Computer Simulation of the Diffusivity of Cement Based

8/3/2019 Computer Simulation of the Diffusivity of Cement Based

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J O U R N A L O F M A T E R I A L S S C I E N C E 2 7 ( 1 99 2 ) 2 0 8 3 - 2 0 9 2

C o m p u t e r s i m u l a t i o n o f t h e d i f f u s i v i t y o fc e m e n t - b a s e d m a t e r i a l sE. J . G A R B O C Z I , D . P. B E N T ZNational Inst i tute of Standards and Technology, Bui ld ing Mater ia ls Div is ion (226/B348),Gaithersburg, Mary land 20899, USA

A d ig i ta l image-based model o f the m ic ros t ruc ture o f cement pas te , coup led wi th exac tt ranspor t a lgor i thms, is used to s tudy the d i f fus i v i ty o f Por t l and cement pas te . The pr in c ip a lvar iab les cons idered are water : ceme nt ra t io , degree o f cem ent hydrat i on and cap i ll a ryporos i ty . Computat iona l methods are descr ibed and d i f fus i v i ty resu l ts are presented, wh ich arefound to agree wi th the ava i l ab le exper imenta l measurements w i th in exper imenta l er ror . Modelcem en t pastes prepared w i th di f feren t wate r : cem ent ratios, and h av ing di f feren t degrees ofhydra t i on, are foun d to have d i f fus i v it i es tha t l ie on a s ing le mas ter curve wh en p lo t ted as afunc t ion o f cap i l l a ry poros i ty . Concepts f rom perco la t i on theory are used to exp la inquant i ta t i ve l y the dependence o f d i f fus i v i ty on cap i l l a ry poros i ty . The e f fec t o f s i l i ca fumeaddi t i on on d i f fus i v i ty i s a l so examined.

1 . I n t r o d u c t i o nT h e t r a n s p o r t p r o p e r t i e s o f p o r o u s m a t e r i a l s h a v ebe e n a sub j e c t o f sc ie n t i fi c i n t e re s t fo r m a ny ye a r s , a ndh a v e b e c o m e t h e f o c u s o f m u c h a t t e n t i o n i n t h e l a s t1 0 - 2 0 y e a r s , p r i m a r i l y d u e t o w o r k c a r r i e d o u t i n t h eo il -w e ll lo g g i n g a n d e x p l o r a t io n c o m m u n i t y o n p o r -o u s s e d i m e n t a r y r o c k s . T h e r e t h e i n t e r e s t h a s b e e n i np r e d i c t i n g t r a n s p o r t c o e f fi c ie n t s b a s e d o n m i c r o s t r u c -r u r a l p a r a m e t e r s o f t h e p o r e s p a c e . T h e s e t r a n s p o r tc oe f f i c i e n t s i nc l ud e t he e l e c t r i c a l c ond uc t i v i t y o f t hepore spa c e [1 ] , t he d i f fus i v i t y o f t he pore spa c e (whi c hi s r e l a t e d t o t h e c o n d u c t i v i t y t h r o u g h a n E i n s t e i nre l a t i on [2 , 3 ], a nd t he f l u i d pe rm e a b i l i t y [4 ] .

C e m e n t - b a s e d m a t e r i a l s a r e p o r o u s m a t e r i a l sw h o s e t r a n s p o r t c o e f fi c ie n t s a r e o f i n te r e s t, b u t f o rd i f f er e n t r e a s o n s t h a n f o r r o c k s . T h e f o c u s o f i n t e r e s ti n r o c k s h a s b e e n f l u i d p e r m e a b i l i t y , w h i c h i s n o tu n r e a s o n a b l e a s o i l i s a f l u id t h a t m u s t b e p u m p e dt h r o u g h a n d o u t o f t h e p o r o u s r o c k s i n w h i c h i t i sf o u n d . I n c e m e n t - b a s e d m a t e r i a l s , t h e t r a n s p o r t o fd i s s o l v e d c h e m i c a l s p e c ie s t h r o u g h t h e p o r e s p a c e i s o fm o re s i gn i f i c a nc e , fo r t he fo l l ow i ng re a sons . F i r s t l y ,m o s t o f t h e p h y s i c o c h e m i c a l p r o c e s se s t h a t d e g r a d ec e m e n t - b a s e d m a t e r i a l s a n d u l t i m a t e l y d e t e r m i n e s e r -v i c e l i f e de pe nd on a supp l y o f i on i c spe c i e s f rome x t e rna l sourc e s [5 ] . The r a t e a t wh i c h t he se spe c i e sc a n m o v e t h r o u g h t h e p o r e s t r u c t u r e l a r g e ly d e t e r -m i n e s t h e r a t e a t w h i c h d e g r a d a t i o n p r o c e e d s . E x -a m p l e s i n c l u d e c h l o r i d e i o n s a t t a c k i n g r e i n f o r c i n gs t e e l i n c o n c r e t e , a n d s u l p h a t e i o n s r e a c t i n g w i t ha l u m i n a t e p h a s e s i n c o n c r e t e t o p r o d u c e c r a c k - c a u s -i n g i n t e r n a l e x p a n s i v e p r e s s u r e s [ 6 ] . S e c o n d l y , t h e r eh a s b e e n m u c h r e c e n t i n t e r e s t i n u s i n g c e m e n t - b a s e dm a t e r i a l s t o c o n t a i n l o w - a n d i n t e r m e d i a t e - l e v e lr a d i o a c t i v e a n d a l s o t o x i c w a s t e [ 7 ] . T h e t r a n s p o r tc oe f f i c i e n t s o f t he se m a t e r i a l s a re t he ke y f a c t o r s t ha twi l l de t e rm i ne t he i r e f f e c t i ve ne ss a s ba r r i e r s .

A l t h o u g h b o t h f l u i d p e r m e a b i l i t y a n d i o n i c d i f f u s -i v i t y a r e i m p o r t a n t t r a n s p o r t c o e f fi c ie n t s f o r c e m e n t -ba se d m a t e r i a l s , t h i s pa p e r foc u se s on t he d i f fus i v i ty . Ad i g i t a l i m a g e - b a s e d g r o w t h m o d e l o f t h e d e v e l o p i n gm i c r o s t r u c t u re o f c e m e n t p a s t e d u r i n g h y d r a t i o n i sc o u p l e d w i t h t w o a l g o r i t h m s f o r c o m p u t i n g t h e c o n -d u c t a n c e o f r a n d o m c o n d u c t o r n e t w o r k s , in o r d e r t oc a r r y o u t t h e c o m p u t a t i o n s t o b e d e s c r i b e d b e l o w .P r e l i m i n a r y a c c o u n t s o f s o m e o f t h is w o r k h a v ea p p e a r e d p r e v i o u s l y [ 8 , 9 ] .

2 . D i g i t a l i m a g e - b a s e d m i c r o s t r u c t u r a lm o d e l2.1 . Cem ent hydrat i onT h e m i c r o s t ru c t u r e o f c e m e n t p a st e i s k n o w n t o b ec o m p l e x [ 1 0 ] . T h i s is n o t s u r p r i s i n g , a s c e m e n t p a s t e i sf o r m e d f r o m a d i s o r d e r e d a q u e o u s s u s p e n s i o n o f ir -r e g u l a r l y s h a p e d c e m e n t p a r t ic l e s, w h i c h u n d e r g o r a n -d o m g r o w t h d u e t o h y d r a t i o n r e a c t i o n s . S i n c e t h eor i g i na l c e m e n t pa r t i c l e s ha ve a wi de s i z e d i s t r i bu t i ona nd a n a ve ra ge s i ze o f 15 -20 I- tm [11] , t he c om pl e xm i c r o s t r u c t u r e o f c e m e n t p a s t e e x t e n d s o v e r m a n yl e n g t h s c a le s , f r o m s m a l l f r a c t io n s o f a m i c r o m e t r e t ot e n s o f m i c r o m e t r e s .

N e g l e c t i n g c h e m i c a l d e t a i l s (w h i c h a d m i t t e d l y i s a nove r s i m pl i f i c a t i on ) t he r e a c t i ve g rowt h p roc e ss t h a tc e m e n t p a r t ic l e s u n d e r g o t o p r o d u c e c e m e n t p a s t e c a nb e t h o u g h t o f in t h e f o l lo w i n g s im p l e w a y [ 1 2 ] . T h es o l i d c e m e n t p a r t i c l e s s u p p l y c a l c i u m i o n s t o t h es u r r o u n d i n g w a t e r t h r o u g h d i s s o l u t i o n o f s u r f ac el a ye r s . The s e i ons t h e n r e a c t w i t h s i l ic a - r i c h su r fa c es o fc e m e n t p a r t i c l es t o f o r m s o l id r e a c t i o n p r o d u c t s ( s u r-f a c e p r o d u c t s ) c o v e r i n g t h e c e m e n t p a r t ic l e s , o r s p o n -t a n e o u s l y n u c l e a t e i n t h e p o r e s p a c e t o f o r m c r y s t a l s( p o r e p r o d u c t s ) , w h i c h c a n t h e n g r o w f u r t h e r b y a c c r e-t i o n . I n c e m e n t p a s t e , t h e m a i n s u r f a c e p r o d u c t ,

