Chain dimension of a guest polymer in the semidilute solution of compatible and incompatible polymers

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Chain dimension of a guest polymer in the semidilute solution of compatible and incompatible polymers

T. Nose

To cite this version:

T. Nose. Chain dimension of a guest polymer in the semidilute solution of compatible and incompatible

polymers. Journal de Physique, 1986, 47 (3), pp.517-527. �10.1051/jphys:01986004703051700�. �jpa-

00210232�

Chain dimension of a guest polymer ^{in the} semidilute solution of compatible ^and incompatible polymers

T. Nose

Department ^of Polymer Chemistry, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo 152, Japan (Reçu ^{le 15 mai} 1985, accepté sous forme définitive ^{le 12 novembre} 1985)

Résumé.

Des chaînes polymériques (degré ^de polymérisation N) ^sont dissoutes dans

solution semi-diluée

en

bon solvant d’un autre polymère (degré ^de polymérisation P, concentration 03A6);

calcule le rayon de giration Rg ^{et le second} coefficient du viriel A2 ^{des chaînes N à} partir ^des modèles de

blobs » thermiques ^{et de}

tration. Le paramètre renormalisé de volume exclu

la forme 03C4(03A6P303BD-1, ~) ^{où 03BD est} l’indice du _rayon

bon solvent (Rg ~ ^{N03BD) et ~ le paramètre} d’interaction entre les deux types ^de polymères. On donne des expressions analytiques pour Rg ^{et A2,} dont les lois d’échelles sont valables lorsque ^la concentration 03A6 dépasse ^la concentration de recouvrement 03A6* ~ ^P1-303BD:

Rg ~ N03BD fR{03A6N303BD-1, 03C4(N03A61/(303BD-1))1/2} ; ^{A2 ~} N303BD-2fA{03A6N303BD-1, 03C4(N03A61/(303BD-1))1/2}.

Ces formes analytiques déterminent plusieurs régimes ^dont

^{discute les} comportements caractéristiques. ^On

trouve

collapse des chaînes N dissoutes dans des chaînes P beaucoup plus ^courtes (N ~ P) lorsque 03A6 = 03A6*.

Abstract.

A small number of « guest » polymeric ^chains (index ^of polymerization N)

^{dissolved in}

^semidilute

solution of

host polymer (index P, concentration 03A6) which is itself in

good solvent; the radius of gyration Rg

and the second virial coefficient A2

^studied

^{the basis} of the thermal- and concentration-blob models. The renormalized excluded

volume parameter

has the form 03C4(03A6P303BD-1, ~) ^where

is the index for the N 2014 depen- dence of Rg ⁱⁿ

good ^solvent (Rg ~ ^{N03BD) and ~ the} interaction parameter between the guest and host polymers.

Analytical expressions of Rg ^{and A2}

^given

^{functions of N, P, 03A6 and ~,} which have the following scaled forms

for 03A6 > 03A6* ~ ^P1-303BD (overlap concentration) :

Rg

N03BDfR {03A6N303BD-1, 03C4(N03A61/(303BD-1))1/2}; A2 ~ N303BD-2 fA{03A6N303BD-1, 03C4(N03A61/(303BD-1))1/2}.

According ^{to these scaled} forms, ^several regions

defined, ^{and the} characteristic behaviour of each region ^{is dis-} cussed. A concentration-dependent collapse ^of

^{N-chain in} small P-chains (N ~ P) has been found around

03A6 = 03A6*

Physique (1986)

Classification

Physics ^Abstracts

36.20C - 61.40K

1. Introduction.

In the semidilute polymer solution, ^where polymer

chains overlap strongly, ^a polymer chain behaves like

a Gaussian chain even in a good solvent because of the

screening ^effect. However, ^monomer distribution within a distance of correlation length is of an excluded- volume type ^where excluded-volume effects are not screened out [1, 2]. ^{In this} paper, ^an attempt ^{will be} made to extend this idea of excluded-volume effects

on chain dimension to a more general ^{case that a} single ^{chain of} polymer ^{A is dissolved} in the semi- dilute solution of compatible ^or incompatible polymer

B with ^a solvent being thermodynamically good ^for

both polymer ^{A and} polymer ^B.

Radius of gyration ^{of a linear} polymer chain, Rg,

in a low molecular weight solvent, ^{exhibits different} dependences ^{on the} degree ^of polymerization ^N according ^{to solvent} goodness [1, 3, 4] :

with v = 3/5 ⁱⁿ good ^solvent region (i),

1/2 ⁱⁿ e-region (ii), ^and

1/3 ⁱⁿ globule region (iii) [5].

