Mean-Field-Theory for Polymers in Mixed Solvents Thermodynamic and Structural Properties

20 Pages·0.22 MB·English

by  A. Negadi

Mean-Field-Theory for Polymers in Mixed Solvents Thermodynamic and Structural
Properties
Amina Negadi1,2, Anne Sans-Pennincks3, M. Benmouna1,2, and Thomas A. Vilgis*1,4
9 1Max-Planck Institut fu¨r Polymerforschung, Ackermannweg 10, D-55122 Mainz, Germany
9
9 2 University Aboubakr Belkaid of Tlemcen, Institut of chemistry, BP 119, Tlemcen 13000, Algeria
1 3 Centre d’etudes de Bruyeres le Chatel, B.P. 12; F-91680 Bruyeres les Chatel, France
n 4 Laboratoire Europ´een Associ´e, ICS, 6, rue Boussingault, F-67083 Strasbourg, France
a
J
February 1, 2008

1
2

TheoreticalaspectsofpolymersinmixedsolventsareconsideredusingtheEdwardsHamiltonian
]
t formalism. Thermodynamic and structural properties are investigated and some predictions are
f made when the mixed solvent approaches criticality. Both the single and the many chain problems
o
s areexamined. Whenthepuremixedsolventisnearcriticality,additionofasmallamountofpolymers
. shifts the criticality towards either enhanced compatibility or induced phase separation depending
t
a uponthevalueoftheparameterdescribingtheinteractionasymmetryofthesolventswithrespectto
m thepolymer. Thepolymer-solventeffectiveinteractionparameterincreasesstronglywhenthesolvent
- mixtureapproachescriticality. Accordingly,theapparentexcludedvolumeparameterdecreasesand
d may vanish or even become negative. Consequently, the polymer undergoes a phase transition
n
from a swollen state to an unperturbed state or even take a collapsed configuration. The effective
o
potentialactingonatestchaininstrongsolutionsiscalculatedandtheconceptofEdwardsscreening
c
discussed. Structural properties of ternary mixtures of polymers in mixed solvents are investigated
[
within the Edwards Hamiltonian model. It is shown that the effective potential on a test chain in

1 strong solutions could bewritten as an infiniteseries expansion of termsdescribing interactions via
v one chain, two chains etc. This summation can be performed following a similar scheme as in the
6
Ornstein-Zernikeseries expansion.
1
2
1
0 1. Introduction
9
9 Polymers in mixed solvents show different properties than in individual solvents raising interesting questions from a
t/ fundamental point of view1–6. Specific features of these systems originate from incompatibility of the two solvents
a
andasymmetryoftheirinteractionswiththe polymer. Ifthesystemexhibits apreferentialadsorptionofonesolvent
m
withthepolymer,adifferenceofsolventcompositioninsideandoutsidethepolymerdomainisobserved7–11. Another
- peculiar behaviorof dilute solutions of polymers in binary mixtures of badsolvents is cosolvency12–15 This behavior
d
n is quite unexpected since two non solvents for the polymer turn out to present together the properties of a good
o solvent which is unusual from the point of view of excluded volume effects. If no specific interactions exist between
c particles in the mixture such as hydrogen bonds or long range electrostatic interaction, one would expect that a
:
v mixed solventexhibits a smooth transition from one solvent quality to the other as the solventcomposition changes
i from 0 to 1 and the solvent power should be an averagedvalue of its individual components. This is not the case in
X
cosolvent systems where sharp changes characteristic of first order transitions take place. These transitions can be
r monitored through variations of radii of gyration, second virial coefficients, solution viscosities ... with the solvent
a
composition.
Inrecentyears,withthedevelopmentofnewtechniquesofpolymercharacterizationusingmolecularlabeling,dyes
and neutron scattering techniques, ...widespread research of polymers in mixed solvents has been witnessed in an
attempt to understand better their fundamental properties. These mixtures were studied by many authors using
different methods such as light and neutron scattering. Mixtures of ordinary and deuterated solvents such as light
and heavy water, ordinary and deuterated organic solvents (benzene, toluene ...) are used quite often in neutron
scatteringmeasurementstoachievemaximumcontrasttakingadvantageofthelargedifferencebetweenthecoherent
scattering lengths of hydrogen and deuterium atoms16.