0022-2461 9 1992 Chapman & Hall 2 0 8 3


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c a l c i u m s i l i c a t e h y d r a t e , i s d e n o t e d C - S - H , a n d t h em a i n p o r e p r o d u c t , c a l c iu m h y d r o x i d e , is d e n o t e d C H ,w h e r e t h e u s u a l c em e n t c h e m i s t r y s h o r t h a n d n o t a t i o ni s C = C a O , S = S i O 2 , H = H 2 0 , A = A 1 2 0 3 , a n dF = F e 2 0 3.T h e r e a s o n t h a t c e m e n t h y d r a t i o n c a n p r o d u c e ar i g i d s o l i d f r o m a v i s c o u s s u s p e n s i o n o f c e m e n t p a r -t i c l e s i n w a t e r i s t h a t t h e h y d r a t i o n r e a c t i o n p r o d u c t sh a v e a l a r g e r v o l u m e t h a n t h e s o l i d r e a c t a n t s . A s t h eh y d r a t i o n p r o c e s s i s n e a r l y a c o n s t a n t t o t a l v o l u m eproc e ss , t he r e a c t i on p rod uc t s c a n f i ll i n t he i n i t i a l l yw a t e r - f i l l e d p o r e s p a c e , e v e n t u a l l y f o r m i n g a r i g i ds o l id b a c k b o n e c a p a b l e o f b e a r i n g m e c h a n i c a l l o a d s .I t i s c o n v e n i e n t t o d e f in e t h e f o l l o w i n g v o l u m e r a t io s .13 i s t he r a t i o o f t he v o l um e o f su r fa c e p rod uc t sp r o d u c e d t o t h e v o l u m e o f c e m e n t r e a c t e d ; a n d [3p i st h e a n a l o g o u s r a t i o f o r t h e p o r e p r o d u c t s . T h e t o t a lv o l u m e e x p a n s i o n f a c t o r i s d e f i n e d a s [ 3T = ~s + [3p.T y p i c a l r a n g e s f o r t h e s e p a r a m e t e r s , f o r v a r i o u st ype s o f P or t l a n d c e m e n t s , a re 1 .6 < [3s < 1 .9 , a nd0 .4 < [3p < 0 .7 [13] . The se pa ra m e t e r s i nc l ud e t here a c t i on o f t r i c a l c i um s i l i c a t e (C3S ) , d i c a l c i um s i l i c a t e( C 2 S ) , a n d t h e l e s s a b u n d a n t a l u m i n a t e p h a s e s . T h ev e r y s m a l l a m o u n t o f f e r ri te p h a s e s p r e s e n t i n c e m e n tw a s i g n o r e d i n [ 1 3 ] . S o m e w h a t s u r p r i s i n g l y , f o r ava r i e t y o f c e m e n t s , t he va l ue o f ~T i s f a i r l y c ons t a n t ,a ro un d 2 .3 +_ 0 .1 [ -13] . In t h e s i m ul a t i o n m o de l t o bedes cr ibe d nex t , [3 i s tak en t o be 1.7, an d [3p i s tak en tobe 0 .61 , so t h a t 13a -= 2 .31 . The se p a r t i c u l a r va l ue sa r e t h o s e r e a l iz e d i n th e h y d r a t i o n o f p u r e C 3 Sc e m e n t [ 1 4 ] .

2 . 2 . M i c r o s t r u c t u r a l m o d e lI n t h e l as t d e c a d e , a n u m b e r o f r a n d o m g r o w t h o ra g g r e g a t i o n m o d e l s h a v e b e e n d e v e l o p e d . T h e s e m o -d e l s , w h i c h e m p l o y v e r y s i m p l e r a n d o m g r o w t h r u l e s ,h a v e b e e n s h o w n t o p r o d u c e c o m p l e x a g g r e g a t e ds t r u c t u r e s , o f t e n w i t h f r a c t a l m o r p h o l o g y . T w o e x -a m p l e s a r e t h e d i f f u s i o n - l i m i t e d a g g r e g a t i o n ( D L A )[ 1 5 ] a n d E d e n m o d e l s [ 1 6 ] . I n t h e l i g h t o f t h e s em o d e l s , i t i s n o t u n r e a s o n a b l e t o s u g g e s t t h a t t h ec o m p l e x m i c r o s t r u c t u e o f c e m e n t p a s te m i g h t b es i m u l a t e d u s i n g a f e w r e l a t i v e l y s i m p l e g r o w t h r u l e s ,w h i c h a r e r e p e a t e d m a n y t im e s . T h e m o d e l u s e d i n th i sp a p e r r e p r e s e n t s a r e a l i z a t i o n o f t h i s a p p r o a c h .

T h e m o d e l o p e r a t e s o n s q u a r e o r c u b i c a r r a y s o fp i xe ls , t yp i c a l l y o f e dge l e ng t h e qu a l t o 500 p i xe l s i n2D, a nd 100 p i xe l s i n 3D, whe re e a c h p i xe l i s a ss i gne dt o a s i ng l e pha se , suc h a s po re spa c e o r c e m e n t .In i t i a l l y , a spe c i f ie d num be r o f c e m e n t pa r t i c l e s( ~ 2 0 0 0 i n 3 D ) a r e r a n d o m l y p l a c e d i n th e u n i t c e lls u c h t h a t n o t w o p a r t i c l e s o v e r l a p , s i m u l a t i n g t h em i x i n g p r o c e s s . P e r i o d i c b o u n d a r y c o n d i t i o n s a r euse d t o e l i m i na t e a ny a r t i f i c i a l e dge e f fe c t s a t t he c e l lw a l ls . T h e p a r t ic l e s m a y b e f r o m a n y s iz e d i s t r ib u t i o n ,wi t h i n t he r e so l u t i on l i m i t s o f t he un i t c e ll (1 -100p i xe l s ) . Mode l pa r t i c l e sha pe s , l i ke c i r c l e s i n 2D o rs p h e r e s i n 3 D , m a y b e u s e d , b u t s i n c e t h e m o d e l i sb a s e d o n a d i g i t a l i m a g e r e p r e s e n t a t i o n o f th e c e m e n tpa r t i c l e s , d i g i t i z e d m i c rogra phs o f a c t ua l pa r t i c l es h a p e s c a n a l s o b e u s e d a s a s t a r t i n g p o i n t i n 2 D .A l l t h e s i m u l a t i o n s d e s c r i b e d i n t h i s p a p e r a r e 3 Ds i m u l a t i o n s .

T h e m o d e l o p e r a t e s b y t h e i t e r a t i o n o f cy c le s . E a c hc yc l e c ons i s t s o f t h re e s t e ps : d i sso l u t i on , d i f fus i on ,a n d r e a c t io n . F i g . 1 d e s c ri b e s t h e g r o w t h p r o c e s ssc he m a t i c a l l y .I n t h e d i s s o l u t i o n s t e p, a n y c e m e n t p i x el s in c o n t a c tw i t h a w a t e r- f i ll e d p o r e - s p a c e p i x e l a t t e m p t t o t a k e as t e p i n a r a n d o m d i r e c t i o n . T h e p i x e l s w h o s e s t e pl a n d s t h e m i n t h e p o r e s p a c e d i s s o l v e , a n d e a c h s u c hp i x e l t u r n s i n t o a r a n d o m d i f f u s e r . T h e p i x e l s w h o s er a n d o m s t e p w o u l d l a n d t h e m i n a s o l i d p h a s e a r e n o ta l l o w e d t o m o v e , a n d s o r e m a i n a t t h e i r o r i g i n a ll o c a t i o n , u n d i s s o l v e d . T h e n u m b e r o f pi x el s t h a t d i s -s o l v e a r e c o u n t e d , a n d t h e c o r r e c t n u m b e r o f e x t r ad i f f u si n g p i x el s a re a d d e d a t r a n d o m l o c a t i o n s w i t h i nt he pore sys t e m , r e p l a c i ng pore spa c e p i xe l s , t o a c -c o u n t f o r t h e c o r r e c t a m o u n t o f s u r f ac e a n d p o r ep r o d u c t f o r m a t i o n . M o r e p r e c is e ly , if n p i x el s d i s so l v ef rom of f t he c e m e n t su r fa c e s , 13pn pore p ro du c t a nd([3s - 1 )n e x t ra su r fa c e p roduc t -d i f fus i ng p i xe l s a rea d d e d t o t h e s y s t e m a t r a n d o m l o c a t i o n s i n t h e p o r es p a c e , i n o r d e r t o a c h i e v e t h e c o r r e c t v o l u m e o fh y d r a t i o n p r o d u c t s a r i s i n g f r o m t h e r e a c t i o n o ft he n d i sso l ve d c e m e n t p i xe l s .

D u r i n g t h e d i f f u s i o n / r e a c t i o n s te p s , th e d i s s o l v e dp ix e ls m o v e b y e x e c u t i n g r a n d o m w a l k s t h r o u g h o u tt h e p o r e s p a c e . S u r f a c e - p r o d u c t p i x e l s c o n t i n u e t om o v e i n t h i s r a n d o m f a s h i o n u n t i l t h e y e n c o u n t e r ac e m e n t s u r fa c e , a t w h i c h p o i n t t h e y r e a c t a n d a t t a c h t o

Dissolvedixel

Dissolution Diffusion Reactionf~- - - I - I - - - ~

1. 7 d i f f u s i n gC S H s p e c i e s

F o r m s

.~ -- V-I ----~

0.61 d i f f u s i n gC H s p e c i e s

Pre cipitates into solidCSH upon contacting asol id C3S or CSH surface

Nucleates into solid CH orprecipitates into solid CHupon contacting a solidCH surface

Figure 1 S c h e m a t i c d i a g r a m o f c e m e n t p a s t e m i c r o s t r u c t u r a l d e v e l o p m e n t a l g o r i t h m .2 0 8 4


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this surface. Once surface products are present , dif-fus ing surface-product p ixel s can react and a t t ach tothese surfaces as well . For any given s tep taken by adiffusing pore -pro duc t pixel , however, there is a non-zero probabi l i ty that i t can nucleate at i ts presentlocat ion . This p robabi l i ty decreases exponent ia l ly asthe num ber of d if fus ing pore-product p ixels decreases[17]. After a pore-product cluster has been nucleated,other diffusing pore- produ ct pixels can aggreg ate on tothe cluster upon contact . When al l diffusing pixelshave reacted or nucleated, the cycle is complete, andthe next cycle begins with a new dissolut ion s tep.Micros t ructura l development i s complete when a l lcement pixels have reacted, or when al l remainingcement pixels are covered by surface product , andtherefore are no longer avai lable for dissolut ion. In3-d s imulat ions, usual ly up to 90 % of the originalcement can be hydrated using these s imple rules .

