Sound and Vibration Damping with Polymers - ACS Symposium

Chapter 1

Sound and Vibration Damping with Polymers Basic Viscoelastic Definitions and Concepts L. H. Sperling

Downloaded by NEW YORK UNIV on April 17, 2015 | http://pubs.acs.org Publication Date: May 1, 1990 | doi: 10.1021/bk-1990-0424.ch001

Department of Chemical Engineering, Lehigh University, Bethlehem, PA 18015

The glass transition of a polymer involves the onset of coordinated chain motion. Dynamic mechanical spectroscopy characterizes the storage modulus, E', the loss modulus, E", and the loss tangent, tan δ, as functions of temperature and frequency. In the presence of mechanical vibrations, the vibrational energy is absorbed by the polymer in the form of heat, the basis for damping with polymers. The phenomenon resembles infrared absorption, where electromagnetic waves increase molecular motion, actually warming the sample. The mechanics of extensional and constrained layer damping are reviewed, including the use of platelet fillers. Methods of engineering the width, height, and position of the transition, such as by plasticization, statistical copolymerization, graft and block copolymerization, and interpenetrating polymer networks are described. Application of current polymer materials to sound and vibration problems are delineated. Sound and vibration damping with polymers utilizes the professions of chemistry, chemical engineering, materials, mechanical engineering, and polymer science and engineering. Each contributes its genius to an improved understanding of the nature of damping. How best shall noisy aircraft, cars, ships, machinery, etc., be quieted? The objective of this book is to provide an up-to-date introduction to the science and technology of sound and vibration damping with polymers, as well as a selection of papers showing the current status of research programs, world wide. Thus, theory, instrumentation, polymer behavior, and engineering systems will be described. The objective of this paper is to introduce those terms and concepts which will appear throughout this book. Basic definitions and concepts will be emphasized. Reviews of both the polymer characteristics (1.2) and the acoustic requirements (3,4) are available. Polymer Structure Polymers are long-chain molecules with molecular weights often measuring in the hundreds of thousands. For this reason, the term "macromolecules" is often 0097-6156/90/0424-0005$06.00/0 © 1990 American Chemical Society

In Sound and Vibration Damping with Polymers; Corsaro, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.

6

SOUND AND VIBRATION DAMPING WITH POLYMERS

e m p l o y e d w h e n referring t o p o l y m e r i c materials. A n o t h e r t e r m , frequently used i n the t r a d e l i t e r a t u r e is "resins," w h i c h goes back before the c h e m i c a l s t r u c t u r e of the l o n g chains was u n d e r s t o o d . T h e t e r m p o l y m e r itself means " m a n y mers," a n d a p o l y m e r is synthesized from its m o n o m e r s , w h i c h are l i t t l e molecules. T h e structure o f p o l y ( v i n y l acetate), widely used in d a m p i n g c o m p o u n d s , is w r i t t e n : CH

9

CH

-

O

C — CH

CH

C H — ....

2

I O — C — CH

3

II O


3

II

o

P o l y m e r s can be organic or inorganic in s t r u c t u r e . Besides p o l y ( v i n y l acetate), above, c o m m o n organic polymers include polyethylene, cis-polyisoprene ( n a t u r a l r u b b e r ) , cellulose ( c o t t o n and r a y o n ) , and p o l y ( m e t h y l m e t h a c r y l a t e ) ( P l e x i g l a s ) . Some i n o r g a n i c polymers include p o l y ( d i m e t h y l siloxane) (silicone r u b b e r ) , and o r d i n a r y w i n d o w glass. T h e glass t r a n s i t i o n (see below) is n a m e d for the softening of glass (1.2). Often p o l y m e r s are made as c o p o l y m e r s , w h i c h means t h a t they c o n t a i n t w o or more kinds o f mers. Sound and Vibrations S o u n d is air or w a t e r b o u r n acoustic v i b r a t i o n s . It is a pressure w a v e t r a n s m i t t e d t h o u g h a i r , w a t e r , or other fluid m e d i a . V i b r a t i o n s are s i m i l a r waves being t r a n s m i t t e d t h r o u g h solid objects. M o s t i m p o r t a n t l y , these pressure waves are a f o r m of energy. It is the r e m o v a l or reduction of this energy, when necessary, t h a t this book is a l l about. W h e n the sound pressure is high e n o u g h , it m a y cause deafness or reduced efficiency when people are exposed to i t . V i b r a t i o n s m a y generate s o u n d , or cause fatigue, and hence m e c h a n i c a l failure. A l l real bodies are n a t u r a l l y d a m p e d , albeit m a n y bodies o f interest d a m p o n l y m o d e s t l y . P o l y m e r s , especially near their glass t r a n s i t i o n t e m p e r a t u r e s , d a m p m u c h more. C o m m e r c i a l l y , polymers m a y be applied t o the surface o f the v i b r a t i n g substrate t o increase d a m p i n g . B o t h single-layer (extensional) and t w o - l a y e r (constrained) layer systems are i n use, albeit for somewhat different purposes. The d a m p i n g increase t h a t can be afforded by a p p l y i n g p o l y m e r i c materials t o a steel reed is i l l u s t r a t e d in F i g u r e 1 (5). T h e w a v e f o r m generated by an u n d a m p e d reed is s h o w n in F i g u r e l a , and a d a m p e d reed is shown in F i g u r e l b . Interestingly, the h u m a n ear is an i n t e g r a t i n g device, d e t e r m i n i n g i n p u t over a short period of t i m e as a single s o u n d . T h u s F i g u r e lb sounds quieter t h a n F i g u r e l a , a l t h o u g h i n i t i a l l y they are nearly the same. T h e h u m a n ear is also a p p r o x i m a t e l y a l o g a r i t h m i c device; this permits hearing at b o t h v e r y low a n d very high sound pressures. B a s i c D a m p i n g C o n c e p t s and Definitions T h e c o m p l e x m o d u l u s , E * , can be expressed as E*

