Cairn Cross 02

7/24/2019 Cairn Cross 02

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T H E F A T E O F R E S I D U A L S O L V E N T I N D R Y I N G C O A T I N G S : C A N I T G E T

T R A P P E D A N D H O W ?

Ri c h ar d Al l an C a i m c r oss , D r e xe l Un i ve r s i ty , Ph i lad e l p h ia , PA

A b s t r a c t

A c om m on goa l i n i n d u str ia l d r y in g o f p o l ym e r so l u t i on c oa t i n gs i s to r e d u c e th e re s id u a l so l ve n t

content RS C) to a spec i f ied leve l . Industr ia l dryers cons is t of a ser ies of zon es op erated at d i f ferent a ir

te m p e r atu r e s an d a i r f l owr ate s to m e e t th e RS C sp e c i f i c a t i on s , an d to p r od u c e d e fe c t - f r e e c oa t i n gs . A

c om m on ob se r vat i on i s th a t, wh e n d r y i n g a t a c on s tan t te m p e r atu re , th e r e s id u a l so l ve n t c on te n t p la te au s

and the drying rate e ffec t ive ly drops to zero. Often the RSC can be reduced by further increas ing the

tem p e r atu r e. In h om oge n e ou s p o l ym e r so l u t i on s ab ove th e g las s t r an s it ion te m p e ratu r e o f th e p o l ym e r ,

th e ob se r ve d p l a te au i n RS C i s ac c u r ate ly p r e d i c te d b y F i c k i an d i f fu s ion wi th a c on c e n tr a t ion -d e p e n d e n t

d i f fu s i o n c o e f f i c i e n t . W e h a v e d e v e l o p e d a s im p l e m o d e l w h i c h p r ed i ct s th e d ep e n d e n c e o f R S C o n

temperature , coa t ing th ickness , an d th e d i f fus ion propert ies of the solut ion . In th is case , so lv ent i s

re tained by the d i f fus iona l res i s tance to mass transfer , and the d i f fus ional res i s tance can be low ered b y

increas ing tem perature .

H owe ve r , th e r e ar e n u m e r ou s c l a i m s th a t th e RS C c an a l so b e r e d u c e d b y u s i n g m i l d e r d r y i n g

cond i t ions , e .g . low er ing air f low, low er ing temperature or part ia l ly saturating the a ir wi th solven t vapor .

S u c h b e h av i or i s an o m a l ou s an d c an n ot b e p r e d i cte d b y F i c k i an d i f fu s ion ; w e c a l l th i s b e h av i or

anom a lou s sk i n n i n g .

W e h a v e m e a s u re d a n o m a l o u s s k i n n i n g in P M M A / a c e t o n e c o a t in g s a n d h a v e

d e ve l op e d a n on -F i c k i an m od e l wh i c h p r e d i c t s th e an om al ou s b e h av i or .

I n t r o d u c t i o n

D r y i n g P e r i o d s

Fi gu r e 1 d e p i c ts an i n d u s tr i al c oa t i n g sys te m c on s i s t i n g o f a c oa t i n g s ta t ion an d tw o z on e s o f a i r

i m p i n ge m e n t d r y i n g ove n s ; so m e d r y i n g a lso oc c u r s b e tw e e n th e c oa t i n g s ta t i on and th e d r y in g ove n s i n

a r eg i on o f u n c on tr o ll e d , s l ow d r y in g . T h e gr ap h i n F i gu r e 1 d e p i c t s typ i ca l p r o f i le s o f ove n an d c oa t in g

temperatures , drying rate , and res idual So lvent in the coat ing as the co at ing p asses from the coa t ing

stat ion through the drying ovens . Th e trends in coat ing tem perature , drying rate, and res idual so lvent

exhib i t 4 character is t ic drying per iods that are often observ ed in drying o f po lym er solut ion coat ings: 1)

wa rm-up , 2) near ly-con stant rate drying, 3) fa l l ing-rate drying, and 4) d i f fus iona l p lateau .

