Computerized analysis of the viscoelastic properties of solid polymers by means of mathematically planned experiments

COMPUTERIZED ANALYSIS OF THE VISCOELASTIC PROPERTIES OF SOLID POLYMERS BY MEANS OF MATHEMATICALLY PLANNED EXPERIMENTS* YU. V. ZELEI~EV, V. g . NOVIKOV, L. N. SAVEL'EVA, A. N. 0VCHn~IKOV

and G. YE. GOLUBKOV All-Union Research Institute for Insulating and Foliated Dielectrics Moscow Textile Institute N. K. Krupskaya District Pedagogical Institute, Moscow

(Received 14 January 1977) The mechanical and dielectric relaxation properties of polymeric systems based on a diane epoxy resin have been investigated by a method involving the use of mathematically planned experiments. The results of calculations are compared with evaluations of experimental parameters of polymeric materials subjected to heat t r e a t m e n t procedures. The amounts of hardening agent consistent with minimal values of the elastic moduli have been determined.

THE viscoelastic properties of epoxy resins depend on their molecular structure and, in particular, on the density of the three dimensional network, which is determined by technological factors such as the type and the ratio of the resin and the hardening agent, and the time and temperature of hardening. From data published in [1, 2] regarding the viscoelastic properties of epoxy polymers it seemed that maxima would appear on curves of these technological factors plotted against properties such as the glass transition temperature Tg, the elastic moduli in the glassy and the high elastic states (El and E~) and temperatures of relaxation transitions appearing in an electric field and in a mechanical field. 2 In such cases the behaviour of polymeric systems can be described in terms of a mathematical model of the type of a second order polynomial k

k

k

Zb.x:,

y=bo+

(1)

where y is the response function, i.e. the property being determined, e.g. the elastic modulus; x~ and x~ are thermal processing factors; be, b~, b~j etc. are coefficients. This paper relates to our investigation of the visco-elastic properties of a diane epoxy resin hardened with isomethyltetrahydrophthalic anhydride. The experiment was conducted with the aid of a mathematical planning method [3] whereby the coefficients of the mathematical model (1) may be found. * Vysokomol. soycd. A19: No. 10, 2328--2333, 1977.

2673

YU.

2674

V.

el al.

ZELE1TEV

As variable factors we selecfed the resin/hardening agent ratio x~, and t h e t e m p e r a t u r e x~ and time of additional heat t r e a t m e n t xa. Variation of these ~echnol~li~a)/ft~o,tora ,w~s C~llrl'~4,.?~lt I ~: ~,h~r~:l?v~.,,. Y'I ~,', ,. '., 'Lt'; ,: W,i

•

Variables

Variation steps

I levels of Ithe variables

--'1.21

0

1

. 1.21 #

[ 1"0 /".'( ,:,:~it$ r)', )e : ".0"6.(1 ~J'j,Q:6;7, . . . .1.0 1 1.33" x~ / 180 | 25 150 i 1~ r lSO L 2o~ i x, ~ .i,~ ..... i:~2I l ,,;l.,t.~L I,., i~,:' I,, I ",i~i : li'
t~

t'Jl

~1

t,

'7

1.40/"

:1o

32

,tt ' >,

!(I .,.:,-.!..i~,,:,', .1; ' To obtain separate evaluations of coefficients bo and b~ we added six star points on coordinates (0, ~ , 6)rand one central p~lnt on coordinates (0, O, 0), where a for the three independent variables is equal to 1.21. Tables 1 and 2 give

th4'<~¢~i~o~','6~ ~5~ifi~':~h' % ' : ~ i ~ i ~ T6~[tti~' ~h'~d~'~¢~rtg~l~s"gIid"~l~ethree

_

"lfLi'IfiJ:~lil_l:

,,

:t.~.l_D.tll,']i¢JiiJ.lii~'

,/*14,\'itki.l,',ql.:,~i,l

fl,o,~tJ,'G'LJl},

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¢~1

I I

i

l','i'~'l

"/1"

Jl

.lilf

,

~,''l,l,

( ~ "I'~F"

I'

a

Arbitrary units ,'~:

(illljtili3

'

'J ~;!f

"t t ) f

~

I 'Jl'l~t

"I

.I, ,:5,,~,. :.d,Natubttltx, tlhi~l,.,:- :.:~ f .

JI I J p , , i , i . .

,~",I t

x 1, ~:ll;

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~:.;!

,., .,:o?.!o~~,,

x~,

X3

:, ,:),(7,j ~,i J~ ,l.~ ,,.lly

:~/i'

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,.