The degree of hydrat ion ach ieved af ter any com-pleted cycle is determined from analysis of the micro-structure. Deg ree of hyd rat io n, ~, is defined as thef rac t ion of the or ig inal cement tha t has been reacted ,so that ~ equals 0 when the cement part icles are fi rs tmixed wi th water , and a t t a ins a value of un i ty whenhydr at ion i s complete . Anothe r param eter tha t i s eas-i ly ca lcu la ted in the model i s the water :cement (w/c)rat io, which is the weight rat io of water to ceme nt inthe in i t i a l cement -water mix ture , a parameter o f tenquo ted in the cement l i terature. If is the solid volum efract ion , and 1 - f i s the water vo lume f rac t ion , thenthe water :cement ra t io i s g iven by

w 1 - f- ( 1 )c 3.2f

where 3.2 is the specific gravi ty of port land cement .The or ig inal poros i ty o f a cement /water mix ture , de-fined as the volum e fract ion of pore space, is then alsoequal to f

3 . Percola t io n pr opert ies of thepore spaceAs cement hydrat ion progresses, the pore space isgra dua lly be ing filled beca use the f acto r [~T is gre atertha n one. The co nnect ivi ty of the pore space as afunct ion of hydra t ion i s a perco la t ion prob lem. In th ispaper the term 'pore space' refers to capi l lary porespace, the water-fi l led space between the cement par-t icles and their react ion products that is left overf rom the or ig inal cement -water mix ture . There aremicropores in the C-S H surface product mater ia l ,which form cont inuous pathways ca l l ed 'ge l pores ' .However , t ranspor t p roper t i es are dominated by themuch larger capi l lary pores as long as they percolate,i .e. form a c ont in uou s pathw ay. If the capi l lary poresclose off , however , then t ransp or t mus t be dom inatedby the much smal ler C-S-H gel micropores . There i sno sharp s ize cut-off betwee n capi l lary an d gel pores .The capi l lary pores are considered to have a s izeranging f rom hundreds of micromet res dow n to t ensof nanomet res , wi th the upper end of the C -S -H gelpore-size dis t r ibut ion overlapping the lower endof the capi l lary pore s ize range [11].

Since the microstructural model is represented asa digi tal image, there is an underlying lat t ice in thes t ructure o f the model . Therefore a l l the computa-t ional techniques developed for lat t ice percolat ionproblems can be carried over to analyse digi t izedcont inuum s t ructures l ike the cement pas te model .Fo r instance, the fract ion of the pore space t hat is partof the pe rcolat ing cluster is easi ly determin ed u sing a'burning algori thm' 1-18].

Recent work us ing the micros t ructura l model [13]has shown that the cap i l l ary pore space of cementpaste does have a percolat ion threshold, at a capi l laryporo si ty qb of abou t 18%, o r qbc = 0.18. This thresh-old qb is indep ende nt of the ini t ial poros i ty orwater :cement ra t io [13] . Also , the C-S-H surfaceproduc t phase i t se l f has a p erco la t ion th reshold , andchanges f rom d i scont inuous to cont inuous a t a vo l -ume f rac t ion of about 17%. The c lose agreement o fthe two th resholds wi th a con jecture by Scher andZallen [19, 20] as to the value of a 'universal ' con-t inuum perco la t ion th reshold of 16% in 3D has beennoted and d i scussed [13] . For typ ical w/c ratios, theC-S-H phase perco la tes qu i te ear ly in the hydrat ionprocess , and i s con t inuous s imul taneous ly wi th thecapi l lary pore space.

The perco la t ion theory-based descr ip t ion of thedependence of d if fusiv ity on cement pas te m icros t ruc-ture will be discussed mor e ful ly in sect ion 6.

4 . C o m p u t a t i o n a l m e t h o d s4.1. P rob lem def in i t ionThe problem being cons idered i s tha t o f a com plete lywater -sa tura ted porous hardened cement pas te . Acon cent rat ion g radien t of dissolved ions exis ts acrossthe sam ple, so tha t the re is a net diffusive flow of ionsthrough the water -sa tura ted pore space . A s teady s ta tei n reg a rd t o an y ad s o rp t i o n ~ l e s o rp t i o n p h en o m en a isassumed to have been establ ished, so that the net f lowis truly diffusive, and indep ende nt of t ime [-3]. Un derthese condi t ions , the Nerns t -Eins te in re la t ion con-nects the electrical cond uct ivi ty of the materia l with i tsdiffusivi ty [2, 3]. Fig. 2 i l lustrates the p hysical co ntentof this relat ion. If D o is the diffusivi ty of the ions beingconsidere d in free water, and cro is the co nduc t ivi ty ofthe solut io n in the pore space, then the resu l t of theNerns t -Eins te in re la t ion i s tha t

Figure 2 Schematic diagram of the physical content of theNernst-Einstein relation relating the diffusivityand electrical con-ductivity of a porous m aterial.2 0 8 5


4/10

w h e r e D a n d ~ a r e t h e m e a s u r e d d i f f u s i v i ty a n de l e c t r i c a l c o n d u c t i v i t y o f t h e f l u i d - s a t u r a t e d m a t e r i a l .D i f f u si o n c o e f fi c ie n t s c a n b e m e a s u r e d e x p e r i m e n t a l l yb y th e a p p l i c a t io n o f a c o n c e n t r a t i o n g r a d i e n t , - V c,a n d e l e ct r ic a l c o n d u c t i v i ti e s b y t h e a p p l i c a t i o n o f ap o t e n t i a l g r a d i e n t , - V V . T h e q u a n t i t y D / D o is a lsos o m e t i m e s c a l l e d t h e d i f f u s i b i l i t y [ 3 ] , o r t h e r e l a t i v ed i f f u s i v i t y [ 8 ] . T h e l a t t e r t e r m i s u s e d i n t h i s p a p e r .

4 .2 . Random conduc t o r ne t wo rkT h e c o m p u t a t i o n a l a p p r o a c h t a k e n i n t h i s p a p e r i s t oe x p l o i t t h e r e l a t i o n s h i p g i v e n i n E q u a t i o n 2 , b yc o n v e r t i n g t h e d i g i t a l - i m a g e m o d e l i n t o a r a n d o mc o n d u c t o r n e t w o rk , a n d t h e n c o m p u t i n g t h e c o n d u c t -i v i ty u s in g o n e o f t w o c o n d u c t i v i t y a l g o r i t h m s t h a tw e r e d e v e l o p e d f o r s i m p l e l a t t i c e p r o b l e m s .

T h e m e t h o d o f c o n v e r t i n g th e d i g i ta l - im a g e m o d e li n t o a c o n d u c t o r n e t w o r k i s a s fo l l o w s , a n d i s i l l u s t ra -t e d s c h e m a t i c a l l y i n F i g . 3 , w h i c h s h o w s t h e r e s u l t i n gc o n d u c t o r n e t w o r k s u p e r im p o s e d o n a n o r ig i n a l ra n -d o m 2 D i m a g e . A f t e r a d i g i t a l i m a g e c e m e n t p a s t em o d e l i s g e n e r a t e d , a 1 p i x e l - t h i c k e l e c t r o d e i s ' g l u e d 'o n o p p o s i n g f a ce s o f t h e u n i t c u b e . A n e t w o r k o f n o d e si s c r e a t e d , w i t h o n e n o d e a t t h e c e n t r e o f e a c h p i x e l .C o n d u c t o r s w i t h c o n d u c t a n c e E q a r e t h e n s e t u pw h i c h c o n n e c t t h e n o d e s i n n e a r e s t - n e i g h b o u r p i x el s ia n d j , w h i c h t h e m s e l v e s h a v e c o n d u c t i v i ti e s ch and cy j ,r e s p e c t i v e ly . T h e c o n d u c t a n c e E ~ j i s d e f i n e d a s t h es e r ie s c o m b i n a t i o n o f E ~ a n d Z j

1Z,j - 1 /2 , + 1 /Zj (3)

w h e r e E ~ i s th e c o n d u c t a n c e o f o n e h a l f o f p i x e l i. T h i sm e a n s t h a t E i = ~i d2/( 0.5d ) = 2 c~id, w h e r e d i s t h ee d g e l e n g t h o f o n e p i x e l . I f p i x e l s i a n d j a r e b o t hc a p i l l a r y p o r e s p a c e p i x e l s , f o r e x a m p l e , t h e n ch = cyj= l , so t ha t Z i j = d . I f e i th e r p ixe l i o r p ixe l j i s a

c e m e n t o r p o r e - p r o d u c t p i x e l , t h e n Z i j = 0 , s in c eeither ~z or c~j is zero.

T h e s u r f a c e - p r o d u c t ( C - S - H ) p i x e l s a r e t a k e n t oh a v e a s m a l l n o n - z e r o c o n d u c t i v i t y , b e c a u se o f t h e

Figure 3 Schem atic diagram of the dig ital image to ran dom co n-ductor netw ork mapping used to co mpu te the electrical conductiv-ity of the cem ent paste model. Conductances of the different ypes ofbonds are give n in the text.2086

s u r fa c e p r o d u c t ' s c o n t i n u o u s g e l m i c r o p o r e s . T h e d i f-f u s i v i ty r e s u l ts w e r e a l l o b t a i n e d u s i n g C ~ c_ s_H= 0 . 0 0 2 5 , w h i c h w a s b a s e d o n e x p e r i m e n t a l d a t a

f r o m c h l o r i d e i o n d i f f u si v i t y m e a s u r e m e n t s I -3 , 2 13 .A s s ig n i n g a b u l k c o n d u c t i v i t y t o th e C - S - H p h a s e i sa n a p p r o x i m a t i o n , s i n c e th e s e p o r e s a r e s o s m a l l t h a tt h e f l o w o f d i ff u si n g i o ns o r t h e m o v e m e n t o f c h a r g e dp a r t i c l e s i n t h e m p r o b a b l y d i f f e r s g r e a t l y f r o m b u l kp r o c e s s e s 1 '-33, a n d t h u s d e p e n d s o n t h e i o n c o n s i d e r e d .F o r e x a m p l e , it is k n o w n t h a t c a e s i u m ( C s i o nd i f f u s i v i t i e s a r e s y s t e m a t i c a l l y s m a l l e r t h a n c h l o r i d ei o n d i f f u si v i ti e s m e a s u r e d i n t h e s a m e m a t e r i a l [ 3 ] ,s o a d if f er e n t v a l u e f o r ~ c - s - n w o u l d b e n e e d e d t od e s c ri b e c a e s i u m - i o n d i ff u si o n . A c o n d u c t o r c o n n e c -t i n g a C - S - H p i x el n o d e t o a c a p i l l a ry p o r e -s p a c en o d e i s t h e n g i v e n a c o n d u c t a n c e of d /200 .5 , i n a c c o r d -a n c e w i t h E q u a t i o n 3 . T h e e l e c t r o d e s a r e c o n s i d e r e dt o h a v e i n f i n it e c o n d u c t i v i t y , w h i c h r e s u l t s i n t h e v a l u eo f E li b e i n g 2 c ~ d w h e n p i x e l i i s o n t h e e l e c t r o d e , a n dp i x e l j h a s f i n it e c o n d u c t i v i t y G ~.