In polyelectrolyte solutions, a low molecular weightsalt is added to increase the ionic strengthwhich weakens the
1
electrostatic forces and lowerstheir range. Salt introduces two additional low molecular weight components (co-ions
and counter-ions) which could present the properties of mixed solvents17.
Recently, mixed solvents were used to study the thermodynamics of polymer brushes18–20. Auroy and Auvray18
observedthe collapse-stretchingtransition of grafted polydimethylsiloxanechains by small angle neutron scattering.
Changing temperature and composition, they were able to control the height of the brush. The special boundary
conditionsintheterminalattachmentsresultintopolymerconformationanddynamicswhicharedifferentfromthose
of the bulk systems. The degree ofswelling of polymer brushes is largelydetermined by the solventquality. Rothers
et al.20 investigatedthe swellingof polystyrene(PS) brushesin toluene-methanolmixtures. By changingthe solvent
composition, they were able to see the collapse-stretching transition in the brush.
Problemsinvolvingpolymersin mixedsolventsarenotonly offundamental interestbut alsothey raiseinteresting
concretequestionsandofferalargespectrumofpracticalapplications. Theincreaseofamountofwaterintopolymer
domainsinnylontechnologycanbemonitoreddirectlybyNMRimagingtechniquesandgivedirectprofilesofsolvent
concentrationandingressdynamics21,22. Researcheffortsare engagedby polymerindustries and rayonmanufacture
to developnew cleanmethods to meet the requirementsof environmentalregulations23. In these efforts applications
of mixed solvent systems are often considered. Polymer synthesis in mixed solvents is another application which is
found to allow for a good control of the polymerization process24,25.
Inthispaper,weareinterestedinthepropertiesofpolymersinmixedsolventsfromthefundamentalpointofview
and present a theoretical investigation based upon the Edwards Hamiltonian formalism26. In the next section, we
review the case of a single chain in mixed solvents. The effective potential acting on the chain is calculated together
with the scattering functions. Phase behavior and chain conformation are obtained from the generalized partition
function in the Edwards Hamiltonian formalism. In section 3, we consider the case of strong solutions and discuss
changesinthephasebehaviorandthescatteringpropertiesduetothepolymerconcentrationofthemedium. Section
4 gives concluding remarks.
2. The single chain in mixed solvents
A. The Edwards Hamiltonian formalism
In this section, we consider infinite dilute solutions of polymers in mixed solvents and use the EdwardsHamiltonian
formalismtodeterminetheeffectivepotentialexertedonachainundervariousconditionsoftemperatureandsolvent
composition. For simplicity, we assume that all the chains have the same degree of polymerization N and the same
length l to avoid complications due to polydispersity. For this idealized system, the Edwards Hamiltonian is
3 N ∂R(s) 2 1 N N
′ ′
β = ds+ V [(R(s) R(s)]dsds
H 2l2 ∂s 2 PP −
Z0 (cid:18) (cid:19) Z0 Z0
N N
+ V [R(s) rA]ds+ V [R(s) rB]ds
AP − i BP − i
i Z0 i Z0
X X
N
1 1
+ V [rA rA]+ V [rB rB]+ V [rA rB] (1)
2 AA i − j 2 BB i − j AB i − j
i,j i,j i,j
X X X
whereβ =1/k T,k istheBoltzmannconstantandTtheabsolutetemperature,R(s)isthechainvariableandsthe
B B
curvilinear coordinates along the chain ( contour variable). The first term in the right hand side (RHS) of equation
(1) is the Wiener measure for random walks (assuming that unperturbated chains are Gaussian) and represents the
chain entropy due to its elasticity. The other terms in the RHS of equation 1 result from two-body interactions
between various particles in the medium either of the same or different species. In this expression,we have used the
assumption that the interaction potentials are short ranged and can be approximated with Dirac delta functions
V (r r ) = V δ(r r ) (2)
ij i j ij i j
− −
where V is the strength of the interaction potential between particles i and j. Using the standard procedures of
ij
functionaltheoryandEdwardsHamiltonianformalism,wederivethethermodynamicandthestructuralpropertiesof
2
the mixture under considerationfrom the generalizedpartition function. Starting from the Hamiltonianof equation
1, one obtains the partition function as a functional integral over all monomers and solvent coordinates
= R(s) drA drBexp β [(R(s),rA,rB] (3)
Z D i i − H i i
Z Z i i
Y Y
Where the superscripts A and B refer to the solvents. It is quite difficult to handle the partition function in the
particlepositionvariable. Rather,itismoreconvenienttotransformintocollectivevariablessuchasparticledensities
ρA = exp( iqrA)
q − i
i
X
ρB = exp( iqrB) (4)
q − i
i
X
Considering the single chain in solution and integrating over solvent coordinates,one obtains the EdwardsHamilto-
nian for the chain subject to the effective potential V (q)
0
3 N ∂R(s) 2 N N
′ ′
β = ds+ ds dsV (q)exp[ iq(R(s) R(s))] (5)
H 2l2 ∂s 0 − −
Z0 (cid:18) (cid:19) q Z0 Z0
X
The explicit form of V (q) is obtained following the procedure described below.