T h e

w a rm u p p e r i o d

corresponds to the in i t ia l trans ients that occur as the co at ing enters a new

d r y i n g z on e . In th is p e r i od e vap or at ive c o o l i n g i s o f te n s i gn i f ic an t an d c an e ve n c au se a d r op in c oa t i n g

temperature. The

nea r l y con s t a n t r a t e

per iod corresponds to a per iod of rapid drying, where the drying

rate i s control led by mass transport in the drying gas and the solvent concentrat ion i s near ly uni form

through the coat ing. Du r ing the near ly-constan t rate per iod , the rate of heat transfer to the coat ing i s

b a l an c e d b y th e r a te o f e vap or at ive c oo l i n g su c h th a t th e c oa t i n g te m p e r atu r e an d d r y in g r a te r e m ai n

near ly constant . H ow ever , as so lvent con centrat ions fa l l , the internal res is tance to so lvent transport by

di ffus ion r i ses and a fa l l ing-rate per iod ensues .

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Initial

Transients

Constant

R a t e

Per iod

Fal l ing R a t e

Per iod

Fal l ing R a t e

Per iod

c

0

.=

o 0.8

g

1.1,.

E

= 0 .6

e ~

0 . 4

o

m

.~ 0 2

I / 1

R Y I N G

R T E

100

90

80

l 70 . . . .

tO

6

o

I i f f u s i o n a l

4

I P l a t e a u s

I 3 o ~

F- -4... \ 0

. I L 0

o . . . .. . s . v . . v . . v . .. .. . v

~ ' . . . . . . . " : ' " " ' ~ " " : : " : ' " : ~ " ~ " . . . .. . . .. . . .. . . .. ' ~ ' ' ~ " ' > ' ~

:

. . . .. . . . .. : " : ' . . . . . . . ? . : : ? ' . - ~ - : " ~ . . . . . . . I . . . . . . ~ : " . ' : . :- - ' . '? - : - : . : .' . ~. . . . . ~ : - - : " . ' : . : - : - : ~ . . . . . . ' : : ': ' . - : - :- : - : ' ? : - - ': ' : : '. ' - ' .. . . . . _ " _ ' ~ - " : ~ - : - : : - : '_ ' ~. . . . . . . . .

. . . . . . . : . . .. . .. . . . . . ' ~ " ~ : ~ , . : , ' . ~ ' " ~ : : . : ' % - - ' - - ; . - : - - .. : i - - - - .~ , . ; ~ : . . . ~ .: - - : - : :: . ' -. .. .. . . ,< ,. ., -, -; ~. .- :- -' ,- ,~ , ~ ~

, , -

Z O N E Z O N E

Figure 1.

Schemat ic o f a two-zone indus tr ia l d ryer wi th typ ica l p rof i les o f tempera ture d ry ing ra te

and residual solvent con tent along the dryer length.

he fa l l ing ra te per io d corresponds to a per iod in which d ry ing becomes d i f fus ion cont ro l led and

drying rates asym ptotically approach zero. In this period the solvent concentrat ion at the surface of the

coating drops sharply to reach equ il ibrium with the solvent vapor in the drying gas and there is typica lly

a steep gradient in solvent concentrat ion near the surface of the coating. In poly me r solution coatings

the fal l ing rate period dominates because the diffusion coefficient for solvent transport through a

poly me r drops by several orders of ma gnitude as solvent departs . An other characteris t ic feature often

observed in drying of poly me r solution coatings is a diffusional plateau. A di f fus iona lp la teau

correspond s to the later part of the fal ling rate period wh ere the dryin g rate becom es negl igible while a

signif icant amo unt of residual solvent remains in the coating.