F i g u r e 3 s h o w s t h e f iv e d i ff e r e n t v a l u e s o f c o n d u c t -a n c e s u se d . N o c o n n e c t i o n i n d i c a te s a z e r o c o n d u c t -a n c e . T h e t h i n d a s h e d l i n e s a r e 0 . 0 0 2 5 d , t h e t h i n s o l i dl i n e s a r e d / 2 0 0 . 5 , a n d t h e t h i c k s o l i d l i n e s h a v e c o n -d u c t a n c e s d . T h e z i g - z a g l i n e d e n o t e s a c o n d u c t o r -e l e c t r o d e c o n n e c t i o n , h a v i n g a c o n d u c t a n c e o f e i t h e r2 d o r 2 (0 .0 02 5 )d , f o r a c a p i l l a r y o r a C - S - H p i x e lc o n n e c t e d t o t h e e l e c t r o d e , r e s p e c t i v e ly .

4 .3 . Cond uctance a lgor i thmsO n c e t h e c o n d u c t o r n e t w o r k i s b u i l t , i t s e f f e c t i v ec o n d u c t a n c e i s c o m p u t e d u s in g o n e o f tw o e ff ic i en tm e t h o d s . I f t h e a v e r a g e c o n n e c t i v i t y o f t h e n o d e s iss m a l l e n o u g h , a r o u n d 1 .5 b o n d s p e r n o d e o n a v e r a g e ,t h e n t h e F o g e l h 0 1 m a l g o r i t h m c a n b e u s e d . T h i s a l -g o r i t h m w a s f i r s t w r i t t e n i n L I S P b y F o g e l h o l m [ 2 2 ]f o r 2 D p r o b l e m s , a n d w a s r e c e n t ly e x t e n d e d t o t h r e e -a n d h i g h e r - d i m e n s i o n a l p r o b l e m s u s i n g a p r o g r a mw r i t t e n i n C [ 2 3 ] . T h i s a l g o r i t h m s y s t e m a t i c a l l y re -d u c e s t h e n e t w o r k d o w n t o t w o n o d e s , w i t h t h ec o n d u c t a n c e o f t h e l as t r e m a i n i n g c o n d u c t o r b e in g t h ee q u i v a l e n t c o n d u c t a n c e o f th e e n t i r e n e t w o r k . I t isv e r y e f fi c ie n t , p a r t l y b e c a u s e t h e e q u i v a l e n t c o n d u c t i v -i t y i s o b t a i n e d w i t h o u t h a v i n g t o s o l v e f o r t h e e l e c t ri cp o t e n t i a l a t e v e r y p o in t . H o w e v e r , t h e s p e e d o f t h ea l g o r i t h m d e c r e a s e s e x t r a o r d i n a r i l y w i t h t h e a v e r a g en u m b e r o f c o n n e c t i o n s p e r n o d e . I n R e f e re n c e 2 3,t h e p r o b l e m c o n s i d e r e d w as t h e c o m p u t a t i o n o f t h ec o n d u c t i v i t y c l o s e t o t h e p e r c o l a t i o n t h r e s h o l d ,P c = 0 .249 , w her e p i s the f r ac t ion of bonds r emain ing ,f o r b o n d p e r c o l a t i o n 1 ,1 8-] o n t h e s i m p l e c u b i c l a t t ic e .H o w e v e r w e h a v e f o u n d t h a t t h e a l g o r i t h m b e c o m e su n a c c e p t a b l y s l o w f o r p > 0 . 2 9 f o r t h e s a m e p r o b l e msolved on a 1003 cubic l a t t i ce .

T h e F o g e l h o l m a l g o r i t h m i s s t i l l u s e f u l f o r t h ec e m e n t p a s t e p r o b l e m , s i n c e t h e r e a r e m a n y r a n g e s o fp o r o s i t y w h i c h y i e l d a n e f f e c t iv e p i n t h e r i g h t r a n g e .H o w e v e r , m o s t o f t h e r e s u lt s r e p o r t e d i n t h i s p a p e rw e r e o b t a i n e d u s i n g a s e c o n d a l g o r i t h m , a c o n j u g a t eg r a d i e n t r e l a x a t i o n a l g o r i t h m [ 2 4 ] .

T h e c o n j u g a t e g r a d i e n t r e l a x a t i o n a l g o r i t h m e s se n -t i a ll y s o l v e s th e c o m p l e t e e l e c t r ic a l p r o b l e m o f t h e


5/10

v o l t a g e d i s t r i b u t i o n i n a r a n d o m m a t e r i a l a c r o s sw h i c h a p o t e n t i a l d i f f e r e n c e i s a p p l i e d . T h e o u t p u t o ft h e a l g o r i t h m is t h e v o l t a g e a t e v e r y n o d e , f r o m w h i c ht h e t o t a l c u r r e n t a n d t h u s e q u i v a l e n t c o n d u c t a n c e i sc a l c u l a t e d . T h e i n p u t i s a n i n i t i a l v o l t a g e d i s t r i b u t i o n ,u s u a l ly t a k e n t o b e 1 a n d 0 a t t h e t w o e l e c t ro d e s , a n dl i n e a rl y i n t e r p o l a t e d a t n o d e s i n b e t w e e n . T h e v o l t a g e sa r e t h e n c y c l i c a l ly u p d a t e d u n t i l K i r c h o f f ' s l a w s a r es a t i s f i e d a t e a c h n o d e w i t h i n s o m e p r e s e t f i n i t e p r e -c i s i o n [ 2 4 ] . A t l o w p o r o s i t i e s , t h e c o n j u g a t e g r a d i e n ta l g o r i t h m is s l o w e r t h a n t h e F o g e l h o l m a l g o r i t h m , a si t g i v e s a l l t h e v o l t a g e i n f o r m a t i o n a s w e l l a s t h ee q u i v a l e n t c o n d u c t a n c e , b u t i t c a n h a n d l e a n y d e g r e eo f c o n n e c t i v i t y a n d e v e n t u a l l y b e c o m e s t h e f a s t e ra l g o r i t h m , a s t h e F o g e l h o l m a l g o r i t h m ' s s p e e d d e -c r e a s e s m u c h m o r e r a p i d l y w i t h i n c r e a s i n g p o r o s i t y .

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Figure 5 Lo garith m (ba se 10) of the relative diffusivity, D/Do,against degree of hydration, ~, for (D) sim ulation and (O) experi-ment for a 0.45 w/c ceme nt paste.

5 . D i f f u s i v i t y r e s u l t s5 . 1 . P l a i n c e m e n t p a s t e r e s u l t sW e h a v e s i m u l a t e d t h e r e l a t i v e d i f f u s i v i t y D / D o fo r 0 .4 ,0 .45 , 0 .5 , and 0 .6 w /c r a t i o c e m e n t p a s t e s , a s a f u n c t i o no f d e g r e e o f h y d r a t i o n , ~ . T h e r e s u l t s, o b t a i n e d u s i n g1 0 03 p i x e l u n i t c e ll s, a r e p l o t t e d i n F i g s 4 - 7 , a l o n g w i t hr e p o r t e d e x p e r i m e n t a l d a t a [ 3 , 2 1 , 2 5 ] . I t s h o u l d b en o t e d t h a t t h e t o t a l c o m p u t e r t i m e u s e d f o r a l l t h er e s u l t s d e s c r i b e d i n t h i s p a p e r w a s o n t h e o r d e r o f1 0 0 h o n a s i ng l e p r o c e s s o r, C o n v e x C 1 2 0 m i n i - s u p e r -c o m p u t e r . A l l s i m u l a t i o n s w e r e r u n u s i n g 3 2 - b i t p r e -c i s i o n r e a l n u m b e r s , w i t h n o s i g n i f i c a n t d i f f e r e n c eb e t w e e n 3 2 - a n d 6 4 - b i t p r e c i s i o n r u n s .

I n F i g s 4 - 7 , t h e o p e n s q u a r e s a r e s i m u l a t i o n r e s u lt sa n d t h e f i l l e d c i r c l e s t h e e x p e r i m e n t a l r e s u l t s . O n ei n t i a l c e m e n t - p a r t i c l e p a c k i n g w a s u s e d a t e a c h w /cr a t i o t o g e n e r a t e t h e s i m u l a t i o n r e s u l t s . T h e f a i r l ys m a l l ( a t m o s t 1 0 - 2 0 % ) v a r i a t i o n b e t w e e n d i f f e r e n ti n i t i a l c e m e n t - p a r t i c l e p a c k i n g s i s l e s s t h a n t h e e x -p e c t e d e r r o r i n t h e e x p e r i m e n t a l r e s u l t s , s o t h a t i t w a sn o t w o r t h a v e r a g i n g o v e r s e v e r a l c o n f i g u r a t i o n s .T h e r e i s r e a s o n a b l y g o o d a g r e e m e n t b e t w e e n s i m u -l a t i o n s a n d e x p e r i m e n t .

I n F i g s 6 a n d 7 , h o w e v e r , t h e r e i s o n e e x p e r i m e n t a ld a t a p o i n t t h a t i s s i g n i f i c a n t l y d i f f e r e n t ( b y a f a c t o r o f2 o r 3 ) . f r o m t h e s i m u l a t i o n r e s u l t s . T h e e x p e r i m e n t a ld a t a p o i n t s d i d n o t h a v e a m e a s u r e d d e g r e e o f h y d r a -

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0.4 0.6 0.8 1.0O~Figure 4 Lo garith m (ba se 10) of the relative diffusivity, D/Do,against degree of hydration, ~, for (~) simulation and (0 ) experi-ment for a 0.4 w/c cement paste.

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Figure 6 Lo garith m (bas e 10) of the relative diffusivity, D/Do,against degree of hydration, ~, for (~) simulation and (O) experi-ment for a 0.5 w/c ceme nt paste.

