分子材料内部的微结构对纤维力学性能具有影响,采用应变梯度弹性(SGE)理论研究了纤维力学性能的尺度效应,并根据高阶应变梯度弹性(HSGE)理论建立了纤维在均匀拉伸变形中的尺度效应模型。
根据理论模型预测,纤维的等效弹性模量存在两种尺度效应:与纤维长度相关的尺度效应(L-SD)和与纤维直径相关的尺度效应(D-SD)。在纤维的拉伸变形建模中,根据SGE可以得到L-SD尺度效应,根据HSGE可以得到D-SD尺度效应,而根据表面效应理论(SE)同样可以得到D-SD尺度效应,但与HSGE模型得到的D-SD不同。由于针对纳米纤维的拉伸实验只有D-SD的数据,L-SD的实验数据尚未见到报道,分别利用HSGE模型和SE模型拟合拉伸实验中的D-SD数据。比较两种模型的拟合结果可以发现,HSGE模型的拟合结果与实验数据比较吻合,而SE模型的拟合结果与实验数据之间存在一定的偏差。
在纤维的弯曲变形建模中,根据SGE和SE建立了纤维在三种弯曲边界条件(悬臂梁、两端固支梁和简支梁)下等效弹性模量的尺度效应模型。根据SGE可以得到L-SD和D-SD两种尺度效应,而根据SE可以得到D-SD尺度效应。分别利用SGE模型和SE模型拟合弯曲实验中的D-SD数据,比较两种模型的拟合结果可以发现,两种模型均能解释纤维弯曲变形的尺度效应,但对于高分子纳米纤维材料,SGE的拟合结果与实验数据更加吻合。
基于对单根纳米纤维力学性能的理论分析,我们进一步研究纤维支架的力学性能。将纳米纤维等效处理为具有特定长度的线弹性杆状结构,假设纤维与纤维之间的交点符合泊松分布,以纤维交点之间的纤维片段作为研究对象,并综合考虑纤维密度逾渗阈值的影响,得到了角度随机分布和角度取向分布两种纤维支架的力学性能计算模型。角度随机分布的纤维支架是各向同性,而角度取向分布的纤维支架是正交各向异性,需要分别研究其平行于纤维取向方向和垂直于纤维取向方向的力学性能。对于角度随机分布的纤维支架,模型的计算结果与有限元模拟情况吻合;而对于角度取向分布的纤维支架,模型的计算结果与文献中的实验数据基本吻合,但存在一定的偏差。
为了研究单根纳米纤维尺度效应对支架力学性能的影响,在支架力学性能计算模型中分别用HSGE和SGE表示纤维在拉伸和弯曲变形中的D-SD尺度效应,得到了包含尺度效应的支架力学性能计算模型。利用该模型分别比较在不考虑尺度效应以及考虑尺度效应两种情况下,支架的弹性模量与其结构参数之间的关系:对于角度随机分布的纤维支架,得到了支架的弹性模量与纤维密度和直径之间的关系;对于角度取向分布的纤维支架,得到了支架两个方向的弹性模量与纤维密度、直径以及角度取向程度之间的关系。
综上所述,高分子纳米纤维的尺度效应比较复杂。根据对单根纳米纤维的力学建模,在不同的实验条件下,纳米纤维的力学性能可能表现出D-SD或者L-SD的尺度效应。根据对纳米纤维支架的力学建模,支架的力学性能不仅与其结构参数(纤维密度、角度取向程度等)有关,而且会受到单根纤维尺度效应D-SD的影响,因此在纤维结构支架的设计与制备中需要综合考虑以上各种因素的影响。
With the rapid development of tissue engineering,nanofibrous scaffolds have been gaining popularity for use as an extra-cellular matrix for cell seeding.Since the fibers used in these scaffolds are nano-scaled,an accurate quantification of their strength characteristics becomes necessary in order to design a scaffold that meets the specifc strength requirements for handling different cell types.Also,it is necessary to quantify the strength characteristics of the scaffold itself as the nanofiber arrangement in terms of fiber orientation and porosity can be a significant factor.Thus,this thesis is concerned with an investigation of the single-strand nanofiber,as well as,the nanofibrous scaffold.For the modeling of the single-strand nanofiber,we have provided experimental data to verify the predictions.As for the scaffold,we presented limited experimental verifications since there are very few reported experiments on the mechanical testing of the scaffold in the literature.