0
B. The effective potential of a single chain V0(q)
The partition function in equation (3) can be written differently by making use of the collective coordinates ρA and
q
ρB
q
3 N ∂R(s) 2 1 N N
= R(s) δρA δρBexp[ ds V dsds′δ[R(s) R(s‘)]
Z D q q −2l2 ∂s − 2 PP −
Z Z q q Z0 (cid:18) (cid:19) Z0 Z0
Y Y
N N
V exp( iqR(s))ρA V exp( iqR(s))ρB
− AP − q − BP − q
q Z0 q Z0
X X
1 1
V ρA 2 V ρB 2 V ρAρB ] (6)
−2 AA | q| − 2 BB | q| − AB q −q
q q q
X X X
wherethe letter P refersto the polymer. The particle conservationcanbe expressedby usingthe followingequation
δρ δ ρ exp( iqr) = 1 (7)
q " q− − i #
Z q i
Y X
and introducing field variables via the auxiliary Edwards random fields Φ
δ[ρq− exp(−iqri)] = δΦqexp[−iΦqρ−q]exp[−iΦq −iqri] (8)
i Z i
X X
With straightforwardmathematical manipulations, one arrives at
3 N ∂R(s) 2 1 N N
= R(s) δρA δρBexp[ ds V dsds′δ[R(s) R(s′)]
Z D q q −2l2 ∂s − 2 PP −
Z Z Z0 (cid:18) (cid:19) Z0 Z0
Y Y N N
V dsexp( iqR(s))ρA V dsexp( iqR(s))ρB
− AP − q − BP − q
q Z0 q Z0
X X
1 1 1 1
−2 (VAA+ S )|ρAq|2− 2 (VBB+ S )|ρBq|2−VAB ρAqρB−q] (9)
q A q B q
X X X
3
For an incompressible mixture, the total mean density fluctuation is zero and one has
N
ρA +ρB + dsexp( iqR(s)) = 0 (10)
q q −
Z0
Substitutingthisresultintoequation(9)andintegratingoversolventcollectivecoordinates,oneobtainsthepartition
function in terms of an effective Hamiltonian given by
H
3 N ∂R(s) 2 N N
′ ′
β = ds+ V (q)exp[ iq(R(s) R(s))]dsds (11)
H 2l2 ∂s 0 − −
Z0 (cid:18) (cid:19) Z0 Z0
wheretheeffectivepotentialV (q)isfunctionofthepotentialsV andthepartialstructurefactorsS (q)andS (q)
0 ij A B
2
V V +V V + 1
V (q)=V +V + 1 2V BB− AB AP− BP SB(q) (12)
0 PP BB S (q) − BP− (cid:16)V +V 2V + 1 + 1(cid:17)
B AA BB− AB SA(q) SB(q)
ThesubscriptsofpotentialsrunoverpolymerP andsolventsAandB. Tomakecontactwithmeasurablequantities,
it is convenient to express V (q) in terms of the Flory-Huggins interaction parameter χ 27
0 ij
V +V
ii jj
χ = V (13)
ij ij
− 2
Combining equations (12) and (13) yields
S−1(q)S−1(q) 2χ S−1(q) 2χ S−1(q)+D
V (q) = A B − AP B − BP A (14)
0 S−1(q)+S−1(q) 2χ
A B − AB
where D is a function of the corresponding Flory-Huggins interaction1 parameters only
D = 2χ χ +2χ χ +2χ χ χ2 χ2 χ2 (15)
AB AP AB BP AP BP − AP− BP − AB
Equation(14)couldbewritteninadifferentformwhichisappealingfromthe pointofviewofthe effectivepotential
acting on a polymer chain in mixed solvents
V (q) = 1 2χ′ = 1 2χ SB−1(q)−χAB−χBP+χAP 2 (16)
0 SB(q) − BP SB(q) − BP− (cid:0) SA−1(q)+SB−1(q)−2χAB (cid:1)
Similar results were obtained by others and in particular by Schultz and Flory1 and by Benoit and Strazielle28
using differentmethods ofthe meanfieldtheory. The EdwardsHamiltonianmethodgivesanotherroutefor deriving
theeffectivepotentialonachaintogetherwiththescatteringproperties. Ithasthemeritofshowingtheresponseofa
polymer chain undergoinginteractionfrom the mixed solventunder variousconditions of temperature, composition,
etc.
