MIT 3.071 Amorphous Maerials 7: Viscoelasiciy and Relaxaion Juejun (JJ) Hu 1
Afer-class reading lis Fundamenals of Inorganic Glasses Ch. 8, Ch. 13, Appendix A Inroducion o Glass Science and Technology Ch. 9 (does no cover relaxaion) Mahemaics is he language wih which God wroe he Universe. Galileo Galilei 2
Where and why does liquid end and glass begin? Wha don we know? Science 309, 83 (2005). 3
Is glass a solid or a viscous liquid? V Solid Elasiciy insananeous, ransien x E x G xy xy Glass Supercooled liquid Liquid Viscosiy ime-dependen, permanen xy xy E : Young s modulus G : shear modulus T m T : viscosiy (uni: Pa s or Poise) 4
Viscoelasiciy: complex shear modulus Consider a sinusoidally varying shear srain xy exp 0 i Elasic response: Viscous response: G G i xy exp xy i 0 i i In a general viscoelasic solid: xy xy 0 exp xy Loss modulus * G i G xy xy xy G * : complex shear modulus * G G i G ig ' " Shear/sorage modulus 5
Phenomenological models of viscoelasic maerials Elasiciy: Hookean spring G xy xy Viscosiy: Newonian dashpo xy xy Models assume linear maerial response or infiniesimal sress Each dashpo elemen corresponds o a relaxaion mechanism 6
The Maxwell elemen Serial connecion of a Hookean spring and a Newonian dashpo Toal sress: Toal srain: G E V E V E E V V 7
The Maxwell elemen E G Consan sress (creep): E V V consa n 0 E V E V G Consan srain (sress relaxaion): consa n 0 E V E exp V 1 exp G Relaxaion ime 8
The Maxwell elemen Oscillaory srain: G E E E V 0 expi V V G ' G" 2 2 G 2 2 1 1 2 2 G 9
The Voig-Kelvin elemen E G E V V Oscillaory srain: G ' G G" G Parallel connecion of a Hookean spring and a Newonian dashpo Toal sress: Toal srain: Consan srain: Consan sress: 0 exp i E E 1 exp G V G V 10
Generalized Maxwell model For each Maxwell componen: G i 1 1 i Gi i 1 G 1 1 i i Gi i i G i 2 G 2 1... i... Toal sress: G 1... i... 11
Generalized Maxwell model G Sress relaxaion: consan 0 Prony series: 1 2 G 1 G 2 G 1... i... G exp Gi i0 i G exp G R In real solids, a muliude of microscopic relaxaion processes give rise o dispersion of relaxaion ime (sreched exponenial) 12
Elasic, viscoelasic, and viscous responses Sress Sress Sress Elasic srain Viscoelasic srain Viscous srain 13
Viscoelasic maerials Mozzarella cheese Human skin Turbine blades Volcanic lava Memory foams Naval ship propellers Image of Naval ship propellers is in he public domain. Various images unknown. Lava image Lavapix on YouTube. All righs reserved. This conen is excludedfrom our Creaive Commons license. For more informaion, see hp://ocw.mi.edu/help/faq-fair-use/. 14
Bolzmann superposiion principle In he linear viscoelasic regime, he sress (srain) responses o successive srain (sress) simuli are addiive Srain Srain Srain 2 1 2 1 1 2 Sress Sress Sress 1 1 G 1 1 2 2 G 2 2 1 2 15
Bolzmann superposiion principle In he linear viscoelasic regime, he sress (srain) responses o successive srain (sress) simuli are addiive Srain i i Viscoelasic response is hisory-dependen d i 0 d ' d ' Relaxaion funcion dicaes ime-domain response Sress G i i i i i d i d d G 0 d 0 d ' ' d ' 16
V Srucural relaxaion in glass Glass ransiion Supercooled liquid Glassy sae Elasic regime Viscoelasic regime Viscous regime T re obs re ~ obs re obs All mechanical and Glass srucure and Srucural changes hermal effecs only affec aomic vibraions properies are hisorydependen are insananeous: equilibrium sae can be quickly reached Ergodiciy breakdown 17
V Srucural relaxaion in glass Glass ransiion Supercooled liquid Glassy sae Elasic regime DN 1 Viscoelasic regime DN ~1 Viscous regime DN 1 T Debroah number (DN): DN re obs he mounains flowed before he Lord Propheess Deborah (Judges 5:5) 18
Comparing sress relaxaion and srucural relaxaion V T Equilibrium sae Zero sress sae Driving force Residual sress Relaxaion kineics Exponenial decay wih a single relaxaion ime Relaxaion rae scales wih driving force Equilibrium sae Supercooled liquid sae Driving force Free volume Vf V V Relaxaion kineics Exponenial decay wih a single relaxaion ime Relaxaion rae scales wih driving force e 19