The d urat ion and magnitude o f each o f these periods during drying of a polym er solut ion coa t ing

depend on the operating condit ions and phys ical propert ies of the coat ing. Ho wev er, in polymer

solut ion coat ings , the internal res is tance to mass transfer caused by a d if fus ion coeff ic ient that drops

several orders of m agnitude during drying leads to a fa l l ing rate period and a d if fus ional p lateau that

controls the f inal res idual solvent . H ence in des ign o f dryers and choice of operat ing con dit ions , i t i s

important to understand the relat ionship betw een dif fus ional res is tance and residual solvent . The

dif fus ional res is tance can be reduced by rais ing d if fus ion coeff ic ients for exam ple by rais ing

temperature) , changing solvents , or reducing f i lm thickness .

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D i f f u s i o n i n P o l y m e r - S o l v e n t S o l u t i o n s

Du ring drying, so lvent must 1) d i f fuse to the surface of the coat ing, 2) evaporate into the

d r y i n g gas , an d 3 ) b e c on ve c te d aw ay fr om th e c oa t i n g su r fac e b y th e d ry i n g gas . T h e ra te o f

e vap or at ion an d r e m ova l f r om th e c oa t i n g su r face i s typ i c a l l y d e sc r i b e d b y a m ass t r ans fe r c oe f f i c i e n t

formulat ion:

E - C ( p s ,~ ~ - P s , - )

1 )

R T

W here E i s the evaporation rate, k i s a m ass transfer coeff ic ien t , p s ~ . - f i s the part ia l pressure of so lvent in

the gas at the coat ing surface , an dp s ,~ i s the part ia l pressure of so lv ent in the drying gas . The so lvent

partia l pressure in the gas at the coat in g su rface i s in equ i l ibr ium with the solv ent in the coat ing at the

surface; th is equi l ibr ium is often descr ibed us ing the solvent ac t iv i ty at the coat ing surface:

P s u y - P s v p a

2 )

Where the vapor pressure , Ps v ap i s on ly a funct ion of temperature and the ac t iv i ty , a , i s on ly a funct ion

of so l ve n t c on c e n tr a t ion .

The mass transfer coeff ic ient , so lvent vapor pressure , and solvent ac t iv i ty comprise the external

res i s tance to so lvent transport. The external res i s tance determ ines drying rate in the warm-up per iod and

constant rate per iod . In the fa l l ing-rate per iod , the solven t part ia l pressure at the coat ing surface

becomes near ly equal to the solvent part ia l pressure in the drying gas , and the value of the mass transfer

coeff ic ien t bec om es unimp ortant . The ma ss transfer coeff ic ie nt i s re lated to the intens i ty of the a ir f low

in the dryer and proport ional to the heat transfer coeff ic ie nt in the drying ga s . In the fa l l ing rate per iod

the heat transfer coeff ic ien t i s often important becau se the rate o f chan ge o f temperature of the coat ing i s

d ic tated by the heat transfer coeff ic ient .

1 . E 0 5

A

N

l = 1 . E 0 6

.1

C

~ 1 . E 0 7

m

Q

0 1 E 0 8

C

0

m

1 . E 0 9

1

1 C

1 . E 1 0

0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6

S o l v e n t M a s s r a c t io n

Figure 2 .

Con c e n tr a t ion an d te m p e r atu r e d e p e n d e n c e o f th e d if fu s i on c oe f f i c i e n t for p o l y v i n y l

ace tate )-to luene solut ions .

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onstant Rate Model

Heat F lo w = Ev ap o r a t iv e Co o l in g

I

Lumped Parameter Model

Temperature Solvent Con c. niform

th r o u g h Coating

d T/d t = H ea t F lo w - Ev ap o r a t iv e Co o l in g

Diffusion Transport Mod el

I n tern a l Ex tern a l M ass Tr an s f e r

Resistances

Pa r t ia l D i f feren t ia l eq u a t io n s f o r Cs T

R e s i d u a

Solvent

Constant

R a t e

Model

/ umped

\ k P a r a m e t e r M o d e l

\ ~ ~ / Diffusion

T i m e

Figure 3.

Hierarchy o f mathem atical models for predict ing drying of polymer solut ion coat ings (a)

f low diagram indicat ing model complexi ty and (b) schematic of residual solvent

predictions from different models.