These regions ^{are defined} by ^the conditions N ^1/2 t> 1

(i), I N1/2 r I I (ii), ^and N 1/2 T - I (iii) [6], respectively, ^{with t} being ^the excluded-volume _para-

meter. So, ^the boundary ^{of these} regions depends ^{on N}

as well as solvent goodness

^{Similar situation} may be expected ^{for mean square distance between i-}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004703051700

and j-segments r2ij > ^within

chain. That is, r2ij >

may follow

with

depending on ’t 1 i - j 11/2 ^{in the} same manner as tN 1/2-dependence ^of

ⁱⁿ Rg. Therefore, ⁱⁿ

^cross-

over region ^a chain does not expand uniformly, but,

excluded volume effects depend ^{on the size of} partial

chain we are looking ^{at, and a} single ^{chain can be}

described as a succession of Gaussian blobs, ^each

made of N, segments, ^with excluded-volume inter- action between the blobs [7]. Here, ^the parameter Nt

defined by I T 1 Nt1/2

^{1, is a cutoff to} separate ^Gaus- sian and excluded-volume regions. ^{This treatment} (called the thermal-blob model) ^was adopted by

Weill and des Cloizeaux [8], ^{and Ackasu and Han} [9]

for interpreting ^{the crossover} behaviour of R. ^and hydrodynamic radius.

In the case of a chain in the semidilute solution, segment interactions beyond ^a certain distance (screen- ing length ç) ^{are screened out} by ^other polymers, and r2ij > ^{behaves like a} Gaussian chain for

r2ij >1/2 > _{ç, i.e., rij = r2ij} >1/2 oc i -j 11/2. Then,

a labelled chain can be described as a succession of blobs, ^each having ^{the size} of ç, ^{with no excluded-}

volume effect between the blobs (called ^{the concen-} tration-blob) [1]. Therefore, segment distribution of

a labelled chain is controlled by ç ^{as well as} N,. ^The

above theoretical considerations are consistent with the temperature-concentration diagram presented by Daoud-Jannink [7] ^{on the basis of} scaling ^laws.

Applying ^{the above} arguments ^{to the} present ternary system, ^a guest polymer ^{in the} semidilute polymer

solution with a good solvent, ^{we can} expect ^{that when} rij is less than the correlation length ç ^{of the matrix}

solution the rid ^exhibits good-solvent type behaviour, whereas, ^when _rid > ç, ^{the chain can be} regarded ^as

a chain in a low-molecular-weight ^solvent with solvent

goodness ^{T, but T in this case is the} excluded-volume parameter ^for interactions between the blobs with dimension ç, depending ^{on the} concentration and molecular weight ^{of matrix} polymer ^{as well as com-} patibility ^of polymer ^{A with} polymer ^{B. The case of a}

labelled chain in the semidilute solution should be a special ^{case that T is} substantially ^zero (N, >> N)

so that the chain is Gaussian when the observational scale is larger ^than ç, ⁱⁿ consistence with a usual

screening ^{effect A main} problem ^{in the} ternary system, therefore, ^{is to evaluate} the excluded-volume _para-

meter

as a function of concentration 0, polymeric

index P of matrix polymer ^and segment-interaction patameter X ^between polymer ^{A and} polymer ^B.

In this _paper- following ^the above arguments, ^we present ^a theory ^{for the} excluded-volume effects in a

guest polymer, ^derive expressions ^and expected scaling

laws for Rg ^{and the} second virial coefficient A2, ^and

examine the dependences ^of R. ^{and A2 on N, P, ql}

and x.

Theoretical and experimental ^studies on Rg ^{and A2}

for the ternary system ^{have not been made} extensively

as yet [11-16]. Joanny et ^al. [11] ^have theoretically

treated a ternary system ^with x

0 and N # P

(i.e., ^{bimodal mixture of} chemically-identical polymers

in good solvent) ^{on the basis of} scaling concepts (or ^blob model) ^to give the radii of gyration ^{as a}

function of N, ^{P and} 41. Recently, ^Tanaka [12] ^has presented ^a theory ^for concentration-dependent ^col- lapse ^{of a} large polymer ^{in a solution of} incompatible

small polymers, predicting ^{a discrete} shrinkage ^{of the} large-polymer ^{chain at a certain} concentration with

increasing concentration of the small polymer. Experi-

mental studies from the above viewpoints ^have just

been started except ^{for a} chemically-identical chain, that is, ^{studies on} polyvinylmethylether ⁱⁿ poly- styrene/good ^solvents by ^Cotts [15] ^and polystyrene

in polymethylmethacrylate/benzene ^{or toluene} by

Kuhn et al. [13], ^{Lin et al.} [14], and Numasawa et al. [16].

Some comparisons ^{of the} present theory ^{with these} experiments ^and existing theories will be made.