,~.,a~

),.;,,.,.,,,~%.

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18",, 7 i IPT, , i~ ~li

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Computerized analyaia d$ ~ ~ c ' l b r ( ~ r t i e s

of solid polymers

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TABLE 3. VALUES OF THE COEFFICIENTS I1~ EQUATIOI~

(2)

Cgef~eient s . . \ Parameter y bo

Tg, °K T~, ° K T m, ° K T~,

I

~8.3

52

b~ 7.9

1"7 1'8

~4.~1 v.9

9.9

,

°K

2.q

2~2.~i 9.8

E ~ X 10 -s, dyn/em ~ E ~ X 10 -~°, t dyn/em ~ t a n ¢fe X 10 "~

!2.~[ 0.3

E:

i~.t o.1

!9"~

oxpor,m0o~a,. , :

t

i

--0"84-0"44 -- 1"15-0'63 ---0-51-0.52 2.7 -- 1.62-2.38 O -- 1 3 . 5 0 - 1 5 . 4 0

1.9y25.9 /- 3.2 ,-',¢5 -2.0 9 ~ ,

i0~1-1.1/-3.4

~

0.8 ~i3

/I

°'k o,

~-0.21

2.0i/ 0.4-0.5!-0.3-.1.0

t ~,

i;

nance °Jr+"~e a~par~._ ~tus.J

O

1"1 1"5

(~.~ iw~.4l/- 2.0 [,~0.3 - 3:o ,~,,o.5

0-9~

-o.11-O.ljC-o.2 -~,1 l ° r a o o| o.~i :.o o",,j, o ,~.

0"5

~,%

baa

P

°

-- 8.90-7.30 -- 2.80-3.00

~

--3.1-2.9

)','

!

,

/i

J",!

~he ' mea~u~lr ' '" g,error { was 1~5~ for t]~e e~stlc" modulus

'

{ ' " (' (

i, •

~ .

.~

x

' ., ,

.

1) the gla~s tra,rsi~n te~peratlL~Jobtairm~l fro~ tl~ l electrical measuremelts by extr~polat'ion. •f the iog plot to~lo a ~nf=~0t

2~f2~/~i'

•

2) the temDerature • m ~ nlcal T~ measurements

. .

t

.

~,

of ~-transltlons ~btamed a t f = 1 0 - o Hz; lb. , ,~ ",

/

e

~'r.o~n electrical T~ and mecha~/ I

.-

,' \

3) the temperature ,of fl-tro(ns,tmns, Tp; 4) the dynamic m~f~ulu~',in the high elastm state 'E~" for T = T m + 5 0 ° and ' T ~ 2fi5°K; in the gl ass- lik e state Ekfor ,

,

Equations for th~ respolas~.su~ceaha~e .~ ~ .~ (, ,' .'~ 2

2

2

y=bo+b~xl+b~x2-}-b3x3+bl~x~+b2~x2+ba~xa-[--b~2XlX~+b13XlXa+b23x~x 3

(2)

Table 3 .gives ~ . ,,values Of. tile c,diieae~ts i~, ec~qatm~ t2) ~gr ~he p~ra~eter~ .in ,question. In,the lest column in Table ~3 we,'~aave,"gh~.degree o~ ~evaatm~ of the par~me~ex J calculated i a aeeo~daaaee wi~J~ eqlaa~n (2) from the. yalp~e o;t •~ h e ~ s a m e ~arame~er" £ou=c~ bv exV6t~ment. The resultant equations axe van0.

Y~. V. ZELEN~V et aL

~2676

with limits of accuracy as stated for a n y values of x~ t h a t are within the limits a c c e p t e d in a given experiment, i.e. in our case for a content of the iso-MTHPA hardening agent ranging from 0.6 to 1.4 mole per epoxy group, for heat treatment in the range from 150 to 210°C (with preliminary hardening at 120 ° for 10 hr) .and times of heat treatment ranging from 16 to 32 hr.

T,°C

200

180 xl

O.g

O'8

1.0 _

1.2

1.4

Ha~'denep, moles

•IO. l. Surface profiles for the elastic modulus in the glass-like state for the epoxy polymer at x8=16 (1, 2, 5, profile a); 24 (3, 6, profile b) and 32 hr (4, 7, profile c) and E~× 10-1'~--2'2 (1, 2, 5) 2.5 (3. 6) and 2"9 dyn/cm= (4, 7).

Figure 1 shows three surface profiles on coordinates x ~ - - x l with x a ~ c o n s t for t h e elastic modulus in the glass-like state. The profiles are ellipsoidal in character, a n d as x a (the time of heat treatment) increases, the contres of ellipses marked by the letters a, b, c are displaced increasingly in the direction of higher tempera-tures of heat treatment and an increasing content of the hardening agent.