As discussed above, the internal diffusional resistance to mass transfer normally controls the

f inal residual solvent in a dr ied polym er solut ion coat ing. In homogeneo us polym er solut ions above

glass-t ransit ion temperature o f the polymer, solvent f lux is normally descr ibed by Fic k 's Law:

J s - - D d Cs

(3)

dx

W he re js is the flux of solvent in the x direction, D is the mutu al diffusion coefficient, and

Cs

is the

solvent concentration. The diffusion coefficient

D Cs, T)

is a material pro perty that cha racterizes the rate

at wh ichso lven t can mov e through the polymer. As show n in Figure 2, the diffusion coeff ic ient is a

strong function of concen tration and temp erature [Zielinski 1992]. The steep drop of diffusion

coeff icients a t low solvent concentration is a dom inant cause o f excessive residual solvent in drying

polymer-solvent coat ings. The diffusional resis tance is proport ional to the square of the coat ing

thickness an d inversely propo rtional to the diffusion coefficient. So, as residual solven t drops in the

falling rate period, diffusion al resistance rises sharply and cau ses the diffusional plateau.

M a t h e m a t i c a l M o d e l s o f Dr y in g Co a t in g s

Figure 3 depicts a hierarchy of drying models in terms of the com plexi ty of the model and the

amount of physical property data needed to solve the models . This sect ion reviews br ief ly the s tandard

models that have been used to predict residual solvent in drying polyme r coat ings.

L u m p e d P a r a m e t e r M o d e l s

In lumped param eter models , the solvent concentration and temperature are assumed uniform

throug h the coating thickness. This leads to substantial simplification of ma ss and energy balances on

the coating and neglects the diffusional resistance to drying. Con stant rate drying models further

simplify the mathem atical problem b y assum ing a pseudo-steady s ta te ; then the coat ing temperature is

determined from a balance of the ra te o f heat t ransfer from the gas phase and the ra te of evaporat ive

cooling due to evaporation. In a constant-rate model, the residual solvent content decreases linea rly to

zero as show n in Figure 3b. A constant-rate m odel is applicable to the start of drying, but loses accurac y

if diffusional resistance becom es significant or if the solvent activity chang es significantly. Con stant

rate models are mo st accurate for drying of particulate system s such as sand and clay [van Brakel 1980].


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Lum ped parameter mo dels a re accura te fo r a longer por t ion of the dry ing t ime than cons tan t ra te

mod els , because they al low for changes in coating temp erature, solvent vapo r pressure, and solvent

activity. Lum ped param eter models are useful for predict ing init ia l drying in the nearly-constant rate

period [Gutoff 1996]

F i c k i a n D i f f u si o n M o d e l s

By far , the most successful drying models for predict ing residual solvent in binary polymer-

solvent coatings have been diffusion models . In these models , solven t transport occurs by diffusion

using Fick 's Law (equation (3)) with a concentrat ion and temperature dependent diffusion coefficient .

Conservation o f mass and en ergy in the coating-substrate system leads to coupled equations for

evolution of the temperature , solvent concentrat ion, an d coating thickness. Diffusion mod els require

solving a non-l inear part ia l differential equation subject to f lux boun dary condit ions at the surfaces o f

the coating. Becau se the coating and substrate are thin, the temperatm'e evolution can typically be

predicted by a lumped-parameter model, because temperature variat ion in coating/substrate is often

small ( i.e . less than 1 C). The result ing equations can be solved nume rically using Finite Difference or

Finite Element techniques, which require a s ignif icant amount of expert ise to apply to these problems.

Never the less , d i f fusion dry ing models have been so lved by many researchers fo r many b ina ry

polymer solution coatings with good comparison to experimental measurements [e .g. Yapel 1988,

Caimcross 1995 , Pr ice 1997 2000 , Alsoy 1998] . W ith advances in comput ing speed and

perform ance, i t is now possible to solve a com plete transient diffusion m odel for drying of a binary

polym er so lut ion coa t ing on a persona l com puter in less than one minute . The pr imary cha l lenge in

apply ing these mo dels i s de te rmin ing accura te va lues o f the phys ica l p roper ties needed in the models .