2. Description of the modeL

2.1 SYSTEM. - Consider a guest polymer (2) ^{in a} polymer ^solution consisting ^of polymer (1) ^and

solvent (0). ^The guest polymer and the matrix polymer

are linear ones made of N and P statistical segments with a length ^a, respectively. The concentration of the matrix polymer is 41 in volume fraction, ^{while the} concentration of the guest polymer ^is always infinitely

dilute. The solvent is good ^{for both} polymers, ^and compatibility ^{of the} guest polymer with the matrix

polymer ranges from good ^to poor. Namely, ^the

interaction parameters _Xij ^between segments ^{of i-}

and j-components ^{are taken as}

where _x is a variable changing ^from negative ^to positive, ^{and the} subscripts 0, ^{1 and 2 denote the} solvent, ^{the matrix} polymer, ^and the guest polymer, respectively.

2.2 EXCLUDED-VOLUME PARAMETER

We define

a quantity g ^{as the} segment ^{number of a} host-polymer

chain within the correlation length ç of the matrix

polymer solution, i.e.,

where vi ^{is the index}

ⁱⁿ good ^solvent region (i).

When the concentration 0 is less than the overlap

concentration 0*, ^{we have}

because we may put ç

R. (matrix polymer)

^aPv’

for 0 0*. For 0 > 0*, ^{we have} [2]

since = aPV1{4>/4>*)Vl/(1- 3Vl) ^with q6* ^{= a3} PI(AP ,)3

pl-3vi.

As mentioned in the introduction, ^a guest ^{chain can} be described as

succession of blobs, ^each having ^the

dimension ç. Within the blob, ^monomer distribution is of a good-solvent type, ^{and each} blob is made of g

segments, ^because equation (1) ^{also holds for}

guest polymer. To evaluate the interaction between the

blobs, ^we define the excluded-volume parameter

and the parameter N, for I i - j I > g, where N

is a cutoff to separate Gaussian and excluded-volume

regions for I i - j I > g. Denoting the renormalized excluded-volume for

unit consisting ^{of n} segments

as V(n) ^{with n} = I i - jl, ^we put

where r3(g) indicates the volume of a blob. According

to the classical theory, the excluded-volume parameter

’t

for the present ternary system ^is given by

for 0 ¹

^a function of P, 0, ^and x (see Appendix).

The classical theory, however, ^cannot take the effect of chain-connectivity ^{into account} correctly, ^{so that}

some modification must be necessary. The renorma-

lized excluded-volume parameter may be expressed ^as

which is obtained by ^a transformation of P - Pb, -> Ob ^{and x -} xb in equation ⁽⁶⁾ [2, 11], ^where Pb ^{is the}

number of blobs in

P-chain, q5b is the volume fraction of the blobs of P-chains, ^and Xb is the interaction _para- meter between blobs of the guest ^{and host} polymers : Pb

P/g ; Ob

(O/P) Pb ç31a3

Og 3vl-l ; Xb

(Xo 11(3 v 1 - 11) g = X (note ^{that the factor} 4>1/(3vl -1) ^{is the} probability for another segment ^{to be in an} adjacent

site [2]). This transformation leads the following ^facts.

(1) Equation (7) has the scaled form :

(2) ^When x

0, ^the renormalized t (Eq. (7)) ^is

reduced to T - 1 /(1 ⁺ Øb Pb) ^{and to T -} g/P ^for 0/0* > ^{1, which is} exactly ^{the same as} the result obtained by Joanny et ^al. [11].

(3) ^From equations (7) ^and (5), ^when x

0, N, changes ^from 4 g ^{to 4} g(N/g)2 (>> N) around ø = ø*

with increasing 0, which is consistent with the normal

screening ^effect.

2.3 MEAN DISTANCE BETWEEN THE i-th AND j-th

SEGMENTS.

Following ^the thermal-blob theory [7-9],

we

assume (n = I i - j I)

Here v2 is the index

for

Gaussian chain, i.e., V2

1/2, ^and v3 is

in the excluded-volume region, i.e.,

V3 ~ 3/5 ^for

> 0 and V3

1/3 ^for

^0.

When the two-body interaction is attractive (T 0)

and dominant (N, N), ^{the chain} collapses ^and

takes the globule state, ^so that in this region ^counter

interaction to the attractive two-body interaction,

which _may be the three-body interaction, ^{should be}

taken into account, ^{which assures the} stability ^of

the system. ^This may be also the case in the 0-region.