C. The scattering functions
In the following subsection the crudest approximation that can be imagined. Although we consider only one chain
in the mixed solution we use a mean field description of the RPA form. We are aware that this approximation is
very bad indeed, nevertheless it is used here to see the principal influence of the binary solvent on the interaction
potentials. Has we used Gaussian chains this approximation would be fine. This case belongs to V =0. Another
PP
possibilityofslightimprovementistotreatthepolymerplusexcludedvolumeinteractionaseffectively”bare”system
and use an appropriate Pad´e approximation including the selfavoiding walk exponent ν 3/5.
≈
Starting from equation 11, after some straightforward manipulations, one could write the Edwards Hamiltonian
in terms of the scattering functions for incompressible mixtures of polymers and mixed solvents. The following
4
conventionalnotation is adopted. For interaction potentials, we keep the subscript P for all the polymer chains but
for the single chain structure factor we use the subscript 1 meaning a single test chain. With this convention, the
Edwards Hamiltonian becomes
β = S−1(q)+V +V +S−1(q) 2V ρ1 2
H 1 PP BB B − BP | q|
q
X(cid:2) (cid:3)
+ V V V +V +S−1(q) ρ1ρA
AP− BP− AB BB B q −q
+1 V (cid:2) +S−1(q)+V +S−1(q) 2V (cid:3) ρA 2 (17)
2 AA A BB B − AB | q|
(cid:2) (cid:3)
The above equation takes a greatly simplified form in the matrix notation
1
βH = 2 ρ~TqS~(q)−1ρ~−q (18)
q
X
where ρ~ is a column vector
ρ1
ρ~(q) = q (19)
ρA
(cid:18) q(cid:19)
ρ~T isitstransposeandS~(q)isasecondranksquarematrixwhoseelementsdependuponthesingleparticlestructure
factors S0(q),S0(q), S0(q) and the Flory-Huggins interaction parameters χ ,χ , and χ .
A B 1 AB BP AP
S~−1(q) = SS−1−1(1q()q)+χSB−1(qχ)−2+χχBP SSB−−1(1q(q))−+χSAB−1−(qχ)BP2+χχAP (20)
(cid:18) B − AB− BP AP A B − AB (cid:19)
Inversion of this matrix gives the partial structure factors S (q), S (q) and S (q)
11 AA A1
S−1(q)+S−1(q) 2χ
S (q)= A B − AB (21)
11 (S−1(q)+S−1(q) 2χ )(S−1(q)+S−1(q) 2χ ) (S−1(q) χ χ +χ )2
1 B − BP A B − AB − B − BP− AB AP
S−1(q)+S−1(q) 2χ
S (q)= 1 B − BP (22)
AA (S−1(q)+S−1(q) 2χ )(S−1(q)+S−1(q) 2χ ) (S−1(q) χ χ +χ )2
1 B − BP A B − AB − B − BP− AB AP
S (q)=S (q)
A1 1A
S−1(q) χ χ +χ
= B − AB− BP AP (23)
− (S−1(q)+S−1(q) 2χ )(S−1(q)+S−1(q) 2χ ) (S−1(q) χ χ +χ )2
1 B − BP A B − AB − B − BP− AB AP
These results can be used to extract the properties of the mixture both in the thermodynamic limit by letting
q = 0 and at finite q where we can explore the chain conformation and the spatial distribution of particles within
the medium. We first consider the thermodynamic limit in the following section.