Free volume model of relaxaion (firs order kineic model) V T Relaxaion kineics a consan emperaure V f V V e exp Vf 0 re Vf Vf re Temperaure dependence of relaxaion E V f f E a V V a re exp 0 exp kt B 0 kt B 20
Model prediced relaxaion kineics Cooling rae: 10 C/s Varying reheaing rae Slope: volume CTE of supercooled liquid 1 C/s 0.1 C/s 10 C/s 21
Model prediced relaxaion kineics Varying cooling rae Reheaing rae: 1 C/s 100 C/s 10 C/s 1 C/s Ficive emperaure and glass srucure are funcions of cooling rae 22
Tool s Ficive emperaure heory V, H 3 Increasing cooling rae Supercooled liquid Ficive emperaure T f : he emperaure on he supercooled liquid curve a which he glass would find iself in equilibrium wih he supercooled liquid sae if brough suddenly o i Wih increasing cooling rae: 2 T f1 < T f2 < T f3 1 T f1 T f2 T f3 T A glass sae is fully described by hermodynamic parameers (T, P) and T f V V T, Tf H H T, Tf Glass properies are funcions of emperaure and T f (srucure) 23
Tool s Ficive emperaure heory V V, H 3 2 1 Increasing cooling rae Supercooled liquid T f1 T f2 T f3 V T, Tf H H T, Tf T Glass propery change in he glass ransiion range consiss of wo componens Temperaure-dependen propery evoluion wihou modifying glass srucure V T Volume:, Propery change due o relaxaion (T f change) T f V g H T C Enhalpy:, T f Pg V T Volume:,, f T V e V g H T C C Enhalpy:,, f T P e P g 24
Tool s Ficive emperaure heory V V, H 3 2 1 Increasing cooling rae Supercooled liquid T f1 T f2 T f3 V T, Tf H H T, Tf T Glass propery change in he glass ransiion range consiss of wo componens dv T, Tf V V dtf dt T T dt V, g V, e V, g Tf f P, g P, e P, g dt d f T dt d dh T, Tf H H dtf dt T T dt Tf f C C C dt d f T dt d 25
Predicing glass srucure evoluion due o relaxaion: Tool s equaion Consider a glass sample wih Ficive emperaure T f P (V, H, S, ec.) Glass P g - P e Supercooled liquid T T f T f - T P g P e P T P e g T Take ime derivaive: T dp P P dt d T T d g e g f dp P P d g g e re f T Assume firs-order relaxaion: dtf Tf T d re Tool s equaion 26
Tool s Ficive emperaure heory Line D'M: Tf T dtf Tf T d re dh T, Tf H H dtf dt T T dt Tf f dt CP, g CP, e CP, g d Tool s equaion f T dt d J. Am. Ceram. Soc. 29, 240 (1946) 27
Difficulies wih Tool s T f heory Riland experimen: wo groups of glass samples of idenical composiion were hea reaed o obain he same refracive index via wo differen roues Group A: kep a 530 C for 24 h Group B: cooled a 16 C/h hrough he glass ransiion range Boh groups were hen placed in a furnace sanding a 530 C and heir refracive indices were measured as a funcion of hea reamen ime Glass srucure canno be fully characerized by he single parameer T f J. Am. Ceram. Soc. 39, 403 (1956). 28
Explaining he Riland experimen n exp re,1 exp re,2 Srucural relaxaion in glass involves muliple srucural eniies and is characerized by a muliude of relaxaion ime scales Tool-Narayanaswamy-Moynihan (TNM) model 29
Wha is relaxaion? 30
Relaxaion: reurn of a perurbed sysem ino equilibrium Examples Sress and srain relaxaion in viscoelasic solids Free volume relaxaion in glasses near T g Glass srucural relaxaion (T f change) Time-dependen, occurs even afer simulus is removed Debroah Number: DN re obs DN >> 1: negligible relaxaion due o sluggish kineics DN << 1: sysem always in equilibrium DN ~ 1: sysem behavior dominaed by relaxaion 31
Modeling relaxaion Relaxaion rae P P P re e Driving force Maxwellian relaxaion models P P0 Pe exp P re P P0 Pe exp P re Bolzmann superposiion principle in linear sysems d G d ' ' d d G ' 0 d 0 d d d ' ' 0 R IRF S e e S R IRF ' ' d ' 32
The Naure of Glass Remains Anyhing bu Clear Free volume relaxaion heory V f a exp 0 kbt Tool s Ficive emperaure heory Uses a single parameer T f o label glass srucure Tool s equaion (of T f relaxaion) Srucural relaxaion in glass involves muliple srucural eniies and is characerized by a muliude of relaxaion ime scales V dtf Tf T d re E The Riland experimen 33
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