In the thermal-blob theory, ^{however, the} assump- tions for r(n) ^{made in} equation (9), ^{i.e., the crossover}

at n

NT: ^{and the} power-law ^{for n} > Nt ^allow

^to

treat the case of L 0 with no explicit evaluation of the counter interaction, ^{and the} assumption ^of V3 = 1/3 (reciprocal of spacial dimensionality) ^{for T 0 assures}

the stability ^{of the system. These} assumptions may be rationalized if the counter interaction is

mild func- tion of 0, ^{P and the} temperature ^{T. We have no} proof

for that at this stage, ^{but we can} show that the three-

body interaction estimated by ^{the mean field} approxi-

mation is

repulsive ^{one with} very week dependence

on

P, 0, ^{and T in a} ternary system (see Appendix). So, here,

adopt the thermal-blob concept, the expres- sion of equation (9), ^{even for}

0, supposing ^{that the}

counter interaction with repulsive ^{force is} substantially independent ^of 0, P, ^{and T.}

2.4 RENORMALlZED EXCLUDED-VOLUME.

As

crude

expression,

^{may assume} the renormalized exclud- ed-volume V(n) ^as

The second equation ^was derived from the requirement

that N, ^{should be} independent of the choice of unit-size

(n) for g ⁿ N,. ^That is, ^{we have} N,ln -

(r3(n)/V(n))2 (similar ^to Eq. (4)), ^then FM - ^± (nlN,)’ /2r3(n) = (n/g)1/2 zr3(n). Equation (9) gives

a

discrete change ⁱⁿ V(n) ^{at n}

g, while a continuous

cross-over at n

N,. ^In reality,

change ⁱⁿ V(n) may be _very sharp ^{around n} = g in general, ^{but not}

discrete.

2.5 RADIUS OF GYRATION.

The definition of Rg

is given by

From equations (11) ^and (9), ^{we have}

for N, ^{N. Here}

^{have used the} approximation

because N is assumed to be large enough.

2. 6 SECOND VIRIAL COEFFICIENT.

The second virial coefficient A2 of osmotic pressure is given by (see Appendix)

which can be calculated from equations (16) ^and (10)

with n

N using the results in section 2. 5, ^where Mo

is the molecular weight ^{of a} segment ^{of the} guest polymer.

Now, equations ^for Rg ^{and A2 are given as functions} of g ^and N, ^{which are evaluated} by equations (2), (3),

and (5) ^with equation (7). Then, Rg ^{and A2 can be}

calculated as a function of 0, P, N, ^and x.

3. Results and discussion

3.1 SCALING LAW.

Expressions ^of Rg ^{and A2}

obtained in the preceding sections have the following

scaled forms :

which is reduced to

where Rgo

^{aNV1, and}

which is reduced to

where.

3.2 ASYMPTOTIC BEHAVIOURS.

According ^{to the}

above scaled forms, ^we may define several regions ^for

the semidilute solution (0/0* > 1), ^each exhibiting

characteristic behaviour of Rg ^{and A2. That is,}

In region I, ^the presence of matrix polymers ^has only

a

very weak influence on the intra- and inter-mole- cular interactions of a guest-polymer ^{chain so that}

the chain is essentially ^of good ^solvent type :

Region ^II is defined as

region ^of 41N 3B11 -1 > 1, i.e., N/P > (41141*)-1/(3B11-1), in the semidilute region (0/0* > 1), ^{where a} guest polymer ^chain overlaps strongly with the semidilute matrix polymers. ^In region 11-1, interactions between blobs are very weak

so that the chain is substantially ^{Gaussian :}

which can be derived from the requirement ^that Rg

^{aNV2 at a fixed 0. Then, equation (22) is reduced}

to

In region 11-2, the excluded-volume effect is domi- nant, and

^can put (note ^Eq. (27))

where the function f§ (x) = ^{1 at x 1,} and the second

equation ^can be derived from the requirement ^of Rg - ^{aNV3 at fixed P and r 1.} Equation (29) ^{is the}

scaled form for the whole _range of region ^II. Then, A2 ^is given ^from equation (22) ^as

More explicit expressions ^{for some} special ^{cases in} region ^II will be shown in the following, ^where

put

vi

3/5, v2

1/2, and V3

1/3.

(1) Suppose ^{the case that the} polymer (1) ^and

the polymer (2) consist of chemically ^{identical mono-}

mers, i.e., x

0. In this _{case, T -} P -1 l/J - 5/4

(l/JIl/J*)-5/4 > 0 (from ^Eq. (7)). ^{Then, for 1} N0514 (pl/J5/4)2, i.e., (l/J I l/J*) - ^{5 /4} NIP (l/J I l/J*)5/4 (Region II-1 ),

and for N0514 gion 11-2),

(2) ^{Near the} 0-point ^{where x is non-zero} but falls in a

certain limited _range, region ^II-1 is realized. That is,

when X ’" à(4)/4>*)-S/4 ^x ₂ ^{or more} precisely 11"1 = p

one has

(3) Suppose ^{the case that 1} > I X I > (4)14>*)-5/4

where

X(1 ⁺ x/2) ~ - x. Then, ^{we have the} following regions :

region ^{11-1 :}

^P-1 0-5/4 I X I (N-’ 0-1/4)1/2 (which requires the condition N/P (0/0*)5/4)

region

3. 3 NUMERICAL RESULTS. - Putting ^the proportiona- lity ^{constant in} equation (7) ^{as 2 so that t}

^{1 when} 41 0*, ^and taking Vt

3/5, v2

1/2 ^and V3

1/3,

numerical values Of T, Rg/Rgo, ^{and A2/A20 were cal-}

culated from the results in section 2. 5 and 2. 6, ^and

are represented ⁱⁿ figures ^{1 ~ 5.}

Figure ¹ shows the excluded-volume parameter given by equation (7) against log 0/0* for various values of _x. With increasing 0/0*, the value of

decreases around 0/0*

^{1 from} unity ^to approach

a constant value which is characteristic for each value of _x. The constant is given by T(0/0* -+ large) = - x(2 ⁺ x), ^that is, t

^{0 for} x

0, t ^{0 for} x > 0, and T > 0 for X 0 in a ^realistic range of _x where x I

is not much greater ^than unity.

