What is Analytical Rheology?

分析流变学是基于其粘弹性响应的测量值确定材料的微观结构的主题。

It is an extension of analytic chemistry in a way similar to other analytic methods predicated on flow properties. One example of the analytical technique that comes under this category is intrinsic viscosity.

Analytic rheology can be used on any material system while the rheological response is based on the microstructure. There are different examples of such rheology-based systems. For instance, based on the measured linear viscoelastic response, the droplet size distribution of a multiphase suspension can be determined. Experimental methods are highly evolved to determine the linear viscoelastic material functions so as to make accurate and reliable measurements in an automated manner. Analytic rheology makes use of this experimental capability to develop advanced interpretation methods. In advanced data interpretation methods such as MWD determinations, the TAOrchestrator software allows the full power of rheological characterization methods. The calculation of the molecular weight distribution for linear flexible polymers based on the measured linear viscoelastic material functions is the specific application of analytic rheology discussed in this article.

流变学MWD确定的好处

It is essential to identify the advantages of having a rheological MWD determination method as there are a number of viable methods available for determining the molecular weight distribution of flexible polymers, including intrinsic viscosity, light scattering, gel permeation chromatography, etc.

There are several advantages to rheologically-based methods for determining the molecular weight distribution. For instance, virtually all conventional MWD methods are predicted based on the ability to readily dissolve the polymer in a solvent at room temperatures. However, most of the commercially used polymers including polypropylene, polyethylene and Teflon are slightly soluble in common solvents at room temperatures. This prevents access to standard molecular weight determination methods used for commercially important polymer systems.

流变方法避免需要进行此类耗时的溶性程序。另外，在获得聚乙烯或聚丙烯熔体的流变学数据方面没有具体的实验困难。流变学方法是必不可少的次级分析方法，该方法还需要一种主要的校准分析方法。在对这些聚合物使用MWD方法后，分析技术的分辨率和敏感性很差，尤其是对于高分子量的尾巴而言。

Characterization of the high molecular weight tail is critical for the characterization of the processability of a commercially available polymer. Rheological methods used for MWD determination are intrinsically sensitive to the high end of the molecular weight distribution, which is evident from the dependence of the linear viscoelastic material functions on molecular weight and molecular weight distribution. For instance, the zero shear viscosity depends on the weight average molecular weight to the 3.4 power. Therefore, any small changes in weight average molecular weight can lead to large changes in zero shear viscosity.

Only a small portion of the linear viscoelastic spectrum consists of information on the molecular weight of the system. These concepts are based on the Bueche-Ferry /1/ theorem. According to the Bueche-Ferry hypothesis, the response of all flexible polymers is similar on sufficiently short time scales, irrespective of the polymer concentration, chain architecture or molecular weight. This shows that the viscoelastic response in the glassy regime is not affected by the MWD. This result is obtained from the fact that the entanglement effect can be observed on relatively large length scales when compared to typical monomeric dimensions.

Eventually, the mechanical response’ dependence on molecular weight is completely lost in the glassy time/ frequency range. Different regimes of dynamic response for a monodisperse polymer melt are shown in Figure 1. The availability of the “glassy modes” in data sets demands a suitable way of choosing these effects out of the experimental data before MWD calculation. The principal complication, however, is that the lower Rouse modes for the high molecular weight components can be combined with repetitive relaxation modes for the lower molecular weight species.

Figure 1.Dynamic responses of a monodisperse linear flexible polymer are identical to those of a monomer in a larger or shorter chain, or a polymer in dilute solution for that matter

混合聚合物融化的规则

A mixing rule is a quantitative relationship between the observed mechanical properties of a polydisperse melt and the underlying microstructure. Tsenoglou and des Cloizeaux recently derived a viable mixing rule for homogeneous systems of well entangled polymers. The mixing rule known as the “double reptation” model is a relatively simple mathematical approximation to a more rigorous and complex molecular theory of polydispersity. The mathematical structure for the double reptation mixing rule is as follows:

(1)

where G(t) is the relaxation modulus that can be determined using different combinations of linear viscoelasticity experiments given below. The F^1/2（m，t）是单分散弛豫函数，代表了小阶段应变后单分散聚合物的时间依赖性分数应力松弛。W（M）是基于重量的分子量分布。可以说，分子量分布的成分将在一定程度上增加模量。该概念应用于分子量分布W（M）上的积分，该积分总和从MWD的每个组件到g（t）的贡献。^1/2(M,t). This describes the “mixing” effect where one component of a complex molecular weight distribution dynamically interacts with all of its neighbors. The TAOrchestrator software consists of options to function under the so called “weight average” mixing rule derived by Marin et al.

(2)

The weight average mixing rule (2) is originally empirical. The physical basis for this mixing rule is determined based on the empirical observation that the relaxation time scales as molecular weight to ~3.4 power coupled with dimensional analysis.

材料依赖性输入参数

可以使用公式（1）计算给定材料的分子量分布，以便为应用提供材料相关的数据。特别是，有必要提供高原模量GN和单分散弛豫函数的形式f^1/2(M,t). The plateau modulus is however tabulated in several references. The monodisperse relaxation function can be used in several forms. In general, the following single exponential form can be used.