1. The thermodynamic limit q = 0
In this limit q =0 and equations (21) and (22) become
(φ−1 χ χ +χ )2
S−1 = φ−1+φ−1 2 χ + B − AB− BP AP (24)
11 1 B − BP 2(φ−1+φ−1 2χ )
(cid:18) A B − AB (cid:19)
(φ−1 χ χ +χ )2
S−1 = φ−1+φ−1 2 χ + B − AB− BP AP (25)
AA A B − AB 2(φ−1+φ−1 2χ )
(cid:18) 1 B − BP (cid:19)
5
For simplicity, we remove the argumentq from the structure factor in the thermodynamic limit q = 0. We consider
a mixed solvent in the vicinity of the critical temperature T where χ is close to the critical value χ and show
c AB c
how a small amount of polymer added to the mixture shifts the criticality condition. We also examine the way in
which the chain responds to the approach towards solvent criticality condition.
In the pure solvent mixture when q = 0, one has
S−1 = 2(χ χ ) (26)
AA c − AB
which can be obtained from equation (25) by letting φ =0; χ is the critical parameter for phase separationwhich
1 c
is function of the solvent composition only
1
χ = (φ−1 +φ−1) (27)
c 2 A B
Ifpolymerisaddedtothe solution,the partialstructurefactorinthe forwarddirectionS becomesfunctionofthe
AA
polymer volume fraction φ and can be written as
1
S−1 = 2(χ′ χ ) (28)
AA c − AB
This result shows that the critical parameter undergoes a shift by an amount ∆χ
c
(φ−1 χ χ +χ )2
∆χ = χ′ χ = B − AB− BP AP (29)
c c − c − 2(φ−1N−1+φ−1 2χ )
1 1 B − BP
′
One observes that the critical parameter χ = χ +∆χ either increases or decreases depending upon the sign of
c c c
∆χ which in turn is fixed by the sign of the denominator in equation (29). This sign is determined by the relative
c
magnitude of χ as compared to 1 + 1 . If χ the interaction between polymer and solvent B is smaller
BP 2φ1N1 2φB BP
than 1 + 1 , the quantity ∆χ is negative and the critical parameter decreases (i.e.χ′ < χ ). This means
2φ1N1 2φB c c c
that the polymer favors solvent demixing since a smaller interaction drives the solvents to the limit of stability.
For example, choosing N = 100, φ = 0.01 and φ =0.495, gives 1 + 1 = 1.5. This means that if χ is
1 1 B 2φ1N1 2φB BP
smaller than 1.5, ∆χ is negative and addition of a small amount of polymer (approximately 1 percent) induces
c
phase separation of the solvent mixture. If χ is higher than 1.5, the system behaves differently since addition
BP
of polymer would result into a positive ∆χ indicating compatibility enhancement of the solvent mixture. The
c
mixture phase separatesonly under a strong repulsionbetweenA and B solvents. These features can be observedin
′
figure 1 where χ is plotted against φ for two values of χ . One observes different tendencies, depending upon
c 1 BP
′
the value of the interaction parameter χ . For the lower value, there is a slight decrease of χ upon addition of
BP c
polymer. If χ is as high as 1.5 expressing a large polymer-solvent B incompatibility, a small amount of polymer
BP
would result into a large increase of ∆χ meaning that the solvents A and B prefer to remain mixed rather than
c
adsorbing on the polymer even though the interaction χ chosen in plotting this figure (χ = 1) is relatively
AB AB
high. This compatibility enhancementofthe mixed solventdue to the presence ofpolymer is quite spectacularsince
′
χ increases by practically an order of magnitude upon addition of a small amount of polymer. Qualitatively, this
c
behavior is consistent with the observation made on PS/(benzene+cyclohexane) by Varra and Autalik29. These
authorsfoundthatastronginteractionsbetweenPSandnonsolvents(cyclohexane)couldleadto areducedsolvent-
solvent interaction and hence a compatibility enhancement of the mixture. Furthermore, one finds that the shift
′
χ χ = ∆χ is only moderately sensitive to the interaction asymmetry between the polymer and the solvents
c − c c
A and B which is expressed by the parameter ǫ = χ χ . The shift in the critical parameter ∆χ is strong
BP AP c
−
upon addition of polymer evenfor smallvalues ofǫ. The interactionasymmetry parameter plays a more crucialrole
in connection with the polymer conformation near the solvent criticality condition as we shall see in the following
section.