Fig. ^1.

Renormalized excluded-volume parameter

as a

function of log (0/0*) for various x-values.

Fig. ^2. - log (Rg/Rgo) ^{and A,IA,O}

^{function of log (l/JIl/J*)}

for various values of N/P ⁱⁿ

^of x

0.

log (R.IR.,,);

b : A21A2o. (The

H ⁱⁿ figure ^{2a indicate} region ^11-2

for N/P

104.)

Figure ^{2 shows} plots ^of log (Rg/Rgo) ^and A2/A2o against log (0/0*) ^{as a function of} N/P ^{for a} chemically-identical polymer combination : x

^0.

In case of N/P 1, ^as 0/0* increases, ^{the chain} begins ^{to shrink} when g ^becomes comparable ^to N, and _goes into an asymptotic region (ølø* > 1) ^of

Gaussian type exhibiting Rg

0-1/8 (Eq. (32)). ^On

the other hand, ^{in case of} N/P > 1, Rg ^{begins to de-}

crease around 0/0*

^{1, and} eventually reaches the

asymptotic region (0/0* >> 1) ^{of Gaussian} type, ^but when (ølø*)5/4 N/P, there exists another asympto- tic region (11-2) ^of good-solvent type, ^where R. OC 0 - 1/4

holds (Eq. (34)), which is indicated by ^{arrows for} N/P

^{104 in} figure 2. For each case, A2 also shows a

characteristic behaviour. In case of N/P ^1, A2 sharply ^{decreased at} (ølø*)5/4

N/P, ^{while in case of} N/P > ^1, A2 gradually decreases around 0/0*

^1,

and reaches region ^11-1 (Eq. (33)), through region ^11-2 (Eq. (35)) ^when N/P > ^{1. In} region ^11-2, A2/A20 ^is expressed by

^{universal function of} 0/0* independent

of N/P. ^{In any cases,} the value of A2 ^{tends to be zero}

when the value of 0/0* ^increases.

Fig. ^3. - log (Rg/Rgo) ^{and A21A2o}

function of log (0/0*)

for various values of N/P ⁱⁿ

^{of x}

^0.1.

log (Rg/Rgo) ;

b : A2/A2o.

Figure ³ represents ^the 0/0*-dependence ^of Rg/Rgo

and A2/A2o ^{for a case of} incompatible polymer ^combi-

nation (x

^0.1 > 0) with various values of N/P.

Similar to the case of _x

0, the chain dimension

begins ^to decrease around c/>Ic/>*

^{1 for} N/P > ^{1 and} around g

^{N for} N/P ^{1 to} eventually ^{reach the} asymptotic region (11-2) of Gaussian type ^{with the} power of - 1/3 : Rg oc 0-1/3 (Eq. (42)). ^{It is quite} interesting ^that Rg ^{decreases more} dramatically ^around 0/0*

1 with increasing ^value of N/P, which will be discussed in detail later. Corresponding ^{to the above}

behaviour of Rg, ^{the A2-value changes from positive to} negative at g

^{N for} N/P ^{1 and} gradually ^de-

creases around 0/0*

^{1 for} N/P > ^1, then increases with the _power of - I(A2 oc - c/> -1), becoming ^very

small in magnitude. ^It should be noted that the con-

centration at which A2

⁰ correspond ^{to the con-}

centration at which Rg ^{begins to shrink. Therefore,}

even at A2

0, ^the chain dimension is still expanded

and almost equal ^{to that in a} good ^{solvent, which is}

due to the fact that repulsive excluded-volume effects still prevail ^between segment pairs within the chain

at A2

^0.

Fig. ^4.

log (Rg/Rgo) ^{and A2/A20}

^{function of log (0/0*)}

for various values of N/P ⁱⁿ

^{of x}

0.1 Curves for

NIP >, 102

indistinguishable from that for N/P

a :

log (RgI Rgo); ^{b : A2/A2o.}

Figure ^{4 shows the results of} R. ^{and A2 for a case of} compatible polymer combination (x

^0.1 0)

with various values of N/P. ^{No essential} change ⁱⁿ Rg

is found, i.e., the excluded volume is always positive,

but the decrease in Rg ^{appears around g}

^{N for} N/P ^{1 and} 0/0*

^{1 for} N/P > ¹ by ^{a certain}

amount corresponding ^{to the} change in r-value from

T(o

0)

^{1 to} 1:(l/JIl/J* >> 1) = I X 1’/’ (Eq. (40)).