(3)

其中λ（m）是单分散系统和k（t）的特征放松时间是温度依赖的系数。对于柔性聚合物，指数X通常为〜3.4。此外，也可以使用诸如doi-eedwards之类的单分散弛豫功能之类的因素。

（4）

当实验数据的准确性不符合预期时，可以使用公式（3）或（4）来识别从等式1的预测MWD的显着差异。基于Arhenius型激活能，K（t）的温度依赖性被建模，忽略了WLF等其他选项的可用性。可以从渡轮或研究文献等标准参考文献中获得材料依赖参数的数据。新型聚合物可能需要实验和校准。

Calculation of Relaxation Modulus Based on Linear Viscoelasticity Data

The relaxation modulus G(t) can be determined from linear viscoelasticity data using several methods. All linear viscoelastic material functions may include equivalent information. However, some of the linear viscoelastic material functions are intrinsically sensitive to short and long time-scales when compared to others. Constrained elastic recovery experiments, for example, are sensitive to long time-scale relaxation processes whereas the dynamic moduli measurements are sensitive to short time-scale relaxation processes. The TAOrchestrator software provides a suitable way of combining these data to enable expansion of the effective dynamic range of the relaxation modulus determination. In general, incorporation of linear viscoelasticity data from different experiments is beneficial to achieve a self-consistent determination of the relaxation modulus over a wide dynamic range.

As discussed earlier, the linear viscoelasticity data on the plateau and terminal regions may contain information related to the molecular weight and molecular weight distribution. Hence, after calculating a relaxation modulus, the short time-scale contributions due to the glassy modes can be eliminated. The TAOrchestrator software achieves this by estimating the MWD. Using the estimated MWD, the Rouse-like glassy response can be calculated by adding the all Rouse modes for all components of the MWD and by subtracting it from G(t). In order to calculate a molecular weight distribution by experimentally determining the relaxation modulus, there are possibilities of achieving two distinct experimental situations such as complete dynamic moduli data and incomplete dynamic moduli data. A data set is said to be complete when the data spans a dynamic time/frequency range from fully terminal behavior through transition to the glassy modes as shown in Figure 1. Research applications may require complete data sets. However, considering the practical circumstance of characterizing of commercial polymers with broad molecular weight distribution, incomplete” data sets are preferred. In such cases, it is impossible to achieve either fully terminal behavior due to exceptionally long relaxation times of the high molecular weight tail. Each of the cases described above have different computational issues which result in restrictions to the method that we explore below.

完整的动态模量数据集

有必要利用所有流变学信息，以在存在从末端区域到高原区域的动态模量数据的情况下反转混合规则。在此阶段的问题是要以稳定且健壮的方式倒入MWD的双重振兴模型。双重仓库混合规则是W（M）的第一类的Fredholm积分方程。有完善的方法来解决这些问题。基于用于通过Mellin Transforms从实验数据中计算出的MWD矩的正则化方法，可以计算分子量分布。结果是一种坚固且稳定的数值方法。

不完整的动态模量数据集

The data set is incomplete, and the double reputation model cannot be inverted without additional information unless the experimental data span the entire frequency range from the terminal to the plateau region.

Prior information on the shape of the distribution can be added into the numerical algorithm. The method involved is assumed to have prior information about the shape of the molecular weight distribution. This assumption is valid for all commercial polymers where the polymer catalyst chemistry determines the MWD shape.

The numerical method effectively fits the predicted dynamic moduli curve from model molecular weight distributions to the measured experimental data in an optimal manner. The method involves manipulating a candidate MWD to obtain an optimal fit to the rheological data. In addition, a number of model molecular weight distributions can be tried along with binary combinations.

通过增加拟合参数的数量，可以提高拟合到测量数据的准确性。但是，观察到大多数商业聚合物模型分子量分布或组合具有令人满意的性能。Taorchestrator软件可促进自动拟合或用户定义的手动拟合数据。此功能使用户能够通过覆盖软件来确切地拟合到所需的数据的特定区域。

Taorchestrator软件由两种类型的模型分子量分布组成。首先是Wesslau或原木正常分子量分布，表明使用Zeigler-Natta催化剂系统产生的添加聚合物或聚合物。第二个是舒尔茨分布，这是一种概括性的分子量分布，该分子分布由通过凝结反应或使用金渐新催化剂产生的更多聚合物组成。

申请of Dynamic Moduli Sets

The dynamic moduli and calculated MWD for a bidisperse system of polybutadiene are shown in Figures 2 and 3. The dynamic moduli data set is complete, and the MWD calculated using the complete data set method was observed to have overlapped with the MWD measured using GPC. However, the rheological MWD and the GPC data were found to be in good agreement with each other.

图2。Dynamic response of a bidisperse

Figure 3.计算出的Bidisperse polybutadiene混合物的MWD

The two metallocene catalyzed binary polyethylene blends are shown in Figure 4. The dynamic moduli data sets are incomplete either in the plateau region or terminal region. Figure 5 shows the results of using the incomplete data set method with a combination of two Schultz model molecular weight distributions. The agreement is also good which indicates the viability of MWD methods for applications in commercial systems.

图4。二元聚乙烯混合的动态响应

Figure 5.计算的二元聚乙烯混合物的MWD

This information has been sourced, reviewed and adapted from materials provided by TA Instruments.

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