The effective potential V of equation (16) is directly related to the classical excluded volume parameter acting
0
on a polymer in the infinite dilute limit. This is made clear by writing the partial structure factor S (q) in the
11
reciprocal form similar to Zimm’s equation
S−1(q) = S−1(q)+V (30)
11 1 0
Letting ∆χ =χ χ one obtains
0 c AB
−
6
1 (∆χ +ǫ)2
0
V = 2χ (31)
0 BP
φ − − 2∆χ
B 0
Foranupper criticalsolutiontemperature mixture,the interactionparameterχ increasesapproachingχ . Inthis
AB c
case, the polymer reacts in different ways depending essentially on whether ǫ is zero or not. This is a clear illustra-
tion of the importance of interaction asymmetry between polymer and solvents in determining the thermodynamic
behaviorofthesolution. Asthesolventmixtureapproachesthecriticalitycondition(∆χ 0),astrikingdifference
0
→
exists betweenthe casewherethe solventspresentthe sameinteractionwiththe polymer (ǫ=0)andthe casewhere
the two solvents interact differently with the polymer (ǫ=0). Equation (31) shows that in the case of a symmetric
6
interaction(ǫ=0), V decreases linearly with ∆χ . However,in the non symmetric interactionproblem (ǫ=0), the
0 0
6
third term in the RHS of equation (31) increases rapidly as ∆χ 0 and V undergoes a rapid decrease. Following
0 0
→
this sharpdecrease,the effectivepotentialvanishesandmayevenbecomenegativeasχ approachesχ . The chain
AB c
undergoesa phasetransitionfromaswollenstate tounperturbeddimensionsorevenacollapsedconfiguration. This
is illustrated in figure 2 where V is represented as a function of ∆χ for ǫ = 0 and 0.2. The upper curve in this
0 0
figure correspondsto the case where the interaction between polymer and solvents A and B are the same and ǫ=0.
It shows that V increases relatively smoothly when ∆χ 0.
0 0
→
One could also introduce an effective polymer-solvent interaction parameter χ via
0
1
V = 2χ (32)
0 0
φ −
B
With
(∆χ +ǫ)2
0
χ = χ + (33)
0 BP
4∆χ
0
Figure 3 represents the variation of the effective polymer-solventinteraction parameter χ as a function of ∆χ for
0 0
the same values of ǫ as in figure 2 and shows similar tendencies. For ǫ = 0, χ decreases slightly when ∆χ 0,
0 0
→
while for ǫ = 0.2, there is an inversion in the variation of χ expressed by a turn up and a sharp increase as the
0
mixed solvent approaches criticality indicating a strong polymer-solvent repulsion.
2. The small q limit
As a first approximation, we consider that q−1 1 which means that by scattering radiation,one observes the
≫
solvent particles as point like and the q- dependence comes entirely from the form factor of the polymer only. This
approximationcouldbeeasilyrelaxedifonewantstoincludeintramolecularinterferenceswithinthesolventparticles
themselves and explore by radiation scattering the internal structure of chains. Moreover, we will assume that the
form factor of the chain could be approximated by
1
P(q) = (34)
q2R2
1+ g
2
Following these assumptions, equation (17) can be written in the form
S (q =0)
11
S (q) = (35)
11 1+q2ξ2
0
Where the correlation length ξ represents the distance over which the polymer configuration is sensitive to solvent
0
fluctuations. It depends upon the polymer radius of gyration R and the excluded volume interaction V according
g 0
to the relationship
R2
ξ2 = g (36)
0 2(1+V φ N )
0 1 1
Figure(4) represents the variation of ξ /R as a function of ∆χ for ǫ = 0 and 0.2. For ǫ = 0, ξ /R decreases
0 g 0 0 g
continuously when ∆χ 0 while for ǫ = 0.2, ξ /R increases suddenly when ∆χ approaches zero. One can
0 0 g 0
→
7
visualize the polymer as a chain of blobs of size ξ . Over distances below the blob size, the chain is sensitive to
0
solventfluctuationsanditssizereducesasaresultofincreasingeffectiveinteractionχ . Overdistancesexceedingthe
0
blob size, the chaindoes not feel the effects of solventfluctuations andshows a comparativelyswollenconformation.
As the solvent approaches criticality when ∆χ 0, the correlation length ξ becomes comparable or larger than
0 0
→
R implying that the entire chain is subject to solvent fluctuations and undergoes a phase transition from a swollen
g
to a more compact conformation. For ǫ = 0 and χ = χ = 0.5, the chain feels theta solvent conditions even
BP AP
though ∆χ 0 and solvents demixing conditions are approached.
0
The blob p→icture for a polymer in mixed solvents was first suggested by Brochard and de Gennes3 who examined
the polymer conformation in a mixture of two good solvents, when the polymer affinity differs substantially for the
two solvents. They were the first to predict that the polymer adopts a collapsedconfigurationnear T even it would
c
be swollen in either of the pure solvents under similar conditions.