Similar to the other cases, the A2-value ^decreases sharply at g

^{N for} N/P 1, ^or gradually ^around 0/.0*

^{1 for} N/P > 1, ^{but tends to take a} positive

constant-value (Eq. (41)) because of compatible ^com-

bination. For the case of N/P 10-1, ^{one can see that}

the A2-value ^{increases once after}

sharp decrease,

then decreases gradually, corresponding ^to region ^I ( g ^N Nt) ^{and II} (N > Nt), respectively. ^The

increase in A2-value ⁱⁿ region ^{I is due to}

good compa-

tibility ^{of the two} polymers.

Figure ⁵ represents ^the N/P-dependence ^of Rg ^and

A2 at 0/0*

10 for various values of x. (Here, note

that Rgo ^{oc N 3/5 and A20}

^{N-1/5.) The N- and}

P-dependences ^of Rg/Rgo ^{and A2/A20 are described}

Fig. ^5. - log (R,IR,,) ^{and A,IA,,}

function of log (N/P)

for various x-values ^at ql/ql*

^10.

log (RgIRgo) ; ^{b :} A2/A20’

as scaled functions of N/P. ^With increasing N/P, ^the

chain dimension Rg ^{goes into the} Gaussian-type region (Region ^II-1: Eq. (38)), and then the excluded volume type region, region ^11-2 (Eqs. ^(40), (42)) ^{when! =1= 0.}

Similarly, ^with increasing N/P ^after N/P

1, ^the A2/A2o-value approaches ^a positive ^{constant value} (i.e., ^a positive excluded-volume type) ^{when t} > 0, ^and

once decreases (being negative) then increases with the

-

4/5-power ^of N/P (Eq. (43)) ^(i.e., A2 ^{oc N -1 : a} negative excluded-volume type) ^{when 0. When}

T ⁼

0, ^{it is} always ^{zero for} N/P > ^1.

3.4 COMPARISON WITH EXISTING THEORIES AND EXPE- RIMENTS.

The present ^{treatment is} essentially

based on the idea of Daoud et al. [2] ^for

^labelled polymer chain in semidilute solution in good ^solvent.

Therefore, when y

^{0 and P}

N,

theory ^is

reduced to the theory ^{of Daoud et} al., giving exactly

the same results for Rg ^{and A2 at the} semidilute limit :

Rg oc N -1/2 l/J -1/8; A2 OC N -1 l/J -1 (Eqs. (32) ^and (33’)).

As pointed ^out already, ^{when x}

0 the excluded- volume parameter

^{has the same form}

^{that of} Joanney et ^al. [11], ^{so that the} present results for

asymptotic behaviours of Rg ^{of N-chain}

^their

results : Rg/Rgo ^{= 1 for 0/0 * 1 ;} Rg/Rgo

(0/0 *) - ’/’

(Eq. (34)) ^for (NIP)’I’ > (P/O* > 1, ^and Rg/Rgo

f ((PI4)*) (NIP)415 1-1/8 (Eq. (32)) ^for 4,10 * > (N/p)4/5 (see also table II in Ref. [11]).

As seen in figure ^{3, we found a} sharp ^{decrease in}

Rg of N-chain in small P-chain around 0 ~ 0* ^with increasing concentration 0 of P-chain. Tanaka [12]

has theoretically predicted

^similar transition from coil to globule ^{at the limit of} N/P ->

According ^to

his theory, it is the first-order transition occurring ^at 4> ~ P-1, whereas, ^{in the} present theory, Rg ^changes drastically ^but continuously at 0 - 0* _ ^{p-4/5. The}

mean-field approximation in Tanaka’s theory may be

responsible for the different P-dependence of transition concentration. In fact the excluded-volume parame- ter T of equation (6) by the mean-field theory changes

from positive ^to negative at 0 - ^{P-1. It seems to be} unlikely ^{that the} collapse ^{would occur at}

^concen- tration 0 - ^{P -1} which is less than the overlap concentration ql * (~ ^{P - 4/5} > P-1). ^The continuity

of Rg ^at the transition in the present theory ^{is due to}

the continuous change in r2ij ) ^{at n}

^{Nt (Eq. (9)).}

As far as the thermal-blob model is used, the transition should be mild, having ^no singularity. ^{So, the} collapse

found here is not

real transition but only

rapid shrinkage of the coil with increasing 0. Retaining

the expression ^{for r,}

^{can construct a} theory ^{for the}

case of

0 without using the thermal-blob model,

for example, ^with using ^the Flory-type calculation of excluded-volume effects [5]

^other approaches [6].