Extension of the Fickian diffusion models to mult icomponent systems (i .e . two or more solvents

or two or more polymers ) requires addit ional phys ical parameters that are diff icult to measure. There is

not currently agreement about the correct equations to describe dependence of diffusion coefficients on

comp osit ion in mu lt icomp onent sys tems [Also y 1999, Zelinski 1999]. This wil l l ikely be an area of

signif icant research effort in the futu re.

N o n F i c k i a n M o d e l s

The Fickian diffusion models discussed above are based on concentrat ion gradients being the

only driving force for solvent diffusion. How ever, in solutions with polym ers that pass through a glass

transit ion during drying, s tresses develop that can also contribute to solvent transport . Stresses aris ing

due to swel l ing in g lassy po lym er coa t ings have been shown to cause anomalous so lven t transpor t in

so~ t ion expe r im ent s - lead ing to so-ca lled Case I I d i f fusion [Thomas W indle 1980 1982 , Fu

Dum ing 1993]. There a re cur ren t ly a couple of compet ing approaches for modify ing the d i f fus ion

mod els to account for the effect of passing thro ugh a glass transit ion. One appro ach is to modify the

concen trat ion-depend ence of the diffusion coefficient in the glassy region [Haj Rohm dane 20 01].

An other appro ach is to develop a new con sti tut ive equation for the solvent transport which includes a

stress-driven diffusion term; then an addit ional equation is required to determ ine how the stress in a

coa t ing deve lops and re laxes dur ing dry ing [Caimcross Dum ing 1996 , Vin jamur 2001 , Edwards 1998

1999].

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Figure 4.

0

m

o a

IJI .

w

w

m

O 8

e~

o 4

0

D

~

0 2

m

0

1 2 3 4 5

T i m e s

Predictions of residual solvent content versus t ime for poly(vinyl acetate)- toluene

coat ings dr ied at a ser ies of different oven tem peratures using a diffusion model solved

with the f mite eleme nt method. At long times, the residual so lvent reaches a di f fus ional

p l a t eau and the only effect ive wa y to reduce the residual solvent is by increasing

temperature. [Vadapalli , 2001 ]

Exam ples o f Predic t ions from ry ing M ode ls

Goa ls o f ry ing M ode ls

The models discussed above have been used to accurately predict residual solvent levels in

drying coatings. In addition to predicting residual solvent, drying mode ls can also predict defects

relevant to coat ing processes . The key issue is developing a mathematical expression for the onset o f

defects, wh ich can be evaluated based on results from the model. Fo r example, Ca imc ross et al. (1995)

predicted onset of bl ister defects due to solvent boil ing; when ever the tem perature o f the coat ing was

higher than the bubble-point temperature o f the coat ing (at i ts current com posi t ion) , the d rying model

indicated formation o f blister defects. Such a criterion does not accou nt for the rate of bubb le growth,

but merely indicates the ini t ia t ion o f bubble form ation. The predict ions m atched q ual i tat ively with

experimental results . Caimcross e t al . (1995) and Price and Caim cross (2000) used the mathem atical

model of bl is ter formation as a constraint on the choice of acceptable operat ing condit ions and used a

drying model to determine the opt imal condit ions to m inimize the residual solvent while avoiding bl is ter

defects.

There are other measures of dryer performance that can be predicted by drying models; for

exam ple energ y usage, suscep tibility to vap or explosion, and overall econom ics. Au st et al. (1997) used

the lower explosion l imit as a constraint on vapor concentrat ions predicted from a d rying model t o

develop additional dryer design heuristics. The limitations impos ed by explosion limits favor highe r air

velocities.

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1 . 4 x 1 0 4

1 . 2 x 1 0 4

E

0

,br

c(u

>

.m .