In our experimental measurements with single-strand polycaprolactone(PCL) nanofibers,we conducted uniaxial tensile and three-point bending tests using the Nano Universal Tensile Machine and diametral measurements using the atomic force microscopy.The tensile test data shows that the elastic modulus of PCL nanofibers is statistically invariant with changing fiber diameters.Our results appear to contradict one set of reported test data for the same fiber type,where it depicts a size dependent trend in the elastic modulus-diameter plot.For the three-point bending tests both our data and reported experimental results are consistent in that the elastic modulus clearly exhibits a significant inverse size-dependent trend with the fiber diameter.
Our modeling effort also begins with the single-strand nanofiber as without a prior knowledge of this basic building block,it is not possible to quantify the mechanical properties of the scaffold.We have developed several computational models for studying a single-strand nanofiber under the 2 types of loading;tensile and bending.Recognizing the non-local interactions between discrete elements of the polymeric material,we employed the strain gradient elasticity(SGE) theory to model the size dependent behavior.To handle the reported tensile size dependency,we expanded on the traditional SGE theory by incorporating 2nd-order strain gradients and termed the resulting approach as higher-order strain gradient elasticity(HSGE).
Our theoretical work indicates that there are 2 kinds of size dependent response; one that pertains to a decreasing fiber length(while still maintaining the fiber aspect ratio) and the other to a decreasing fiber diameter.We referred the former as L-SD (length size-dependency) and the latter as D-SD(diameter size-dependency).The L-SD is observed in our SGE tensile predictions while the D-SD is seen in our HSGE predictions.Since all the reported experiment data(including ours) are designed to measure only D-SD,the agreement between the model predictions and experimental results is very good.In this thesis,we have also presented a modified surface effect (SE) approach to study size-dependency in polymeric nanofibers and showed that its predictions also matched the measured data but not as closely as our HSGE model.
In the bending size dependency work,we developed computational models using both SGE and SE theories for 3 types of boundary conditions:cantilever,fixed-fixed and simply supported.Application of SGE to the bending deformation of nanofibers produces both L-SD and D-SD,while the application of SE leads to D-SD only.Both D-SD models are able to track the bending size-dependent behavior;however the SGE model appears to produce a better fit with the measured data for polymeric nanofibers.
With the knowledge gained from the modeling of single strand nanofibers,we now consider the micromechanical modeling of a nanofibrous scaffold.The nanofibers are assumed to be linear-elastic straight rods of constant length.However, in the scaffold the nanofibers are not viewed as distinct elements;instead,it is the individual fiber segment between any 2 fiber crossings that is considered.Further,the length of a fiber segment is assumed to obey a Poisson distribution of fiber-to-fiber contacts.In our 2-D scaffold model we studied 2 kinds of fiber distribution - random and aligned,and obtained its elastic modulus that included an exponential decay term to capture the percolation effect for both these 2 cases.For the randomly distributed nanofibers,the elastic modulus predictions agree with results from a finite element analysis to a reasonably good degree of accuracy.For the aligned nanofibers the scaffold is orthotropic in nature and thus,the elastic modulus has 2 components-parallel and perpendicular to the fiber orientation.A comparison of the 2 moduli with published experimental data shows that although the model is able to capture the data trend the agreement still has minor errors.
We also incorporated size dependency into the micromechanical model of the scaffold.Specifically,we introduced HSGE to capture tensile D-SD and SGE for bending D-SD.We presented several plots of the scaffold's elastic modulus versus the fiber density and fiber diameter for the randomly distributed fibers with and without size-dependency.Likewise,we did the same for the aligned fibers but with an additional parameter - the fiber orientation.
To summarize,we note that size dependency in polymeric nanofibers is a complex phenomenon.Our research on single strand nanofibers indicates that it can manifest either as a diameter-based and/or a length-based size dependency.The phenomenon in a scaffold is even more complex as it can be affected by additional parameters such as fiber density and fiber orientation.
引文
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