Microphase separation transition in uniform deformed interpenetrating polymer networks

Microphase separation transition inuniform deformed interpenetratingpolymer networks

Michael Schulz*Fachbereich Physik, Martin-Luther Universita« t, 06099 Halle, Germany

andHarry L. FrischDepartment of Chemistry, State University of NewYork, Albany,12222, NY USA(Received 6November 1996; revised 1 July 1997)

The microphase separation transition in IPNs is an important phenomenon, which can be explained as acoupling between a network chain interaction and the e�ect of irreversibly closed crosslinks, which preventa usual macrophase separation. We study in the present paper we study the in¯uence of external defor-mation on the microphase separation based on the random phase approximation and classical Landau±Ginzburg theory. # 1998 Published by Elsevier Science Ltd. All rights reserved.

(Keywords: microphase separation transition; IPNs)

INTRODUCTION

Interpenetrating polymer networks1±6 (IPNs) con-sist of two or more crosslinked polymers. They aremore or less intimate mixtures of two or more dis-tinct, crosslinked polymer networks held togetherby permanent entanglements with only accidentalcovalent bonds between the polymers, i.e. they arepolymeric `catenanes'7±9. The entanglements inIPNs must be of a permanent nature and are madeby crosslinking.Due to the thermodynamic incompatibility of

polymers, polymer blends have a multiphase morph-ology. However, if mixing is accomplished simul-taneously with crosslinking, phase separation maybe kinetically controlled by interlocking permanententangled chains10,11.The theoretical description of IPNs includes two

parts: the ®rst part is the investigation of IPN-for-mation as a result of the combination of chemicalnetwork growth and di�usion controlled dynamicsof the monomers and pre-gel network units. Espe-cially in the case of a repulsive interaction betweenthe network components, the ®nal structure of thenetwork is determined by the competition betweenthe irreversible chemical reactions and the reversi-bleÐbut progressively frozenÐphase transition.

The second part is the description of the equili-brium properties of an IPN with a given topology.Although the crosslinks and entanglements of thenetwork chains are irreversibly ®xed, the composi-tional structure depends strongly on the tempera-ture or what is equivalent to the Flory±Hugginsparameter ��T� describing the strength of inter-chain interaction. Normally a blend of polymerchains undergoes a phase transition, if this �-para-meter becomes large enough. In comparison, thetopological restrictions of an IPN (as a result of theirreversibly formed crosslinks) prevent the totalseparation of such materials. This behaviour issimilar to that of a melt of block-copolymers12.Thus the phase separation becomes incomplete, i.e.only microphases of the order of the mesh size ofthe network are created. Furthermore the quen-ched stochastic disorder of the topology leads to anadditional disturbance of the microphase separation,so that a regular order of the microphases is com-pletely suppressed or at least limited to small scales.However, the formation of microphases is well

known from the microphase separation transitionof block copolymers. If a microphase separationoccurs in such materials, the microdomains form aregular macrolattice13±18. The theory of the micro-phase separation transition for copolymers wasdeveloped in the last decade19±21. A mean ®eldapproach for the MST was taken by Leibler12,

Computational and Theoretical Polymer Science Vol. 7, No. 2, pp. 85±94, 1997

# 1998 Published by Elsevier Science Ltd

Printed in Great Britain. All rights reserved

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*To whom correspondence should be addressed

COMP. AND THEOR. POLYMER SCIENCE Volume 7 Number 2 1997 85

using the random phase approximation (RPA).Here, the thermodynamic potential was construc-ted for a system of diblock copolymer by consid-ering weak perturbations of the homogeneoussystem. It is remarkable, that only two parameters(the product of the Flory parameter and the chainlength �N and the composition f) controls themicrophases in this theory. A generalisation of thisRPA was given by the Hartree±Fock approach22.which considers the e�ects of the order parameter¯uctuations. This theory is facilitated by a trans-formation of the block copolymer Hamiltonian toa general form previously analysed by Brazovs-kii23. Numerical simulations24 con®rm the Hartreecorrections to the RPA.Another interesting system is a two component

simple network, which is formed by crosslinkingtwo linear polymers A and B in their coexistenceregion and then cooled down to a temperature,where the MST should take place. This case wasrecently considered theoretically25,26 and experi-mentally27. Such a system can be described by asimple Hamiltonian (without nonlinear terms)