0

( n

r a m

0

. l

( /)

G)

r

1 0 4 '

8 x 1 0 5,

6 x I 0 5 '

4 x 1 0

P s e u d o - S t e a d y S t a t e

. , ,

M o d e l

T r a n s i e n t

D i f f u s i o n M o d e l

3 1 0 3 2 0 3 3 0 3 4 0 3 5 0

T e m p e r a t u r e K

Figure 5 . Com p ar i son o f p r e d ic t i on s o f r e s id u a l so l ve n t a t lon g t i m e i .e . i n th e d i f fu s i on a l p l a teau )

ve r su s ov e n te m p e r atu r e for a p se u d o-s te ad y-s ta te d i f fu s ion m od e l an d a c om p l e te

tr an si e nt d i ffu s i on m od e l . Pr e d i c ti on s for a p o l y v i n y l ac e ta te )- to l u e n e c oa t i n g wi th 100

l .tm dry f i lm th ickness [Vad apal l i , 20 01 ]

A p h i l o sop h i c a l goa l o f m od e l i n g i s to a i d in d e ve l op i n g i n tu it i on ab ou t th e a f fec t o f op e r a t i n g

parameters on dryer performance and to reduce the number of exploratory exper iments required to

d e s i gn / op t i m i z e a d rye r. W i th a l l m o d e l s , i t i s n e c e s sar y to tak e i n to ac c ou n t th e p h ys i ca l p h e n om e n a

th at ar e n o t i n c l u d e d i n th e m od e l wh e n ap p l y i n g th e c on c l u s i on s o f th e m od e l to p r ac ti c a l p r oc e s se s .

P r e d i c t i o n s o f R e s i d u a l S o l v e n t a t V a r i o u s O p e r a t i n g C o n d i t i o n s

Fi gu r e 4 d i sp l ays p r e d i c t ion s o f re s i d u a l so lve n t m ass o f so l ve n t p e r m a ss o f c oa t i n g ) i n a d r y i n g

p o l y v i n y l ac e ta te ) -to l u e n e c oa t i n g i n a s i n g l e -z on e d r ye r u s i n g a F i c k i an d i f fu s ion d r y i n g m od e l .

Po l y v i n y l ac e ta te ) h as a g l a s s t r an s it ion te m p e r atu r e o f ab ou t 32 C an d F i c k i an d i f fu s i on h as b e e n ~

sho wn to be accurate in th is system at the temperatures cons idered here . The predic t ions sh ow that as

the oven temperature increases the drying rate and f inal res idual so lvent content decrease monotonical ly .

In addi t ion , a l l the predic t ions sho w a d i f fus ional p lateau in the fa l l ing rate per iod wh ere the drying rate

b e c o m e s n e g l i g i b l e I t i s apparent from these predic t ions and others that once a d i f fus ional p lateau i s

reached, the only e ffec t ive w ay to reduce the res idual so lvent i s by increas ing the a ir and coat ing)

temperature.

The mode l predic t ions a l so show that the pr ior temperature h is tory does not affec t the res idual

solven t leve l in the d i f fus ion al p lateau i .e . it does n ot matter ho w quiCkly the temperature was raised to

the f inal temperature , the res idual so lvent leve l in the d i f fus ional p lateau only depends on the f inal

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0 . 0 9

C

o 0 . 0 8

g 0 0 7

~ 0 0 6

I ~ 0 . 0 5

u_ 0 04

0 .0001 0 .001 0 .01 0 .1 1

V e l o c i t y a c r o s s c o a ti n g m / s )

Figure 6 .