=

E ' +

I E "

n

(1)

where E ' is the storage m o d u l u s , and E is the loss m o d u l u s . These quantities are i l l u s t r a t e d s c h e m a t i c a l l y i n F i g u r e 2. Here, a ball is d r o p p e d o n a perfectly elastic floor. It recovers a distance equivalent to E ' , a measure of the energy stored elastically d u r i n g the collision of the ball w i t h the floor. T h e q u a n t i t y E " represents the equivalent energy lost (as heat) d u r i n g the collision of the b a l l w i t h the floor.


1. SPERLING

Viscoelastic Definitions and Concepts

T h u s , the b a l l a c t u a l l y heats up d u r i n g the collision. heat gained, H , is

7

T h e e q u a t i o n expressing the

H = *E"el

(2)

where e represents the m a x i m u m deformation of the b a l l . A further q u a n t i t y is the loss tangent, Q


tan 6 = E " / E '

(3)

where the r a t i o of the t w o m o d u l u s quantities represents an e x t r e m e l y useful d a m p i r g quantity. T h e t w o quantities, E " and t a n 8 are usually the p r i m e p a r a m e t e r s of interest for d a m p i n g . If these quantities are s m a l l at a given t e m p e r a t u r e a n d frequency, d a m p i n g w i l l be s m a l l , a n d vice-versa. T h e several quantities are applied to crosslinked p o l y s t y r e n e i n F i g u r e 3. A t the glass t r a n s i t i o n t e m p e r a t u r e , T g , w h i c h is about 120°C at 110 H z , the storage m o d u l u s begins to decrease, w h i l e b o t h E " and tan 6 go t h r o u g h m a x i m a , respectively, as the t e m p e r a t u r e is increased. F o r d y n a m i c d a t a of this k i n d , the glass t r a n s i t i o n t e m p e r a t u r e is frequently defined as either the t e m p e r a t u r e of the loss m o d u l u s peak, or the t e m p e r a t u r e of the t a n 6 peak. Since a l l of these quantities are frequency dependent, it is convenient to define the glass t r a n s i t i o n t e m p e r a t u r e as the t e m p e r a t u r e of the m i d d l e o f the Y o u n g ' s m o d u l u s , E , elbow at ten seconds. Since this is equivalent to 0.1 H z , the s t a n d a r d value of T g can be converted t o a n y desired frequency. F i v e Regions of V i s c o e l a s t i c B e h a v i o r T h e log m o d u l u s - t e m p e r a t u r e or log m o d u l u s - l o g frequency plots of a m o r p h o u s p o l y m e r s show five distinct regions. A s shown in F i g u r e 4, at l o w t e m p e r a t u r e s a p o l y m e r w i l l be glassy w i t h a Y o u n g ' s modulus of about 3 x l 0 Pa. A s the t e m p e r a t u r e is increased, it w i l l e x h i b i t a glass-rubber t r a n s i t i o n . P h y s i c a l l y , the p o l y m e r softens i n the glass t r a n s i t i o n region, going from a glassy, plastic m a t e r i a l to either a r u b b e r y or l i q u i d m a t e r i a l above T g , depending on m o l e c u l a r weight and c r o s s l i n k i n g . A s s h o w n already i n F i g u r e 3, the storage m o d u l u s drops about three orders of m a g n i t u d e t h r o u g h about 30°C increase i n t e m p e r a t u r e . Semi-crystalline polymers e x h i b i t t w o t r a n s i t i o n s : m e l t i n g , w h i c h involves e x a c t l y the same concepts as ice or i r o n m e l t i n g , and the glass-rubber t r a n s i t i o n , w h i c h behaves quite differently. T h e t w o should not be confused ( ! ) . T h o s e polymers w h i c h are not c r y s t a l l i n e because o f irregular structure or other reasons o n l y e x h i b i t the glass t r a n s i t i o n . 9