Computerized analysis of viscoelastic properties of solid polymers

2677

c'~, 1o"°,a~.~/~ ~

2"G 2.4

I

g8 2-0 O.G

0.8

~i ~ 400I

l.O

1-2

14

xl,mo/es

__ 3

~

e

~oo~ ,

,

,\,

f

380

0.7

0.8

17

I-3

['5 x/,mole~

Fzo. 2. Respor~se surface profiles for the elastic modulus ir~ the glass-like state for the epoxy polymer (a-c) and for the glass transition temperature (d-J) at fixed temperatures of 150 (1), 180 (2) and 200 ° (3) the times of heat treatment being 16 (a, d), 24 (b, e) and 32 hr (c, f).

2678

Yr.

V.

ZE'LEI~EV

et

a~.

In Fig. 2a we have surface profiles on coordinates E'~=q~(xl) with x2, x3 : c o n s t . These appear in the form of curves with a minimum. Similar profiles for T m and E~ are exhibited in Figs. 2b and 3, in the form of curves with a maximum, and are described b y an equation of the type of

y = a +bxl +cx~l

(3)

When it is desired to show the relationship between any of the parameters under s t u d y and factors whereby it is being influenced, e.g. to shed light on one

....

,

,\,

_

Z.O l'G 1

0"7

0.9

1.3

14

1.5 .:U/ , m r~,/g.,"7,

FIG. 3. Response surface profiles for the elastic m o d u l u s in the high-elastic svate for t h e e p o x y p o l y m e r at t e m p e r a t u r e s of 150 (1), 180 (2) a n d 200 ° C (3), the times of heat t r e a t m e n t beiag 16 (a), 24 (b) a n d 32 hr (c).

of the ratios oi the starting components relative to the temperature x2 or the time xa of heat trea¢ment, we do so b y an analysis of equation (3) as the latter enables us to describe appropriate response surface profiles by equations: a) fi)r determining the effect cf the temperature of heat treatment v , ~';

,

;

2

y = a t o x~-pc x~, where

t

2

2

a =:bo~blxl-Fbaxa-Fbilxl-Fbaaxa-FblaXlXa; b'~blex1~-b~b23x3; c':b2~ X1, ~ a = c o n s t ;

Computerized analysis of viscoelastic properties of solid polymers

2679

b) for determining the influence of the time of heat treatment y=a"~b"x3~c"x 2

with xl, x2----const. I t can be seen from the Figures that the parameters t h a t are to be investigated have a maximum on curves plotted against the ratio of the components; the conditions of heat treatment are rather less influential, and the curves have a flatter appearance. This stems from the application of preliminary heat treatment for l0 hr at 120 ° which allows time for most of the chemical processes to reach completion, which is in agreement with the results of investigations reported in [4, 5J by authors investigating the extent to which the dynamic properties of epoxy resins is influenced by variable amounts of diamines under various conditions of heat treatment. Thus it has proved possible, with the aid of foregoing equations describing t h e behaviour of an epoxy polymer based on the diane resin and isomethyltetrahydrophthalic anhydride, to determine the parameters of viseo-elastic properties, and the extent to which the lat~er are influenced by the conditions of heat treatment. I t has been shown t h a t the amounts of the hardening agent at which the parameters in question have a maximum are non-uniform. Thus for T m and T: a maximum appears when the content of hardening agent amounts to 1.07-1-08 mole per epoxy group. The maximum for the fl-process T~ is outside the limits of the xl values adopted in the present experiment, and amounts to some 2 moles. The minimal value of e'~ corresponds to a hardener content of 0.95-0.97 mole, and the E~ minimum to 0.8-0.9 mole. Translated by R. J. A. HENDRY REFERENCES

1. G. Ye. GOLUBKOV, V. I. YELINEK and B. K. ARTEMYEV, Elektrotekhnika, No. 8, 33, 1966 2. R. C. ARRIDGE and J. H. SPEAKE, Polymer 13: 443, 1972 3. V. V. NAT,ITIOV and N. A. CHERNOVA, Statisticheskiye metody planirovaniya ekstremalnykh eksperimentov (Statistical Methods of Planning Extremal Type Experiments). Izd. "Nauka", 1965 4. T. HIRAI and D. KLINE, J. Appl. Polymer Sei. 17: 31, 1973 5. T. HIRAI and D. KLINE, J. Appl. Polymer Sci. 16: 3145, 1972