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Figure 7 Logarithm (base 10) of the relative diffusivity, D/Do,against degree of hydration, ~, for ([]) sim ulation and (O) experi-ment for a 0.6 w/c cem ent paste.

t i o n , a s o n l y t h e i r a g e s w e r e r e c o r d e d . C o n s e q u e n t l yt h e d e g r e e o f h y d r a t i o n w a s e s t i m a t e d a s f o l lo w s . T h e2 8 - d a y - o l d s a m p l e s m e a s u r e d i n R e f e r e n c e 3 w e r ea s s i g n e d ~ = 0 .7 , t h e 6 0 - d a y - o l d s a m p l e s m e a s u r e d i nR e f e r e n c e 2 1 w e r e a s s i g n e d ~ = 0 .8 , a n d t h e 1 8 0 - d a y -.o l d s a m p l e s m e a s u r e d i n R e f e r e n c e 25 w e r e a s s i g n e d cz

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= 0 .9 . The se va l ue s o f ~ we re no t p i c ke d t o g i ve t heb e s t a g r e e m e n t w i t h s i m u l a t i o n , b u t r a t h e r w e r e b a s e do n d a t a p r e s e n t e d i n R e f e r e n c e 1 1 . T h i s c o r r e l a t i o nb e t w e e n ~ a n d a g e w a s a s s u m e d t o b e i n d e p e n d e n t o fw/c r a t i o . H o w e v e r , i t i s k n o w n t h a t h i g h e r w/c r a t i oc e m e n t p a s t e s c a n h y d r a t e f a s t e r t h a n l o w e r w/c r a t i op a s t e s [ 2 6 ] , s o t h a t t h e a p p a r e n t d i s a g r e e m e n t in F i g s6 a n d 7 c o u l d j u s t b e d u e t o a n i n c o r r e c t l y a s s i g n e dd e g r e e o f h y d r a t i o n . I n a l l f a ir n e s s, i t s h o u l d b e s t a t e dt h a t t h i s c a v e a t a l s o a p p l ie s t o t h e p o i n t s t h a t a g r e e dwe l l wi t h t he s i m ul a t i on r e su l t s .5 . 2 . S i l i c a f u m e r e s u l t sC on de n se d s i l ic a fum e (a sm a l l , 0 .2 -0 .4 l am , h i gh l yr e a c t i v e , a l m o s t p u r e a m o r p h o u s s i l i c a m a t e r i a l ) i sb e i n g i n c r e a s i n g l y u s e d a s a m i n e r a l a d m i x t u r e i nc onc re t e wh e re a l ow c h l o r i de d i f fus i v i t y i s de s i r e d[ 2 7 ] . I t s e ff e ct o n t r a n s p o r t p r o p e r t i e s i n c o n c r e t e o rm o r t a r i s p r o b a b l y p a r t l y d u e t o m o d i f i c a ti o n o ft h e s a n d - c e m e n t p a s t e i n t e r f a ci a l z o n e [ 1 0 , 2 8 ] , a n dp a r t l y t o m o d i f i c a t i o n o f b u l k c e m e n t p a s t e m i c r o -s t r u c t u r e [ 1 3 ] . B e n t z a n d G a r b o c z i [ 1 3 ] s h o w e d h o wt h e i n c o r p o r a t i o n o f si li c a f u m e i n t o t h e c e m e n t p a s t ec o u l d b e s i m u l a t e d . T h e s i l ic a f u m e r e a c t s w i t h t h e C Hp h a s e t o p r o d u c e m o r e ( p o z z o l a n i c o r s e c o n d a r y )C - S - H , w h i c h h a s a l a r g e r v o l u m e t h a n t h e o r i g i n a lC H a n d s i li c a f u m e c o m b i n e d . T h e r e f o r e , u s i n g s il ic af u m e t e n d s t o f u r t h e r r e d u c e t h e c a p i l l a r y p o r o s i t y o f ac e m e n t p a s t e r e l a t iv e t o t h a t o f a p l a i n p a s te .

T h e i n c o r p o r a t i o n o f s i li c a f u m e h a s b e e n s i m u l a t e da t t w o d i f f e re n t w a t e r - t o - s o l id (w/s) at ios, 0 .6 and 0.4 ,w h e r e w/s r a t i o i s de f i ne d s i m i l a r l y t o w/c r a t i o i nE q u a t i o n 1 , b u t w i t h t h e w e i g h t o f c e m e n t r e p l a c e d b yw e i g h t o f c e m e n t p l u s s il ic a fu m e . C e m e n t ( 1 0 a n d2 0 w t % ) h a s b e e n r e p l a c e d b y s i l ic a f u m e , w h i c h k e e p sth e w/s r a t i o c o n s t a n t , p e r m i t t i n g a f a i r c o m p a r i s o nw i t h p l a i n c e m e n t p a s t e a t a n e q u i v a l e n t w/c r a t i o[13] . S i l i c a fum e ha s a l owe r spe c i f i c g ra v i t y t ha nc e m e n t ; s o t h a t t h e f r a c t i o n o f s o l id v o l u m e i n i t i a l lyt a k e n u p b y t h e s i li c a f u m e i s g r e a t e r t h a n i t s f r a c t i o nb y w e i g h t.

F i g . 8 show s t he s i m u l a t e d d i f fus i v i t y r e su l t s fo r 0 ,1 0 a n d 2 0 % r e p l a c e m e n t o f t h e c e m e n t b y s i l ic a f u m e ,fo r w/s = 0 .6 0 . T h e 0 % d a t a a r e t h e s a m e a s s h o w n i n

F i g . 7 , c on ne c t e d w i t h s t r a i gh t - l i ne se gm e nt s t o f a c i l it -a t e c o m p a r i s o n w i t h t h e s i li c a f u m e r e s u l ts , w h i c h d on o t l ie a t e x a c t l y th e s a m e d e g r e e s o f h y d r a t i o n . B o t ht he 10 a nd 20% re su l t s l i e be l ow t he 0% d i f fus i v i t i e s ,wh i c h i s a s e xpe c t e d s i nc e t he r e a c t i on o f s i l i c a fum ew i t h C H r e d u c e s t h e c a p i l la r y p o r o s i t y . H o w e v e r , i t isa l it t l e su rp r i s i ng t h a t t he 10% s i li c a fum e pa s t e s ha v ea l owe r d i f fus i v i t y a t a l l de gre e s o f hydr a t i o n pa s t 0 .2t ha n t he 20% s i l i c a fum e re su l t s . Th i s r e su l t c a n bee a s i l y e x p l a i n e d b y s t u d y i n g F i g . 9 . I n t h i s F i g u r e ,c a p i l l a r y p o r o s i t y a g a i n s t w e i g h t p e r c e n t a g e s i l i c af u m e , t a k e n a s a p e r c e n t a g e o f t o t a l o r i g i n a l s o l idw e i g h t , is p l o t t e d f o r v a r i o u s d e g r e e s o f h y d r a t i o n ,us i ng r e l a t i onsh i ps de ve l ope d i n Re fe re nc e 13 , fo r w/s= 0 .60 . The r = 0 .2 c u rve shows t ha t t he c a p i l l a ry

p o r o s i t y i s a l m o s t i d e n t i c a l fo r 1 0 a n d 2 0 % s il ic a fu m er e p l a c e m e n t , w h i c h i s t h e r e a s o n t h a t t h e t w o d i f f u s -i v i t i e s i n F i g . 8 a t r = 0 .12 a re a l m os t i de n t i c a l . F o r

> 0 .2 , t hou gh , t he c a p i l l a ry poro s i t y i s a l wa y s sm a l -l e r f o r t h e 1 0 % t h a n f o r t h e 2 0 % s i li c a f u m e p a s t es .Phy s i c a l l y , t h i s i s be c a use t he re i s a t r a de - o f f wh e nr e p l a c i n g c e m e n t w i t h s i li ca f u m e . L e s s c e m e n t m e a n st h a t t h e r e w i l l b e l e s s C H p r o d u c e d t o r e a c t w i t h t h es i l ic a fum e . At sm a l l s i l i c a fum e f ra c t i ons , t he re i s m oret h a n e n o u g h C H t o r e a c t w i t h a l l t h e s i li c a f u m e , b u ta s c e m e n t c o n t e n t d e c r e a s e s a n d s i l i c a f u m e c o n t e n ti fi cr e as e s , t h e r e c o m e s a p o i n t w h e n t h e r e i s t o o m u c hs i li c a f u m e t o r e a c t w i t h t h e C H p r o d u c e d , a n d p a r t o ft he s i l i c a fum e be g i ns t o a c t a s a n i ne r t f . i ' e r wh i c hc a n no t f il l po re spa c e a s e f fe c t ive l y a s r e a c t i ve c e m e n t[ 1 3 ] . T h i s i s t h e p h y s i c a l e x p l a n a t i o n f o r t h e m i n i m ai n t h e p l o t s s h o w n i n F i g . 9 . I t s h o u l d a l s o b e n o t e dt ha t t he g re a t e s t d i f f e renc e s in d i f fus i v i t y be t we e n t he1 0 a n d 2 0 % r e s u l t s a r e a t i n t e r m e d i a t e d e g r e e s o fhy dra t i o n , 0 .4 < a < 0.6, w he re t he g re a t e s t d i f fe r -e nc e s i n c a p i l l a ry poros i t y a re a l so found i n F i g . 9 .Th i s e xp l a na t i on su f f i c e s fo r t he w/s = 0 .60 da t a , a st h e c a p i l l a r y p o r e s p a c e a l w a y s r e m a i n s c o n t i n u o u s ,s o t h a t t h e r e l a t iv e d i f f u s iv i t y i s a l w a y s d o m i n a t e d b yt he c a p i l l a ry pore spa c e . Di f fe re nc e s i n c a p i l l a ry por -os i t y a re t he n d i r e c t l y a nd e a s i l y r e l a t e d t o d i f f e re nc e si n r e l a t i ve d i f fus i v it y . Ho we v e r , t he w/s = 0.40 resul t sa r e s o m e w h a t d i f f er e n t, b e ca u s e t h e c a p i l l a r y p o r o s i t yfa l l s be low qbc = 0.18.

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Figure 9 C a pi l l a ry poros i t y a ga i ns t pe rc e n t a ge of t o t a l o r i g i na lsol ids , by weight , of s i l i ca fume m ixed wi th the original cem ent a tw/s = 0.6, for var iou s degrees of hyd ratio n: [Z, 0.2; (D, 0.4; ~, 0.6;ZX, 0.8; V , 1.0.