A s the t e m p e r a t u r e is raised still further, the p o l y m e r becomes r u b b e r y , m a r k e d by the r u b b e r y p l a t e a u . If the p o l y m e r is crosslinked, it w i l l behave like a rubber b a n d i n this region. Because of extensive chain entanglement, v e r y high molecular weight p o l y m e r s also e x h i b i t a r u b b e r y plateau and c o n c o m i t a n t elastic b e h a v i o r . A s the t e m p e r a t u r e is increased still further, the r u b b e r y flow a n d l i q u i d flow regions are encountered. Since the glass t r a n s i t i o n region is the most i m p o r t a n t for sound and v i b r a t i o n d a m p i n g , special effort w i l l be expended on describing this p h e n o m e n o n . T h e glass t r a n s i t i o n region is m a r k e d by the onset of long-range c o o r d i n a t e d molecular m o t i o n . Some 10-50 backbone atoms are i n v o l v e d . A c c o r d i n g t o m o d e r n theories o f p o l y m e r m o t i o n , the chains begin to reptate back a n d f o r t h along their length at T g , resembling the m o t i o n s of a snake, see F i g u r e 5. H o w e v e r , a c c o r d i n g to increasing a m o u n t s o f recent d a t a , r e p t a t i o n as such m a y not be the m o t i o n d i r e c t l y associated w i t h T g a n d d a m p i n g . T g involves shorter range m o t i o n s ; for e x a m p l e , b r a n c h i n g suppresses r e p t a t i o n , but has l i t t l e effect on T g . A l s o , the area under the




b Fig. 1. Oscilloscope traces of vibration decay. Characteristics of an undamped reed(a), and a damped reed(b). Steel reed damped with an IPN constrained layer system. (Reprinted with permission from ref. 5. Copyright 1975 Wiley.)

F i g . 2. T h e ball a c t u a l l y gets w a r m e r o n b o u n c i n g . T h e c r i t i c a l t i m e is t h e t i m e the b a l l remains i n c o n t a c t w i t h floor; this defines t h e equivalent frequency.

10

0

b

8

HI

-1

0

o U)

tan

v

Master Curve at T*

Relaxation Modulus

Log time or -Log frequency F i g . 8. S c h e m a t i c representation o f the c o n s t r u c t i o n o f a master curve a t T . D a t a are slid h o r i z o n t a l l y either left o r right u n t i l they o v e r l a p d a t a at t h e t e m p e r a t u r e o f interest. A master curve can be used t o predict results a t v e r y l o n g o r short times, o r e q u i v a l e n t l y , v e r y short o r long frequencies. 3


1. SPERLING

13


where e represents the s t r a i n . constant stress,

E q u a t i o n (5') can be i n t e g r a t e d under c o n d i t i o n s of

E

e< ^)

(6')

(1 - t / r )

(6")

e = f (l In terms of the r e t a r d a t i o n t i m e , e = | where r

2

2

equals 77/E.

T h e o r i e s o f Effective D a m p i n g . If one assumes t h a t a range o f temperatures a n d / o r frequencies w i l l be encountered, then the area under the d a m p i n g c u r v e ( E or t a n 6) determines the effectiveness of the p o l y m e r better t h a n the hight of the t r a n s i t i o n alone, see F i g u r e 11. F o r E " , the effective area is called the loss area, L A . T h i s is d e t e r m i n e d after s u b t r a c t i n g the b a c k g r o u n d , as i n any spectroscopic e x p e r i m e n t . T h e r e are t w o theories t o determine the q u a n t i t y L A . T h e first is the phenomenological m e c h a n i c a l t h e o r y . O n e f o r m o f the e q u a t i o n reads (6-9), ™


; /

E"dT JT

= (E'

- E' )

G

R

G

R ;

7s E

(^ a)

2

Tg 6

(7)

a c t

where E ^ a n d E n represent Y o u n g ' s m o d u l i in the glassy a n d respectively, j u s t before and after the glass t r a n s i t i o n . A n o t h e r form o f the equation reads (10), 2