3. Many chains in mixed solvents
A. The effective potential of a test chain
In this section, we discuss the problem of many chains by extending the Edwards Hamiltonian formalism to strong
solutions of polymers and mixed solvents. The effective potential acting on a test chain together with partial
structurefactorsarecalculatedasafunctionofpolymerconcentrationandmixedsolventproperties. TheHamiltonian
formalism is extended by taking into account the effects of the many chains present in the medium. This gives the
following result for the partition function including summations over all chains designated by the letters α and β
= R (s) dr~A dr~Bexp[ 3 N ∂Rα(s) 2ds
Z D α i i −2l2 ∂S
Z Z α Z0 (cid:18) (cid:19)
Y Y Y X
1 N N
′ ′
V δ(R (s) R (s))dsds
PP α β
−2 −
αβ Z0 Z0
X
N N
V δ(R (s) rA)ds V δ(R (s) rB)ds
− AP α − i − BP α − i
α Z0 i α Z0 i
X X X X
1 1
V (rA rA) V δ(rB rB) V δ(rA rB)] (37)
−2 AA i − j − 2 BB i − j − AB i − j
ij ij ij
X X X
The subscriptα andβ run overallchains in the solutionexcepta testchain designatedby the subscript1 (meaning
single). Transforming space coordinates into collective density variables leads to the partition function
3 N ∂R (s) 2 1
= R (s) δρA δρBexp 1 ds V ρ1 2
Z D 1 q q −2l2 ∂s − 2 PP| |
Z Z Z0 (cid:18) (cid:19)
Y Y 1
V ρP ρ1 (V +S−1(q))ρP 2
− PP | q|| q|− 2 PP P | q|
q q
X X
−VAP ρ1qρA−q−VAP ρPqρA−q−VBP ρ1qρB−q−VBP |ρPq||ρBq|
q q q q
X X X X
1 1
−2 (VAA+SA−1(q))|ρAq|2− 2 (VBB+SB−1(q))|ρBq|2− VABρAqρB−q (38)
q q q
X X X
Where ρA and ρB, ρP and ρ1 are the collectives coordinates for the solvents A and B, the polymer P and the test
q q q q
chain 1, respectively. In order to identify the effective potential acting on a test chain together with the partial
scattering functions for the multicomponent mixture under investigation, it is convenient to write the partition
function in matrix form
8
3 N ∂R (s) 2 1
= R (s) δρ~ exp[ 1 ds V ρ1 2]
Z D 1 q −2l2 ∂s − 2 PP | q|
Z Z Z0 (cid:18) (cid:19)
Y exp[− 2V~(q)ρ~q− ρ~−q1M~X(q)ρ~−q] (39)
X X
where ρ~ and V~(q) represent the column vectors
q
ρA
q
ρ~ = ρB (40)
q  q 
ρP
q
 
1V Nexp( iqR(s)ds)
2 AP 0 −
V~(q) = 1V Nexp( iqR(s)ds) (41)
 2 BPR0 − 
1V Nexp( iqR(s)ds)
 2 PPR0 − 
 
and M~ (q) is a squareRthree-by-three matrix
1(V + 1 ) 1V 1V
2 AA SA(q) 2 AB 2 AP
M~ (q) =  12VAB 12(VBB+ SB1(q)) 12VBP  (42)
1V 1V 1(V + 1 )
 2 AP 2 BP 2 PP SP(q) 
 
Integrating over the collective variables ρ , one obtains the partition function as follows
q
3 N ∂R(s) 2 1
= R (s)exp ds V ρ 1 2+ V~−1M~ −1V~ (43)
Z Z D 1 "−2l2 Z0 (cid:18) ∂s (cid:19) − 2 PP q | q | q q q q#
X X
Recalling that = R (s)exp β yields
1
Z D − H
3 N ∂RR(s) 2 N N
′ ′
β = ds+ V(q) dsds exp( iq(R(s) R(s))) (44)
H 2l2 ∂s − −
Z0 (cid:18) (cid:19) q Z0 Z0
X
where one can identify the effective potential V acting on a test chain. If the volumes occupied by monomers and
q
solventmoleculesarethesame,thenV ,V andV areapproximatelyrepresentedbythesamequantitydenoted
PP AA BB
V∞ andonecanintroduce the Flory-HugginsinteractionparametersusingV∞ =Vij -χij wherethe subscriptsiand
j run over A, B and P. All polymer chains in the medium have the same interaction parameters and no distinction
existsbetweenthetestchain1andtheothers. TheeffectivepotentialV(q)isnotonlyduetothesolventsbutalsoto
allthe chainspresentinthe medium. Forincompressiblemixtures,the potentialV∞ goestoinfinity andone obtains
S−1(q)S−1(q)+D 2χ S−1(q) 2χ S−1(q)