However, whether the collapse is of the first-order or

mild shrinkage ^{is still}

open question ^{both theo-} retically ^and experimentally [17-19].

The results for Rg ^{and A2 in case of x}

^{0 with}

P

N have been basically ^verified by experiments

with neutron scattering [2]. P-dependences ^of Rg ^and

A2 ⁱⁿ chemically-identical chains have been measured

by Kirste and Lehnen [20], but, ^for binary systems with no solvent so that direct comparison ^{with the} present ^{results cannot} be made. For

compatible polymer combination (r > 0), polystyrene (PS) ⁱⁿ polyvinylmethylether (PVME)/toluene ^{with N} > P, Cotts [15] ^{has found that, with} increasing ^concentra-

tion of PVME, A2 ^{decreases more} rapidly ^{in the} range

0 > 0* ^than in 0 ø*, ^with A2 ex ø-1 ^{similar to}

the case of _x

0. Our theory ^{does not} conflict with this fact, suggesting

very small x-value ^for PS/PVME.

For

incompatible polymer combination (r > 0), polystyrene ⁱⁿ polymethylmethacrylate (PMMA)/ben-

zene (or toluene), Kuhn et al. [13], Rosen et al. [14]

and Numasawa et al. [16] have found that A2 ^vanishes

at

certain concentration of PMMA which _may

correspond to 0*. (Kuhn et al. showed that the con-

centration 4>0 ^{at which} A2

⁰

proportional ^to

the reciprocal of intrinsic viscosity [11]. ^Note [q]

0*-’.) ^{Rosen et} al. and Numasawa et ale pointed ^out

that

at A2

⁰ (0 = 00), the chain dimension of PS is almost the same as or slightly less than that in _pure benzene, being ^much larger ^{than the} unper- turbed dimension. This fact is quite consistent with the present ^result (Fig. 3). On the other hand, ^Kuhn

et ale and Rosen et ale have found that the ^concen- tration 00 ^was independent ^of temperature ^{and mole-} cular weight ^N and also that A2 ^was represented by

a

single ^{master curve of}

function of 0/00 ^for

fixed N with various P over the wide range of N/P.

Their findings ^{seem to} be in conflict with

theory (see Fig. 3) ^which predicts that, ^{in a certain} range of

N/P 1, 00 should have N-dependence ^and _A2 ^{is not}

a

universal function of 0/0*. However, ^{it is} supposed

that the N/P range and/or ^N range they ^used

^not appropriate ^for detecting ^such theoretical predictions.

In fact, Numasawa et ale have found for P > N - g

that the temperature ^{at which} A2

0 increases with

increasing ^{N and} appears ^to level off at higher ^mole-

cular weights (N), in consistence with the theory.

It should be noted that the present theory ^{is too}

crude for quantitative descriptions ^of A2 in ^real poly-

meric systems, ^{because of}

very crude approximation (Eq. (10)) ^{with a discrete} n-dependence ^of V(n).

Concentration-dependent collapse of N-chain in a

small P-chain has not been found experimentally ^for synthetic ^flexible polymers ⁱⁿ organic solvents, ^but

DNA in polyethylene oxide/water [21]. Experiments

are really interesting and needed to verify ^{the occur-}

rence of collapse and examine the P-dependence ^of

the transition concentration.

4. Conclusions.

A theory ^for Rg ^{and A2 in a} ternary semidilute ^solution has been presented by combining ^and extending ^the

ideas of thermal- and concentration-blobs. The fol-

lowing results have been obtained :

(1) ^An expression ^{of the} renormalized excluded- volume parameter

^{has been} presented ^as

^function

of P, ql, ^{and X} (Eq. (7)).

(2) Chain dimension Rg ^{and A2}

^{given in analy-}

tical forms as functions of N, P, 0, ^and x.

(3) ^These expressions ^of Rg ^{and A2} have the scaled forms represented by equations (17) - (22). According

to these scaled forms, ^several regions ^{are defined for} ø > 0*, ^each exhibiting its characteristic behaviours of Rg ^{and A2} (Eqs. ^{(24) -} (31)).

(4) ^A concentration-dependent collapse, ^{which is}

not discrete but _very sharp, ^{has been} predicted ^for

N >> ^P around 0 = 0*.

Appendix

CALCULATION OF OSMOTIC PRESSURE FOR A TERNARY SYSTEM.