E " d ( l / T ) = (E'

Q

- E' ) ( R / A E R

a c t

) TT

rubbery

states,

2

(8)

G

These t w o equations are useful in p r o v i d i n g energies o f a c t i v a t i o n , A E j . (11), but require three parameters for useful predictions o f a c t u a l L A values. A n alternate a p p r o a c h makes use o f a group c o n t r i b u t i o n analysis (7.9) approach. T h e g r o u p c o n t r i b u t i o n analysis m e t h o d has p r o v e d useful for the d e t e r m i n a t i o n o f s o l u b i l i t y parameters f 12.13). and has p r o v i d e d an a p p r o a c h for a host o f other c h e m i c a l properties (14). In a g r o u p c o n t r i b u t i o n analysis, each m o i e t y in the c h e m i c a l or p o l y m e r contributes a d d i t i v e l y t o the p r o p e r t y in question. B y analogy w i t h the s o l u b i l i t y parameter a p p r o a c h , the loss area, L A , for the area under the loss m o d u l u s - t e m p e r a t u r e curve in the v i c i n i t y o f the glass-rubber t r a n s i t i o n is given b y (7.9) n (LA).M. a c

LA =

E

(9)

1=1 where M - represents the molecular weight o f the i ^ m o i e t y , a n d M is the molecular weight of the whole mer. T h e a c t u a l analysis of L A v i a this m e t h o d w i l l be e x a m i n e d further i n a C h a p t e r i n this book by F a y , et a l . T h e r e are several hidden assumptions in this t h e o r y . T h e first is t h a t the m o d u l u s i n the glassy state is c o n s t a n t . T h i s is t r u e for a wide range of p o l y m e r s , w i t h E Q in e q u a t i o n (7) being about 3 x l 0 P a (3xl0 d y n e s / c m ) . Second, t h a t E is m u c h smaller t h a n E Q , SO its value can be i g n o r e d . U s u a l l y , E ' drops about three orders o f m a g n i t u d e t h r o u g h T g . Because the decline in E ' w i t h t e m p e r a t u r e is not k n o w n q u a n t i t a t i v e l y , the equivalent t h e o r y for t a n 6 has yet to be w o r k e d o u t . n

9

1 0

2

R

Damping and Dynamic Mechanical Spectroscopy. Dynamical spectroscopy means d a t a taken w i t h cyclical d e f o r m a t i o n o f the sample.


mechanical O f t e n , the


14


Element

Element

4-Element Model

F i g . 9. M o d e l s for a n a l y z i n g d a m p i n g . T h e springs follow H o o k e ' s l a w a n d c o n t r i b u t e m o d u l i , while the dashpots follow N e w t o n ' s l a w a n d i n d i c a t e viscous contributions.

Log E" tan $

Log t or -Log N F i g . 10. T h e quantities E a n d t a n 6 peak when t h e n a t u r a l frequency o f c h a i n m o t i o n equals t h e e x t e r n a l v i b r a t i o n a l frequency. A t the peak, the r e l a x a t i o n t i m e , r a p p r o x i m a t e l y equals the t i m e o f the experiment (or the inverse o f the frequency). / ;

1 ?

Background

T, Log t, or -Log N F i g . 11. G r o u p c o n t r i b u t i o n analysis leads t o the d e t e r m i n a t i o n o f t h e effective area under the loss m o d u l u s - t e m p e r a t u r e curve. A s i n a n y spectroscopic e x p e r i m e n t , b a c k g r o u n d must be s u b t r a c t e d , a n d the i n s t r u m e n t c a l i b r a t e d .


1. SPERLING


15

e x p e r i m e n t involves a s i n u s o i d a l m o t i o n . T h e r e are several w a y s o f representing this i n f o r m a t i o n , depending o n the objective. If an a p p l i e d stress varies w i t h t i m e i n a s i n u s o i d a l m a n n e r , the stress, represents the a n g u l a r frequency i n radians, equal to 27rxfrequency. H o o k i a n solids, w i t h no energy d i s s i p a t e d , the s t r a i n , e, is given b y e = e

sin ut


Q

For

(11)

where the quantities w i t h subscript zero represent the a m p l i t u d e values. F o r real m a t e r i a l s w h i c h e x h i b i t d a m p i n g , the stress a n d s t r a i n are not i n phase, the s t r a i n l a g g i n g b e h i n d the stress by the phase angle, 6, the same angle w h i c h appeared i n t a n

Sound and Vibration Damping with Polymers - ACS Symposium

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