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F ig . 10 show s the d i f f us iv i ty r esu l t s f or a w/s r a t i o o f0 .4 0 . A g a i n , th e 1 0 % s i li c a f u m e p a s t e s h a v ~ a b o u t t h es a m e d i f fu s i v i t y a s t h e 2 0 % s i li c a f u m e p a s t e s a t t h el o w e s t d e g re e s o f h y d r a t i o n . T h e 0 % s il ic a f u m e d a t ai s f r o m F i g . 4. F o r i n t e r m e d i a t e d e g r e e s o f h y d r a t i o n ,t h e 1 0 % r e s u l t s a re s y s t e m a t i c a l l y l o w e r . F ig . 1 1s h o w s t h e c a p i l l a r y p o r o s i t y p l o t t e d a s a f u n c t i o n o fs i li c a fu m e w e i g h t f r a c t i o n f o r t h e 0 . 4 w/ s r a t i o p a s t e s ,w h i c h e x p l a i n s t h e i n t e r m e d i a t e d e g r e e o f h y d r a t i o nr e s u l ts o f F i g . 1 0 , i n t h e s a m e w a y a s F i g . 9 e x p l a i n e dt h e r e s u l t s o f F i g . 8 . F o r d e g r e e s o f h y d r a t i o n g r e a t e rt h a n 0 .6 , h o w e v e r , t h e 1 0 a n d 2 0 % s i li c a f u m e r e s u l t sc o n v e r g e t o t h e v a l u e D/Do = 0 . 0025 ( log lo(0 . 0025) =- 2 .6 ), a n d a r e s y s t e m a t i c a l l y a b o v e t h e p l a i n c e m e n t

p a s t e r e s u l t s . T h i s i s b e c a u s e i f t h e r e i s e n o u g h s i li c af u m e t o c o n v e r t b a s i c a l l y a l l t h e C H t o s e c o n d a r yC S H , a t h i g h d e g e e s o f h y d r a t i o n w h e r e t h e r e i sv e r y li t tl e u n r e a c t e d c e m e n t o r c a p i l l a r y p o r e s p a c el e f t , t h e c e m e n t p a s t e w i l l c o n s i s t a l m o s t e n t i r e l y o fC - S - H . T h e C - S - H p h a s e h a s b e e n m o d e l l e d a sh a v i n g a r e l a t i v e d i ff u s i v i ty o f 0 . 0 0 25 , s o t h a t t h e b u l kv a l u e o f r e l a t iv e d i f f u s i v it y fo r t h e p a s t e w i ll b e t h es a m e . F o r l e s se r a m o u n t s o f s i li c a f u m e r e p l a c e m e n t ,s o t h a t n o t a l l th e C H i s r e a c t ed , t h e m i n i m u m v a l u e o f

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Figure 11 Cap illary porosity against percentage of total o riginalsolids, by w eight, of silica fume mixed with the original cement, atw/s = 0.4, for various degrees of hydration; D, 0.2; O, 0.4; ~, 0.6;, 0.8.

DID o w i l l v a r y c o n t i n u o u s l y b e t w e e n 0 . 0 0 1 , t h e m i n -i m u m f o r p l a i n c e m e n t p a s te ( t o b e d e r i v e d in S e c t i o n6 . 3 ) , a n d 0 . 0 0 2 5 , t h e v a l u e o b t a i n e d f o r c o m p l e t er e a c t i o n o f c e m e n t a n d C H .

T h e a b o v e r e s u l t s i m p l y t h a t s i l i c a f u m e r e p l a c e -m e n t o f c e m e n t c a n r e d u c e t h e d i ff u si v it y o f b u l kc e m e n t p a s t e b y r e d u c i n g t h e c a p i l l a r y p o r o s it y , b u t a tl o w p o r o s i t i e s a n d h i g h d e g r e e s o f h y d r a t i o n , s i li c af u m e r e p l a c e m e n t c a n a c t u a l l y i n c r e a s e d i f f u s i v i t y b yr e p l a c i n g i m p e r v i o u s C H w i t h m i c r o p o r o u s C - S - H .S i l i c a f u m e i n c o n c r e t e c o u l d a l s o r e d u c e t h e d i f f u s -i v i ty b y re d u c i n g t h e p o r o s i t y o f t h e s a n d - c e m e n tp a s t e i n t e r fa c i a l z o n e , i f t r a n s p o r t t h r o u g h t h ec o n c r e te w e r e d o m i n a t e d b y p a t h w a y s c o n n e c t in gt h r o u g h t h e i n t e r f a c i a l z o n e r e g i o n s .

A n a d d i t i o n a l p o s s i b i l i t y n o t c o n s i d e r e d i n t h i sw o r k i s t h a t s i li c a f u m e m o d i f i e s t h e m i c r o s t r u c t u r e o ft h e C - S - H g e l p h a s e , c h a n g i n g i t s e ff e c ti v e d i f fu s i v i tyr e l at i v e t o t h e C - S - H i n p la i n p a s te . R e d u c t i o n s i n th eC / S r a t i o o f t h e C - S - H , f r o m 1 .7 to 1 .4 , h a v e b e e no b s e r v e d i n c e m e n t s c o n t a i n i n g p o z z o l a n i c a d m i x -tur es l ike s i l i ca f ume [ 29 , 30 ] , a s w e l l a s an inc r ease int h e p o l y m e r i z a t i o n o f t h e C - S - H g e l p h a s e [ 3 1 ] .T h e s e s t r u c t u r a l c h a n g e s c o u l d m e a n t h a t a d i f f e r e n te f f ec t iv e b u l k d i f f u si v i t y s h o u l d b e a s s i g n e d t o t h eC S - H p h a s e i n c e m e n t p a st e s c o n t a i n i n g s i li c a f u m e.A n a d d i t i o n a l d i ff ic u l ty e x is t s w h e n c o m p a r i n g e q u a la g e s p e c i m e n s o f s i li c a f u m e - m o d i f i e d c e m e n t p a s t e , a si s u s u a l l y d o n e , i n t h a t s i l i c a f u m e m a y a c c e l e r a t e t h eh y d r a t i o n p r o c e s s [ 3 1 ] , so t h a t s p e c i m e n s o f t h e s a m ea g e , b u t w i t h d i f f e re n t a m o u n t s o f si l ic a f u m e , m a yh a v e d i f f e re n t d e g re e s o f h y d r a t i o n . T h u s t h e p r e s e n c eo f si li ca f u m e m a y m o d i f y t h e r e l a t i o n s h i p b e t w e e n a g ea n d d e g r e e o f h y d r a t i o n u s e d a b o v e .

6 . C e m e n t p a s t e d i f f u s i v i ty d e p e n d e n c eo n p o r e s t r u c t u r e

6 . 1 . C a p i l l a r y p o r o s i t y c o n t r i b u t i o nS i n c e t h e m i c r o s t r u c t u r a l m o d e l g i v e s a d e t a i l e dq u a n t i t a t i v e p i c t u r e o f t h e c e m e n t p a s t e p o r e s t r u c t u r ea t a n y d e g r e e o f h y d r a t i o n , i t c a n b e u s e d t o d e t e r m i n et h e d e p e n d e n c e o f d i ff u s iv i ty o n p o r e s t r u c t u r e i n af u n d a m e n t a l w a y.

O f t h e tw o p h a s e s t h a t c o n t r i b u t e t o t h e d i f fu s i v i ty ,t h e c a p i l la r y p o r e s p a c e a n d t h e C - S - H g e l p h a s e , t h ec a p i l l a r y p o r e s p a c e i s f i r s t c o n s i d e r e d . T h e c a p i l l a r yp o r e s p a c e p e r c o l a t i o n t h r e s h o l d o c c u r s w h e n t h ec a p i l l a r y p o r o s i t y is 1 8 % , e x p r e s s e d a s a p e r c e n t a g e o ft o t a l v o l u m e . F o r p o r o s i ti e s w e ll a b o v e t h i s t h r e s h o l d ,t h e d i f fu s i v it y s h o u l d b e d o m i n a t e d b y t h e c a p i l l a r yp o r e s p a c e , s in c e it s c o n d u c t i v i t y i s s o m u c h h i g h e r ( b ya f a ct o r o f 4 0 0) t h a n t h e C - S - H , a l t h o u g h t h e C - S - Hc o n t r i b u t i o n i s s t i l l n o t t o t a l l y n e g l i g i b l e . T o s e p a r a t et h e c o n t i b u t i o n s o f t h e t w o p h a s e s , t h e d i f f u s iv i t y w a sc o m p u t e d w i t h a c - s - H = 0 [ 8 ] . T h e d i ff u s iv i ty o fc o u r s e t h e n t e n d s t o w a r d s 0 a s t h e c a p i l l a r y p o r o s i t ya p p r o a c h e s t h e p e r c o l a t i o n t h r e s h o l d o f 1 8 % . I nF i g . 1 2, t h e l o g a r i t h m o f d i f f u s iv i t y i s p l o t t e d a g a i n s tthe lo ga r i th m of the qu an t i ty (qb - 0 .18 ), w he r e (~ i s thec a p i l l a r y p o r o s i t y . F r o m p e r c o l a t i o n t h e o r y , i t i s e x -p e c t e d t h a t s u c h a p l o t w i ll r e s u lt i n a s t r a i g h t l i n e w i tha s l o p e o f a b o u t t = 2 [- 22 ] a s ~ a p p r o a c h e s ~ c = 0 .1 8 ,

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-4.2-2.4

i i i i i I I

f /~ 1 I I I I I-2.2 -2.0 -1.8 -1.6 -1.4 -1.2 -1.0Lo g ( ~ - 0 . 1 8 )Figure 12 Plot oflogxo D/Do)agains t loglo (~ - ~Pc),where ~ is thecapillary porosity, and ~c = 0 .18 is the ap proximate percolationthreshold for the capillary pore space [13]. The so lid curve s a best-fit straight line, and the simu lation data points were obtained usingthe Fogelholm algorithm [21 ]. No C-S -H contribution totransport.s ince the d i f f us iv i ty i s exp ec te d to sca le as (~ - 0 . 18 )i n t h i s r e g i o n [ 1 8 ] . T h i s s c a l i n g i s e x p e c t e d f r o m t h ec o n c e p t o f t h e u n i v e r s a l it y o f c ri t ic a l t r a n s p o r t e x p o -n e n t s [ 1 8 ] . A v e r y g o o d s t r a i g h t l i n e is i n d e e d f o u n d ,w i t h t h e c o r r e c t s l o p e o f a b o u t 2 .0 . T h e c o m p l e t ee q u a t i o n o f t h e l in e i s

D - 1.8 (qb - 0.18 ) 2 (4)Dob a s e d o n a l e a s t - s q u a r e s f i t . E v e n t h o u g h t h i s f u n c -t i o n a l f o r m i s r e q u i r e d t o h o l d o n l y f o r qb n e a r q bc , i t isk n o w n u s u a l l y t o h o l d f a r t h e r a w a y a s w e l l [ 3 2 ] , s ot h a t E q u a t i o n 4 is a d o p t e d t o c h a r a c t e ri z e th e c o n n e c -t e d f r a c t io n o f th e c a p i l la r y p o r e s p a c e' s c o n t r i b u t i o nto dif fu s ivi ty for al l 0 .18 < qb < 0 .60. Fig. 12 dem on -s t r a t e s t h a t t h i s f u n c t i o n a l f o r m d o e s i n d e e d h o l dr a th e r f a r aw ay f r om qb = 0 .18 .