Exper imen ta l m easu remen t o f t r app ing sk inn ing in PMM A-ace ton e coa t ings d r ied in a

Hig h Airf low Dryin g Experiment . The dry f i lm th ickness is approxim ate ly 15 ~tm and

th e d ry ing tempera tu re i s 50 C. [Vin jamur 2001 ]

temperature) . Becau se the d iffusion mod els require s ignif icant computat ional effor t , a s impler metho d

was d evelope d to est imate the res idual so lvent level us ing a pseud o s teady-s ta te m odel (PSS). In the

PSS model , the d iffusional res is tance to solvent t ransport to the coat ing surface was assumed to be

contro l led by a th in low-concen tra t ion layer near the coat ing surface; below th is layer , the solvent

concentra t ion prof i le was presum ed to be fa ir ly f iat . Beca use the PSS mod el assume s a s teady-s ta te

profi le , the model can only predic t the res idual so lvent content a t long t imes and cannot determine how

long i t takes to get there . F igure 5 shows a comparison b etw een predic t ions from the PSS mod el and a

transient d iffusion model . The PSS m odel consis tent ly predic ts a h igher res idual so lvent level , but

reproduce s the t rends accura te ly . Con sequently , the PSS m odel could be used as a rough, quick

calcula t ion of what ove n temperature is neede d to achieve a desired res idual so lvent content .

A n o m a l o u s S k i n n i n g

A cla im com mo nly reported in the drying l i te ra ture is tha t rapid drying can lead to formation o f a

skin that t raps solvent with in the coat ing . Furtherm ore i t is of ten c la imed that s lower drying can avoid

the skin and resul t in lower f inal res idual so lvent levels; we wil l refer to th is observat ion as anomalous

s nning becau se i t is contrary to in tu i tion - i .e . in anomalous skinning, low er dr iv ing force leads to

higher so lvent removal . The predic t ions in F igures 4 and 5 dem onstra te a type of sk inning wh ere the

res idualsolvent content s tops changing (on pract ica l t ime scales) while there is s t i l l s ignif icant res idual

solvent in the coat ing , but the resul ts in F igures 4 and 5 are not anomalous.

Ho we ver , F ick ian d ry ing mode ls (w i thou t chemica l reac t ions o r phase t rans it ions) canno t p red ic t

anom alous sk inning as def ined above - i .e . even though Fick ian d iffusion models predic t the d iffusional

pla teaus d isplayed in Figure 4 , the res idual so lvent a lways decreases when the drying condit ions become

mo re severe . Indeed, despi te the num erous c la ims o f anom alous skinning, there are very few

experim enta l s tudies dem onstra t ing i t, and a ll of the reported exp erimen ta l resul ts of anom alous

skinning show a very s l ight affec t , with in the range of experimenta l error [Powers & Coll ier 1990] .

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R e c e n t l y V i n j a m u r C a i m c r o s s ( 2 0 0 1 2 0 0 2 ) m e a s u r e d a n o m a l o u s s k in n i n g i n p o l y ( m e t h y l

m e th ac r y l a te ) -ac eton e c oa t i n gs . PM M A i s a g l a s sy p o l yr ne r at r oom te m p e r atu re an d i s we l l k n ow n for

an om al ou s d i f fu s ion b e h av i ou r i n sor p t ion e xp e r i m e n ts . F i gu r e 6 sh ows m e asu r e d r e s id u a l so l ve n t

c on te n t i n PM M A-a c e ton e c oa t i n gs d r y i n g i n a H i gh A i r f l ow D r y i n g E xp e r i m e n t th a t e n ab l e s ac c u rate

c on tro l o f th e d r y in g gas f l owr ate. In th e se e xp e r i m e n ts , th e r e s id u a l so l ve n t r e ac h e d a m i n i m u m at an

i n ter m e d i a te a i r ve l oc i ty o f 0 .01 m / s an d b e c am e s i gn i f ic an t l y h i gh e r a t h i gh e r a ir ve l oc i t i e s .