V(q)= A B − AP B − BP A (45)
S−1(q)+S−1(q) 2χ +S (q)(S−1(q)S−1(q)+D 2χ S−1(q) 2χ S−1(q))
A B − AB P A B − AP B − BP A
Here D is defined in equation (15). It is interesting to note that the effective potentials acting on chains in infinitely
dilute solutions V (q) and in strong solutions V(q) are related via
0
V−1(q) = V−1(q)+S (q) (46)
0 P
This result is characteristic of the series expansion in the Ornstein-Zernike model30. It suggests that one can write
the total potential V(q) as an infinite series of interaction terms. The order of these terms depends on the number
ofintermediate chains involvedinthe interaction(seefigure5a). Eachdiagramcorrespondsto aparticularterm in
the series expansion
V(q) = V (q) V (q)S (q)V (q)+V (q)S (q)V (q)S (q)V (q)+... (47)
0 0 P 0 0 P 0 P 0
−
9
TheseseriescanbesummedupinasimilarwayasintheOrnstein-Zernikeexpansion(seefigure5b). Thesummation
gives
V(q) = V (q) V (q)S (q)V(q) (48)
0 0 P
−
which is equivalent to
V(q) = V (q)[1+V (q)S (q)]−1 (49)
0 0 P
B. The Edwards screening
For completeness, we remark that the Edwards screening can be recovered. The effective potential for many chains
in equation (47) can also be written in the following form
V2(q)
V(q) = V (q) 0 (50)
0 − S−1(q)+V (q)
P 0
Using Edwards prescription for the form factor of Gaussian chains
φ N
P P
S = (51)
P 1+ 1q2R2
2 g
where the radius of gyration is related to the degree of polymerization N and segment length l by the known result
R =l N/6. Substituting equation 51 into 50 yields
g
Vp q2ξ2
= (52)
V 1+q2ξ2
0
Where ξ is the Edwards screening length
l
ξ = (53)
√12V φ
0 P
In the infinite dilute limit, φ goes to zero and the screening length ξ tends to infinity. As one would expect, the
P
effective potential V tends to V in this limit. Figure 6 represents the ratio V/V versus qξ as given by equation 52.
0 0
The ratio V/V is small for qξ less than 1 indicating screening of the potential field acting on a test chain. When qξ
0
increases the ratio V/V tends to 1 and the effective potential is practically the same as in infinite dilute solutions.
0
Locally at large q, the test chain behaves as if it were isolated and is essentially not perturbed by the presence of
other chains in the medium. It feels only the fluctuations of the solvent mixture in its immediate vicinity.
ThisanalysisofscreeningpresentedhereandfollowingtheEdwardsprescriptionisonlyqualitative. Amoreprecise
treatmentisneededifonewantstohaveaquantitativeevaluationofthe screeningeffects. Suchanimprovementcan
be made using other methods such as renormalizationgroup theory methods or field theoretical tools31,32,6.
C. The scattering functions
The scattering matrix can be readily extracted from the partition function in the matrix notation
S−1(q)+S−1(q) 2χ S−1(q) 2χ S−1(q) χ χ +χ
S−1(q)= 1 S−1(q)B 2χ− BP S−1(q)B+S−−1(q)BP2χ SB−1(q)−χAB−χBP+χAP (54)
 S−1(q)B χ − χBP+χ S−P1(q) χ B χ−+χBP SB−1(q)−+SAB−1−(q)BP2χ AP 
B − AB− BP AP B − AB− BP AP A B − AB
 
The diagonal elements of the scattering matrix S(q) define the contribution to the scattering signal due to dif-
ferent constituents in the mixture. Three quantities are particularly relevant for our purpose in the present work.
Considering the solvent B as a background medium, one can obtain the diagonalelements of equation (54) denoted
S (q), S (q) and S (q). The properties of these partial structure factors are briefly discussed below.
11 PP AA
10