6 .2 . C - S - H c o n t r i b u t i o nW h e n t h e c a p i l la r y p o r o s i t y f al ls b e l o w 1 8 % , t h e n t h ed i ff u si v i ty w il l b e c o n t r o l l e d b y t r a n s p o r t t h r o u g hC - S - H g e l p o r e s . H o w e v e r , t h e r e i s s ti ll s o m e c a p i l l a rypor .e space le f t, in the f o r m of i so la ted c lus te r s . Th ep h y s i c a l p i c t u r e o f t h e d o m i n a n t d i f fu s i v e f l o w p a t h -w a y s i n t h i s r e g i m e c o n s i s t s o f i s o l a t e d c a p i l l a r y p o r ec l u s te r s l i n k e d t o g e t h e r b y C - S - H g e l p o r e p a t h w a y s .A l t h o u g h t h e C - S - H p h a s e i s i ts e lf c o n t i n u o u s , p a t h -w a y s t h a t a l s o i n c l u d e t h e m u c h m o r e c o n d u c t i v ec a p i l l a r y p o r e s s h o u l d b e m o r e i m p o r t a n t t o t h e t o t a ld i f fu s i v i ty . T h i s p h y s i c a l p i c t u r e i s s im i l a r t o t h a tp r o p o s e d b y A t k i n s o n a n d N i c k e r s o n [ 3 ] . C a p i l l a r ypor os i ty , qb, w i l l s t il l be an ap pr op r ia te va r ia b le in th i sr e g i m e , w i t h t h e d i f fu s i v i ty c o n t i n u i n g t o d e c r e a s ew i th ~ . F or ~ < 0 . 18 , then , th e d i f fus iv i. ty i s f i t t ed w i tha n A r c h i e ' s la w [ 3 3 ] f o r m , a q ~" , w i t h a a n d m c o n -s t a n t s , b u t m o d i f i e d b y h a v i n g a c u t - o f f v a l u e R m i ,,w h e r e R m i n i s t h e v a l u e o f t h e r e l a t i v e d i f f u s iv i t y w h e nt h e c a p i l l a r y p o r o s i t y i s z e r o . F o r s a n d s t o n e r o c k s , f o r2 0 9 0

w h i c h A r c h i e ' s l a w w a s d e f i n e d , t h e c r i t i c a l v a l u e o ft h e p o r o s i t y i s a p p r o x i m a t e l y z e r o , so t h a t t h e t r a n s -p o r t p r o p e r t i e s a n d q~ a p p r o a c h z e r o s i m u l t a n e o u s l y[ 3 3 ] . ( A r c h i e 's l a w i s re a l l y o f t h e s a m e f u n c t i o n a l f o r ma s E q u a t i o n 3 , ( ~ - ~ c ) m, b u t f o r s a n d s t o n e r o c k s,~bc ~ 0 . A t k i n s o n a n d N i c k e r s o n [ 3 ] , w h e n d e m o n -s t r a t i n g t h a t A r c h i e ' s l a w d i d n o t a p p l y t o c e m e n tp a s t e , d i d n o t k n o w t h a t q~c = 0 .1 8 , r a t h e r t h a n 0 . 0, f o rc e m e n t p a s t e . ) H o w e v e r , a z e r o c a p i l l a r y p o r o s i t yc e m e n t p as te w o u l d b e c o m p o s e d o f C - S - H , C H , a n du n r e a c t e d c e m e n t , w h i c h w i l l h a v e a n o n - z e r o r e l a t i v ed i f f u s i v i t y R m i n b e c a u s e o f th e c o n n e c t e d g e l p o r e s o ft h e C - S - H p h a s e . T h e v a l u e o f R m i n i s no t a f i t t ingp a r a m e t e r , b u t c a n b e c a l c u l a t e d u s i n g c o m p o s i t et h e o r y a n d t h e k n o w n v a l u e o f d ~c - s- H , as d e s c r i b e d inSec t ion 6 . 3 .

I f w e c o n s i d e r t h e p u r e c a p i l l a r y p o r e - s p a c e d i f f u s-i v i ty a b o v e qb = 0 .1 8 , a n d t h e C - S - H / c a p i l l a r y p o r e -s p a c e p a t h w a y s f o r a l l v a l u e s o f d~, t o b e r o u g h l y i np a r a l l e l , t h e n a r e a s o n a b l e f u n c t i o n a l f o r m f o r t h er e l a t i v e d i ff u s i v it y as a f u n c t i o n o f c a p i l l a r y p o r o s i t y ,w hic h i s w e l l jus t i f i ed p hys ica l ly , is

D - 0.001 + aqb 2 + H(qb - 0.18)D ox 1.8 x (qb - 0.18 ) 2 (5)

w h e r e H i s a f u n c t i o n s u c h t h a t H ( x ) = 1 for x > 0 ,H ( x ) = 0 f o r x < 0 , t h e e x p o n e n t m o f t h e A r c h i e ' s la wt e r m i s a s s u m e d t o b e 2 b e c a u s e o f t h e u n i v e r s a l i t y o fe x p o n e n t s m e n t i o n e d a b o v e [ 1 8 ] , a n d a is a p a r a m e t e rto be f i t t ed to d a ta po i n t s hav ing q~ < 0 .18 . A f te r th i sf i t i s c a r r i e d o u t , E q u a t i o n 5 b e c o m e s

D - 0.001 + 0.07qb2 + H ( , - 0 .18 )D ox 1 .8 x (q b- 0 .18) 2 (6)

F i g . 1 3 s h o w s E q u a t i o n 6 p l o t t e d a l o n g w i t h a l l t h es i m u l a t i o n d a t a p o i n t s f r o m F i g s 4 - 7 . E q u a t i o n 6g i ve s a r e a s o n a b l y g o o d d e s c r i p t i o n o f t h e r e la t i v ed i f f us iv i ty D / D o o v e r t h e c a p i l l a r y p o r o s i t y r a n g e0 < qb < 0 .6 . E q u a t i o n 6 m u s t b r e a k d o w n a t s o m epo in t f or d~ > 0 .6 , a s it does n o t g ive the co r r ec t qb ~ 1l imi t o f DID o = 1 .

-0.5-1.0

"~ -1.5- 2 . 0

0-2.5-3.0-3.5 0 I I I I I I0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6Copi t to r y por os i ty

0 , , ,

0 .7Figure 13 Logarithm (base 10) of the relative diffusivity D/D oagainst capillary porosity for all p lain cement paste simulation datapoints. The solid line is giv en by Equation 6.


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6.3. Zero porosity diffusivity derivationT h e c u t - o f f v a l u e o f t h e p l a i n c e m e n t p a s t e r e l a t iv ed i f f us iv i ty a t 4 ) = 0 o f 0 . 001 i s jus t i f i ed by the use o f ar e c e n t e q u a t i o n t h a t g i v e s a p e r c o l a t i o n t h e o r y - b a s e dd e s c r i p t i o n o f t h e e f f e c ti v e c o n d u c t i v i t y o f a t w o -c o m p o n e n t c o m p o s i t e [ 3 4 ]. F o r p l a in c e m e n t p a s t ew i t h n o s i l i c a f u m e , a t 4 ) - - 0 t h e t w o c o m p o n e n t s a r eC - S - H , w i t h D I D o = 0 . 0 0 2 5 , a n d t h e c o m b i n a t i o n o fC H a n d a n y u n r e a c t e d c em e n t , w i t h D I D o a s s u m e d t ob e z e r o f o r t h i s p h a s e . T h e e q u a t i o n t o b e s o l v e df or the e f f ec t ive r e la t iv e d i f f us iv i ty , Din~Do, f o r t h ec o m p o s i t e i s

A X 2 + B x - x L x n = 0 (7)w h e r e A o c ( 1 - vc)/vc, vr = 0 . 16 i s th e pe r co la t i ont h r e s h o l d i n t e r m s o f v o l u m e f ra c t i o n v f o r C S H ,w h i c h i n t h i s c a s e i s t h e h i g h c o n d u c t i v i t y p h a s e ,x = (DIn /Do) 1/~, t = 2 i s t h e u n i v e r s a l c r i t i c a l e x p o n e n tf o r c o n d u c t i o n / d i f f u s i o n p e r c o l a t i o n p r o b l e m s i nt h r e e d i m e n s i o n s [ 2 3 ] , x L = 0 i s t h e c o n d u c t i v i t yo f t h e l o w - c o n d u c t i v i t y p h a s e , x H = ( ~ c - s _ n ) 1/t= (0 . 0025) l /t , an d B = x n( 1 - v - A v ) + x L ( A v + v- - A ). S i n c e x L = 0 , E q u a t i o n 7 b e c o m e s

O m - ( - - B / A ) ' = ( - B / A ) :Do( ~ c - s - H ( 1 -- V -- Av ) 2= A2 (8)

F o r w /c r a t ios l e s s than 0 . 41 , i t i s theor e t ica l ly poss ib let o a c h i e v e z e r o c a p i l l a ry p o r o s i t y f o r d e g r e e s o f h y d r a -t i o n < 1. U s i n g t h e C S - H v o l u m e e x p a n s i o n f a c t o r13 , t h e C H v o l u m e e x p a n s i o n f a c t o r 13p, a n d E q u a t i o n1 , w hic h r e la tes w /c r a t i o a n d i n i t ia l c e m e n t a n d w a t e rv o l u m e f r a c t i o n s , i t i s p o s s i b l e t o s h o w t h a t t h e v o l -u m e f ra c t io n s o f C - S - H a n d ( C H + u n r e a c t e dcem ent ) a t 4 ) = 0 a r e g iven in t e r m s of w /c r a t i o b y

3.2[~w/cU C _ S _ H = (1 + 3.2w/c) (13 - 1)13 - 1 + 3.2 ([~p - 1)w /c

/ ) C H+ . . . . . t -- (1 + 3 . 2 w / c ) ( ~ T - - 1 ) ( 9 )U si ng Eq ua t io ns 8 and 9 , an d 13 - - 1 . 7 and 13p = 0 .61 ,w e f i n d t h a t D, , /D o = R m i n e q u a l s 0 . 0 0 1 2 f o r a w /cr a t io 0 . 4 , 0 . 00098 f or w /c = 0 . 35 , and 0 . 0008 f or w /c= 0 .3 , t h u s j u s t i f y i n g t h e c h o i c e o f 0. 0 01 a s a r e a s o n -a b l e a p p r o x i m a t i o n f o r a n y w /c r a t i o l e s s t h a n 0 . 4 1 ,

w h e n n o s i l i c a f u m e i s p r e s e n t . W i t h s i l i c a f u m e p r e -s e n t , a m o r e r e a s o n a b l e v a l u e o f t h e c u t - o f f v a l u e i s0 .001 < Rmin < 0 .0 0 25 ,. d e p e n d i n g o n t h e a m o u n t o fs i l i c a f u m e r e p l a c e m e n t , a s d i s c u s s e d i n s e c t i o n 5 . 2a b o v e .

