Tag Archives: viscoelasticity

什么是“松弛时间”?

不知道是不是我所在的学院主要是一个化学背景的学院,所以我遇到很多同学——一度也包括我自己——对什么是“松弛时间”是很困惑的。学生可能一律是在上《高分子物理》课的时候接触到这个概念的。但是,高分子物理,是讲粘弹性力学模型的时候突然用这个词的,大致上就是有个粘壶,有个弹簧,列个方程,里面含有粘度与模量的比值,书上就直接说这个比值“叫做松弛时间”,令人无语。粘度是粘度,模量是模量,怎么会出来时间?不是要发生一件事情,有开始,有结束,才能测量出时间来的吗?这个时间既然叫做“松弛”,那就是发生了“松弛”这件事。所以问题应该是“松弛”是一件什么事情?主语是谁?过程是怎样的?结果是什么?这些,不光在《高分子物理》课本上没有说清楚,在很多流变学的书里面也没说清楚。

然后学生进入实验室之后,特别是我们组,例如遇到动态光散射实验,发现又有“松弛时间”。动态光散射没有粘度,没有模量,怎么又叫“松弛时间”?整天这个“松弛时间”那个“松弛时间”动不动就“松弛时间”,不同场合所出现的“松弛时间”是不是一个意思?同一个材料,做动态光散射的那个“松弛时间”,跟做应力松弛的那个“松弛时间”相不相等?这些问题都没法从“粘度除以模量”的已有知识进行回答,就会令人觉得所有“松弛时间”都是属于另一个世界的东西。“松弛时间”这一概念似乎是阻碍我们院的研究生从“反应、过柱、挤出、打红外打核磁做DSC做拉力”的“民工式”材料学研究进入到真正的物理化学研究的最大门槛。为了搞懂一个词,叫人回家通读《平衡态统计物理》,似乎有点过份。但是,一个仅知道“松弛时间等于粘度除以模量”的学生,叫他怎么去理解在不同场合下出现的“松弛时间”?更别说自主地正确使用这一物理量了。

我看到俄罗斯流变学家Alxander Malkin的流变学教材Rheology: Concepts, Methods and Applications在第2章介绍粘弹性的时候,专门用一块小字号的阅读材料来解释什么叫松弛时间——而且是作为物理学的一般性概念来介绍。很简短明快,值得借鉴。

Relaxation time — general concept in physics

The concept of relaxation has a general meaning for many physical phenomena. It is a reflection of an idea of restoration of equilibrium state from a non-equilibrium condition, regardless of the reasons which caused the departure from equilibrium. For example, this can be concentration fluctuation caused by purely statistical reasons as was considered by Maxwell. Let the equilibrium value of some physical parameter be X, current value of this parameter be X, and let it be supposed that the rate of approach of equilibrium is proportional to the distance from the equilibrium. This assumption immediately leads to the following first-order kinetic equation:
\frac{dX}{dt}=-k\left(X-X_\infty\right)
where k is a kinetic rate constant with the dimension of reciprocal time.

The parameter X in the initial state equals to X0. Then, the solution of this equation is
\frac{X\left(t\right)-X_\infty}{X_0-X_\infty}=e^{-kt}

Now, if X=0, then the simplest form of this equation is
X\left(t\right)=X_0e^{-kt} (*)

The last two equations describe the relaxation process, and the value of
\theta=k^{-1}
is called the relaxation time. Its value characterizes the rate of approch of the equilibrium (but not the complete time necessary to reach this equilibrium because it is infinitely large according to equation *.

以上这段话不仅从正面简短介绍了“松弛时间”的概念,还适时解答了一般人很容易产生的困惑,包括本文开头提出的那几个疑问。在一个教材里面,加上这么一小段话,并不影响教材的篇幅或者印刷,但是却对学生今后的发展作用重大。由于这一介绍松弛时间一般性的定义,因此它不会像“粘度除以模量”那样无法延伸到其他场合。所有场合下出现的“松弛时间”,都可以拿以上那段话的内容去理解。这才是学习应该达到的效果。

Alexander Yakovlevich Malkin是俄罗斯的流变学家。很老了,可惜关于他的故事了解得不多。也许郑融老师会对他有所了解?

为什么G''是负值?

主题回归到流变学一下。

众所周知(线性)粘弹性是处于Hookian弹性和Newtonian粘性这两个极端之间的性质。对Hookian弹性体(Hookian弹簧模型)施加正弦形变,测到的是同相的正弦应力;向Newtonian粘流体(Newtonian粘壶)施加正弦形变速率,测到的是同相的正弦应力。向一个线性粘弹性样品施加正弦形变,测到的是相位角为[eq]\delta[/eq]的正弦应力:[eq]\gamma_0 \sin \left ( \omega t+\delta \right )[/eq]。

小幅振荡力学响应

小幅振荡力学响应

最为人熟悉的流变学测试(也是最多不懂流变的人知道要做的测试)就是[eq]G'[/eq]、[eq]G”[/eq]~[eq]\omega[/eq]曲线,这个也叫做“粘弹谱”。我曾经详细解释过[eq]G'[/eq]和[eq]G”[/eq]到底是怎么来的。典型线型聚合物粘弹谱如下图:

线型聚合物粘弹谱

线型聚合物粘弹谱

下图是我配制的合成锂藻土Laponite凝胶的粘弹谱,发现[eq]G”[/eq]在高频处出现负值。下面是对这个不正常现象的解释。

Laponite凝胶粘弹谱

Laponite凝胶粘弹谱

一台应变控制型流变仪若要给出这样的图,就要给样品施加一定频率[eq]\omega[/eq]和幅度[eq]\gamma[/eq]的正弦应变[eq]\gamma \left ( t \right ) = \gamma_0 \sin \left ( \omega t \right)[/eq],然后夹具所连接的力学传感器记录样品向夹具施加的转矩,并根据夹具形状和Gap值换算成应变[eq]\sigma \left ( t \right )[/eq],在线性粘弹性条件下[eq]\sigma \left ( t \right )=\sigma_0 \sin \left ( \omega t + \delta \right )[/eq]。仪品仅需且必须准确测量出两个值:[eq]\sigma_0[/eq](从在转矩数据中找到的最值换算)和[eq]\delta[/eq](通过比较转矩数据和应变数据的相位差[eq]\Delta t[/eq]换算,见第1幅图)。其中,[eq]\Delta t=\frac{\delta}{\omega}[/eq]。其他诸如[eq]G'[/eq]、[eq]\eta'[/eq]之类的,都是从这两个数据算出来的。例如[eq]G”=\frac {\sigma_0}{\gamma_0} \sin \left( \omega \right )[/eq]。

如果样品非常接近Hookian弹性体,应力和应变的时间曲线接近同相,相位角[eq]\delta[/eq]的值是非常小的,如果同时频率[eq]\omega[/eq]的值很大,那么相位差[eq]\Delta t[/eq]的值就会非常小。按照第一幅图的情况,,这么大的时间差就算人也能测得很准。但是,在Laponite凝胶粘弹谱中,样品在高频区非常接近Hookian行为,[eq]\delta[/eq]小于2°,这时,如果要求仪器仍然把[eq]\delta[/eq]测准,就等于要求准确分辨小于万分之几秒的时间差。以为例:

仪器是通过比较应力和应变两条曲线在横轴上的位值差来获得[eq]\Delta t[/eq]的。理论上[eq]\Delta t[/eq]再小也总大于0。但如果待测值小于仪器误差,仪器可能就搞不清两条曲线谁前谁后了,就会出现负值的[eq]\Delta t[/eq],从而出现负值的[eq]\delta[/eq]。虽然[eq]\delta[/eq]出现负值,但绝对值总之并不会太大,所以[eq]\delta[/eq]在第四象限,[eq]\cos\left(\delta\right)\gg 0,\sin\left(\delta\right)<0[/eq],所以“遭殃”的就会是[eq]G''[/eq]。

总之,[eq]G”<0[/eq]的实质是,Hookian样品损耗角[eq]\delta[/eq]的高频值已超过仪器的测量能力,测出来的值不管是正是负,都在误差范围内,已经不可靠了。

How does Google suggests for "rheology"?

Study showed recently that Google search suggestions may be misleading. The study took the word “nanotechnology” for example and showed that Google frequently directs searcher of this word to topic of health impact of nanotechnology. News reporters said this means that Google may scramble our perception of science reality.

Although I don’t buy the logic that not knowing any topic about the health impact of nanotechnology at all is helpful for a positive public image of nanotechnology, I am still interested what Google suggests for “rheology”. There is a relatively new tool in Google search — the wonder wheel. When searching for “rheology”, you can start a wonder wheel of it and explore the second-order wheel of each suggestion to “rheology”.

wonder wheel.jpg

To my surprise, “thixotropic” seems much more suggestible than “viscoelastic”. Does this mean people want to know about the former more frequently than the latter? Also more suggestible than “viscoelastic” is the simpler concept “viscosity”. “Viscoelastic” is even not in the suggestion list.

“Viscoelastic” does appears in wider list of suggestion, though.

related searches.jpg

Viscoelasticity is a fundamental concept in rheology. The study of this property started from the very beginning of the history of rheology until today. However, many that are new to rheology find it much harder to accept than the concept of viscosity or even thixotropy.

The concept of viscosity is simple. The famous Newtonian definition appears exclusively in every textbooks. And starting from this, it is quite easy to understand what is non-Newtonian. Indeed, the Newtoinan/non-Newtonian type of classification is very convenient for fluids — things that can flow, but most rheological problems is concerned by things with fluidity that depends. The abyss most students of rheology really struggling against is things that cannot be properly characterized by only viscosity but need to intorduce the measures of G‘ and G”. I have been asked too many times about the “real, real meaning” of these moduli. Even experienced materials science researchers may not understand why bother uses this complicated framework of measure to characterize a piece material.

No bother indeed, in the practical context. There are two rheometers in my research group. Colleagues from other collages or institutes often find me for rheometry. 90% of the cases are requests for simple viscosity vs shear rate curves. Only when a requester wants to “add more plots and depth” to his/her paper would he/she asked for dynamic, that is, G‘/_G_” tests. In industry, a viscosity curve gives enough information for production in most cases.

The also popular “thixotropy” should thank the industry, too. Thixotropy is hard to characterize in a scientific way (ready for structural modeling) till even today, but materials of this property are essential for the existence of the paint industry, where an ad hoc thixotropic loop test is enough in most cases. More interestingly, while the concept of viscosity cannot meet the cases where the fluidity of materials is dependent, the concept of thixotropy is just about the duality of “flow/don’t flow”, somewhat complementing the former. In the practical sense, it seems that simple extreme concept like flow and not-flow (i.e. Newtonian vs Hookian) is enough in dealing rheological problems. That’s why Google does not suggest “viscoelastic” when searched for “rheology”. This may be misleading, however, when the user wants to know about rheology academically.