Taking the solvent (0) and the matrix

polymer (1) ^as diffusible components, ^{and the} guest polymer (2) ^{as a} non-diffusible component,

^have the following equilibrium conditions :

where gk(67r, 60i) is the chemical potential ^{of k-com-}

ponent per mole when the _pressure and the i-compo-

nent volume-fraction Oi deviate from their mean

values by ^{the small amounts 67r and} 6ql;. ^{From equa-}

tion (A. 1),

^have

that is,

where T is the absolute temperature, ^{P the} pressure, and

C8J-lk/8P)T,/J ^{the partial molar volume of k-com-}

ponent. On the other hand, ^{from the} condition Y- ^oi

^1,

^have

i

If

have expressions ^for gk (k

0, 1) ^as

function of Oi (i

0, 1, 2),

^can compute 07E/O(P2 ^from equa- tions (A. 2) ^and (A. 3). ^{Here we} adopt ^the Flory-Huggins theory, ^which gives ^the following expressions ^{for the}

present ^{case where} Po = 1, Xoi

X02

0, ^and x12

X [22] :

where Api is the chemical potential ^of i-component per mole, the chemical potential ^{of the} pure liquid being

taken as a standard, R ^the gas ^constant and Pi is the number of segments ^{in an} i-component ^chain.

From equations (A. 2) ^and (A. 3) ^with eqs. (A. 4) ^and (A. 5), ^{we obtain}

Here, ^{we have assumed} V o

V 1/ PI- Expanding equation ^{(A. 6) in} respect ^of 412 ^at 412

^{0, we} eventually ^have

where CPiO is CPi{CP2

0) and CPI0 + .!É.oo

1. (Therefore, CPI0 is cP in the test.) Since (IIRT) (ðnlðCV) = (1/M2) ⁺ 2 A2 ^{C’ +}

^{and C’}

M20 02/VO, then the second virial coefficient A2 ^is given by

Here, M2 is the molecular weight ^{of the} guest polymer ^and M20 is the molecular weight ^{of its} segment, being equivalent ^to Mo ⁱⁿ equation (16).

To know

nature of the three-body interaction for ^{a case} of _x > 0 that will give

negative at 0 > 0*,

the third virial term of7r is calculated by taking the second derivative of equation (A. 6) ⁱⁿ respect Of (k 2 ^at 02

⁰

under the condition of 4>O/Pl

^constant (constant composition of the matrix solution), ^for simplicity. ^That is,

we have from equation (A. ⁶⁾

The w-value is positive (repulsive interaction) ^and changes ^with PI 4>1 ^as follows when x > 0, ^which may give

the nature of the three-body interaction :

The cv-value exhibits a very mild change ^with 01 PI ^{and T,} having ^no singularity ^at A2

^0. (Note that X may

change gradually ^with T.)

References

[1] ^DE GENNES, ^P. G., Scaling Concept ⁱⁿ Polymer Physics (Cornell University ^{Press, Ithaca} N.Y.), ^1979.

[2] DAOUD, M., COTTON, ^J. P., FARNOUX, B., JANNINK, G., SARMA, G., BENOIT, H., DUPLESSIX, R., PICOT, C.,

DE

GENNES, ^P. G., Macromolecules 8 (1982) ^804.

[3] FLORY, ^P. J., Principles of Polymer Chemistry (Cornell University Press, Ithaca, N.Y.), ^1953.

[4] YAMAKAWA, H., ^Modern Theory of Polymer ^Solutions (Harper ^{and Row,} N.Y.),1971.

[5] ^DE GENNES, ^P. G., ^J. Physique ^{Lett. 36} (1975) L-55 ; ibid. 39 (1978) ^L-299.

[6] This criterion for the region (iii) ^{has not been esta-}

blished yet

other than the logarythmic-term

correction [5]. ^See for instance ALLEGRA, G., GANAZZOLI, F., Macromolecules 16 (1983) ^1311.

It should also be noted that three-body interactions must be taken into account for describing

^detail-

ed nature of the collapse ^and interpreting ^the

0398- and the globule regions

correctly [5].

JOANNY and BROCHARD, ^J. Physique ⁴² (1981) 1145,

have theoretically predicted

strong contraction of

polymer ^{chain in}

^{melt of} incompatible polymer ^{chains similar to the usual} collapse ^of

single ^{chain in}

poor solvent.

[7] DAOUD, M., JANNINK, C., ^J. Physique ³⁹ (1978) ^331.

[8] WEILL, ^G.

CLOIZEAUX, J., ^J. Physique ⁴⁰ (1979) ^99.

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KUHN, R., CANTOW, ^{H. J.,} Makromol. Chem. 122

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[17] ^NIERLICH, M., COTTON, ^J. P., FARNOUX, B., ^{J. Chem.}

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[18] SUN, ^S. T., NISHIO, L, SWISLOW, G., TANTKA, T.,

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[19] PERZIINSKI, R., ADAM, M., DELSANTI, M., ^J. Physique

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[20] ^{KIRSTE, R.} G., LEHNEN, ^B. R., Makromol. Chem. 177

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[21] LERMAN, ^L. S., Proc. Natl. Acad. Sci. 68 (1971) 1886 ; LAEMMLI, ^U. K., ^{ibid. 72} (1975) ^4288.

[22] ^KURATA, M., ^Modern Industrial Chemistry (Asakura

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