7 . C o n c l u s i o n sI n o r d e r t o u n d e r s t a n d t h e d e p e n d e n c e o f d i ff u s iv i tyo n t h e m i c r o s t r u c t u r e o f a p o r o u s m a t e r i a l l ik e P o r t -l a n d c e m e n t p a s t e , t h e r e m u s t f i r s t b e q u a n t i t a t i v eu n d e r s t a n d i n g o f m i c r o s t r u c t u r e , a s w e ll a s m e t h o d sf o r c a l c u l a t i n g t h e d i f f u si v i t y f o r a g i v e n m i c r o s t r u c -t u re . B o t h t h e s e g o a ls h a v e b e e n a c h i e v e d b y u s i n g ar a n d o m g r o w t h m o d e l f o r g e n er a t io n o f t h e m i c r o -s t r u c t u r e o f th e c e m e n t p a s t e, a n d b y u s i n g e x a c t

a l g o r i t h m s a p p l i e d t o t h e u n d e r l y i n g d i g i t a l - i m a g el a t t i c e o f t h e m o d e l t o c a l c u l a t e t h e d i f f u si v i t y f o r ag i v e n m i c r o s t r u c t u r e . I t s h o u l d b e e m p h a s i z e d t h a t t h eq u a n t i t a t i v e r e p r e s e n t a t i o n o f th e m i c r o s t r u c t u r e w a sa c h i e v e d b y b a s i n g t h e m o d e l o n a d i g i t a l im a g e , a n dt h a t t h e t r a n s p o r t a l g o r i t h m s u s e d w e r e o n l y ap p l i c-a b l e t o a l a t t i c e s t r u c t u r e , a s u s e d h e r e . T h e r e s u l t so b t a i n e d a r e s u m m a r i z e d b e l o w .

1 . T h e i o n i c d i f fu s i v i t y o f c e m e n t p a s t e c a n b ec a l c u l a t e d w i t h a l g o r i t h m s a p p l i e d t o t h e d i g i t a li m a g e - b a s e d m i c r o s t r u c t u r a l m o d e l , w i t h c a l c u l a t e dv a l u e s o f s t e a d y - s t a t e c h l o r i d e i o n d i f fu s i v it i e s i n r e a s -o n a b l e a g r e e m e n t w i t h e x p e r i m e n t a l d a t a . T h e c a l c u -l a t i o n t e c h n i q u e s a r e e q u a l l y a p p l i c a b l e t o a c t u a l a s t om o d e l m i c r o s t r u c t u r e s .

2 . T h e c h l o r i d e d if f u s iv i t y o f p l a i n P o r t l a n d c e m e n tp a s t e c a n b e e x p r e s s e d a s a f u n c t i o n o f c a p i l l a r yp o r o s i t y o n l y , w i t h t h e f u n c t i o n a l f o r m b e i n gD / D o = 0.001 + 0.074)2 + H (4) - 0.18) 1.8 (4) - 0.18) 2,w h e r e H ( x ) = 0 f o r x < 0 , a n d 1 f o r x > 0 . T h i sr e l a t i o n s h i p i s d o m i n a t e d b y t h e p e r c o l a t i o n p r o p e r -t i es o f t h e c a p i l l a r y p o r e s p a c e a b o v e t h e c r i t ic a lc a p i l l a r y p o r o s i t y 4 ) c = 0 . 1 8 , w h e r e t h e c a p i l l a r y p o r es p a c e b e c o m e s d i s c o n n e c t e d , a n d b y s i m p l e A r c h i e ' sl a w - t y p e p o w e r - l a w b e h a v i o u r b e l o w 4 ) o w h e r e t h ep a t h w a y s th r o u g h C - S - H g el p o r e s d o m i n a t e t het r a n s p o r t . T h e a b o v e f u n c t i o n a l f o r m m u s t b r e a kd o w n a t p o r o s i t i e s s o m e w h a t h i g h e r t h a n 0 . 6 , a s i td o e s n o t g i v e t h e c o r r e c t 4 ) = 1 l i m i t.

3 . T h e p h y s i c a l p i c t u r e o f d i f fu s i v e t r a n s p o r t i nc e m e n t p a s t e i s a s f o l lo w s . A b o v e 4 ) = 4 )c , t r a n s p o r t i sm a i n l y t h r o u g h c o n t i n u o u s c a p i l l a r y p o r e s , w i t h as m a l l e r a m o u n t o f f lo w t h r o u g h p a t h w a y s o f c a p i l l a ryp o r e s l i n k e d b y C S - H g e l p o r e s . B e l o w 4 ) c , t h ed o m i n a n t p a t h w a y s a re n o w m a d e u p o f i s o la t e dc a p i l l a r y p o r e c l u s t e r s l i n k e d t o g e t h e r b y C - S - H g e l -p o r e c o n n e c t i o n s , w h i c h d e t e r m i n e t h e f l o w r a t e [ - 3 ] .

4 . T h e m i n i m u m v a l u e o f D / D o f o r c h l o r i d e i o n sd i ff u si n g t h r o u g h p l a i n P o r t l a n d c e m e n t p a s t e h a sb e e n p r e d i c t e d t o b e a b o u t 0 . 0 0 1 , o b t a i n e d w h e n t h ec a p i l l a r y p o r o s i t y i s z e ro . W i t h s i li c a f u m e p r e s e n t , t h ec u t - o f f v a l u e b e c o m e s 0 . 0 0 25 .

5 . F o r l o w w /s r a t ios , w hen suf f ic ien t s i l i ca f ume i sp r e s e n t t o r e a c t w it h m o s t o f th e C H p r o d u c e d , a n dw h e n t h e c a p i l l a r y p o r o s i t y i s m u c h l e s s t h a n4)c = 0 . 18 , so tha t the d i f f us iv i ty i s co nt r o l led by th eC - S - H p h a s e , t h e a d d i t i o n o f s il ic a fu m e c a n i n c r e a set h e r e l a t i v e d i f f u s i v i t y b y c o n s u m i n g C H (D/ D o = 0),a n d r e p l a c i n g i t w i t h C - S - H ( D / D o = 0.0025).

A c k n o w l e d g e m e n t sT h e a u t h o r s w o u l d l i k e t o t h a n k t h e N a t i o n a l S c i e n c eF o u n d a t i o n S c ie n ce a n d T e c h n o l o g y C e n t e r f o r A d -v a n c e d C e m e n t - B a s e d M a t e r i a ls f o r p a r t i a l s u p p o r t o ft h is w o r k ; P . M . D u x b u r y f o r s u p p l y i n g a w o r k i n gc o p y o f a 2 - d c o n j u g a t e g r a d i e n t c o d e ; a n d C . J . L o b bf o r s e v e r a l h e l p f u l c o n v e r s a t i o n s .

R e f e r e n c e s1. L . M . S C H W A R T Z a n d J. R . B A N A V A R , Phys. Rev. B, 3 9

( 1 9 8 9 ) 1 1 9 6 5 .

2091


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2 . V . K . S . S H A N T E a n d S . K I R K P A T R I C K , Adv. Phys. 20 ,(1971) 325.

3 . A . A T K I N S O N a n d A . K . N I C K E R S O N , J. Mater. Sci. 19(1984) 3069.

4 . A . H . T H O M P S O N , A . J . K A T Z a n d C . E . K R O H N , Adv.Phys . 36 (1987) 625.

5. J . P O M M E R S H E I M a n d J . R . C L I F T O N , Mater . Const . 18(1985) 21.

6 . W . C . H A N S O N i n "T h e C h e m i s t ry o f S ul p h a te - R e s is t in gP o r t l a n d C e m e n t s " , e d i t e d b y E . G . S w a n s o n ( U n i v e r s i t y o fT o r o n t o P r e s s, T o r o n t o , 1 9 68 ).7 . A . A T K I N S O N , Nucl . Chem. Mgt 5 (1985) 203.

8 . D . P . B E N T Z , D . B . G I N G O L D , E . J . G A R B O C Z I , C . J .L O B B a n d H . M . J E N N I N G S , Ceram. Trans. 16 (1991) 211.

9 . E . J . G A R B O C Z I a n d D . P . B E N T Z in " M a t e r i a l s S ci e n c e o fC o n c r e t e " V o l . I I , e d i t e d b y J . S k a l n y ( A m e r i c a n C e r a m i cSociety , Westvi l le , 1991) in press .

1 0. K . L . S C R I V E N E R i n " M a t e r i a l s S c i en c e o f C o n c r e t e " V o l. I ,e d i t e d b y J . S k a l n y ( A m e r i c a n C e r a m i c S o c i e t y , W e s t v i l l e ,1989) p . 127 .

1 1. S . M I N D E S S a n d J . F . Y O U N G , " C o n c r e t e " ( P r e n t i c e -H a l lE n g lew o o d Cl i f f s , N e w Je r s ey , 1 9 8 1 ) Ch . 4 .

1 2. E . M . G A R T N E R a n d J . M . G A I D I S in " M a t e r i a l s S c i e n c e o fC o n c r e t e " V o l . I , e d i t e d b y J . S k a l n y ( A m e r i c a n C e r a m i cSociety , Westervi l le , 1989) p . 95.

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