Vi n jam u r (2001) a l so d e ve l op e d a m od e l o f n on -F i c k i an so l ve n t tr an sp or t i n p o l ym e r c oa t i n gs

wh ich pass through a g lass trans i t ion dur ing drying. In essen ce the mod e l accoun ts for ho w stress

gr ad ie n ts in th e g l as sy p o l ym e r c on tr ib u te to so l ve n t t r an sp or t. T h e n o n -F i c k i an m od e l p r e d i c t s

an om al ou s sk i n n i n g as sh ow n i n F igu r e 7 . At h i gh gas f l owr ate s th e c oa t i n g te m p e r atu r e r i se s r ap i d ly to

th e ove n te m p e r atu r e an d a th i n low-c on c e n tr a t i on l aye r for m s a t th e su r fac e o f th e c oa t in g . T h i s l ow-

c on c e n tr a t ion l aye r d e v i a te s s i gn i f i c an t l y f r om w h at i s p r e d i cte d b y F i c k i an d i f fu s i on m od e l s , b e c au se

th e c on c en tr a t ion p r o f i l e sh ow s a s i gm oi d a l sh ap e ; i n a F i ck i an m o d e l , th e c on c e n tr a t ion gr ad ie n t a l ways

b e c om e s s te e p e r towar d s th e c oa t i n g su r fac e .

At l ow gas f l owr ate s , th e c oa t i n g te m p e r atu r e r i se s m o r e s l ow l y to th e ove n te m p e r atu r e an d th e

surface layer of low concentrat ion i s th icker; th is l eads to lower overal l res idual so lvent content at low

a i r fl ows . Vi n jam u r ' s m od e l m a tc h e s q u a l ita t i ve ly wi th e xp e r im e n ta l m e asu r e m e n ts o f an om a l ou s

sk i n n i n g i n PM M A-ac e ton e c oa t i n gs .

C h a l l e n g e s t o D r y i n g M o d e l s

M ath e m at i c a l m od e l s ar e an e f fe c t i ve w ay to p r e d ic t d r y i n g b e h av i or o f p o l ym e r so l u t i on

c oat i n gs. W h i l e d e ve l op i n g an d so l v i n g a fu n d am e n ta l d i f fu s i on -b ase d m od e l o f d ry i n g r e q u ir e s

s i gn i fi c an t e xp e r t i se i n n u m e r i c a l m e th o d s , th e r e su lt i n g m od e l s c an b e so l ve d q u i c k l y on s tan d ard

p e r son a l com p u te r s . T h e m ai n b ar r ie r to ap p l y i n g th e se m od e l s to m an y p r ac t ic a l c oa t i n g p r oc e s se s i s

th e s ign i f ic an t n u m b e r o f p h ys i c a l p r op e r ti e s n e e d e d i n th e m od e l s . For b in ar y p o l ym e r so l u t i on s , th e r e

are publ i shed methods to predic t the d i f fus ion propert ies , but to obtain accurate predic t ions requires

sop h i s t ic a te d te c h n i q u e s for m e asu r i n g d i f fu s i on c o e f f i c ie n ts .

Fu r th e rm or e , m an y p r ac ti c a l sy s te m s ar e m u l t i com p on e n t , w i th s e ve r a l so l ve n ts , p o l ym e r s , an d

ad d i t ive s . Ap p l y i n g b i n ar y m od e l s to m u l t i c om p on e n t sys te m s h as som e t i m e s b e e n su c c e s s fu l , b u t th e re

are m an y c ase s wh e r e m u l t i c om p on e n t d i f fu s i on i s i m p or tan t , an d th i s is an ac t ive r e se ar ch ar e a . Al so ,

m a n y c oa t i n gs c on ta i n c u r i n g c om p on e n ts , an d i t i s n o t d i f f ic u l t to au gm e n t m ath e m at i c a l m od e l s o f

d r y in g to i n c lu d e c u r i n g r e ac t ion s . H o we v e r , th e r e l a ti on sh ip b e tw e e n e x te n t o f r e ac t ion an d

c on c e n tr a t ion -d e p e n d e n t d i f fu s ion c oe f f i c i e n ts ar e n o t we l l e s tab li sh e d .

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in this paper. Prior collaboration with Peter Price from 3M w as also a critical part of developing the

material presented here.

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