H �Xq

�c ÿ �� q2 � C

q2

� �j��qj2 �1�

This Hamiltonian have near the maximum of thecorresponded structure factor the same analyticalstructure as the Hamiltonian used by Brazovski23,e.g. (1) is also valid for the MST of block copoly-mers, only the constants have an other meaning.Under consideration of nonlinear terms (whichfollows particular by the expansion of the mixingfree energy part26,28 in powers of the relative den-sity ¯uctuations), it is possible to determine thein¯uence of the ¯uctuations by using a renormali-sation group approach29. In result of these calcu-lations a small deviation from the mean ®eldtheory was predicted.Here we present a mean ®eld description for the

equilibrium state of IPNs which allows thedescription of the phase transition and the equili-brium properties of such networks. The quenchedcharacter of the topological e�ects (topologicalstructure of the subnetworks and their mutualentanglements) are represented by e�ective para-meters in such a theory.A ®rst mean ®eld theory of the equilibrium and

phase stability based on phenomenological argu-ments30 consists of the construction of an e�ectivefree energy density F by the free elastic energydensity of the subnetworks Fsub and the interactionenergy density Fint between monomers of bothtypes

F � Fsub � Fint �2�

The discussion of (2) leads to an understanding ofthe homogeneous (non±separated) phase region inthe limit of small crosslinking. Note that becauseof the very high molecular mass between perma-nent entanglements the contributions to the freeenergy of these loops between the subnetworks areirrelevant for the homogeneous region. On theother hand, the neglect of this part of the freeenergy leads to an instability of the inhomogeneous(microphase separated) region. Thus we the mustconsider the elastic e�ects of the subnetworks andthe permanent network loops between the subnet-works preventing a macrophase separation.

THE LOCAL DISPLACEMENT FIELD

The formation of IPNs leads to an irreversibly®xed topology of the network chains. Thus, theinteraction between the monomers leads to achange of the local structure, but not of the topo-logical constraints. On the other hand, the struc-ture of an IPN depends on the distribution ofcrosslinks and ®xed entanglements. Hence, a mean®eld description of the equilibrium state of IPNsneeds a reference state, which considers the topol-ogy of the underlying IPN. A reasonable referencestate is the local structure in the high temperaturelimit because all interactions which are based onenergy scales becomes irrelevant and only thetopological induced interactions remain relevant.This reference structure is a result of irreversiblethermodynamic processes during the formation ofthe IPN. In general, IPNs undergo both nucleationand growth31, and spinodal decomposition duringthe synthesis of both subnetworks, in which spino-dal decomposition often predominates32. A part ofthe resulting inhomogeneities will be ®xed bycrosslinks and mutual entanglements and cannotbe deleted by su�ciently high temperatures. Thesimplest case is the reference state with an idealhomogeneous structure. Such a state can beobtained by a simultaneous formation of bothsubnetworks during a very short time starting froma homogeneous initial mixture. The following cal-culations are based on this reference structure, butan extension based on inhomogeneous structures ispossible and will be discussed in a subsequentpaper.In general, the position of the monomers at ®nite

temperatures is changed in comparison to thepositions in the reference state. In the case of a two-component IPN (components A and B) which weconsider here we can introduced the displacement-

86 COMP. AND THEOR. POLYMER SCIENCE Volume 7 Number 2 1997

Uniform deformed interpenetrating polymer networks: M. Schulz and H. L. Frisch

®elds uA�r� and uB�r� which describes the displace-ment of monomers of type A and B, respectively,located at the position r in the reference state. Bothdisplacement ®elds can be written as

uA�r� � w�r� � �1ÿ f �v�r� and

uB�r� � w�r� ÿ f v�r� �3�

with a possible common displacement w�r� and therelative displacement v�r� of the subnetworks. Theaveraged composition ratio f corresponds to thevolume fraction of component A in the referencestate. Here we restrict our calculations to the caseof a uniform external pressure, i.e. w�r� can writtenas w�r� � !r. Thus, the deformation tensor

"A;Bij �1

2uA;Bi;j � uA;Bj;i �

Xl

uA;Bl;i uA;Bl;j

!

with uA;Bi;j � @uA;Bi =@xj becomes

"ij � ! 1� !2

� ��ij � 1

21� !� � vA;Bi;j � vA;Bj;i

� �� 1

2

Xl

vA;Bl;i vA;Bl;j

�4�

with vAi � �1ÿ f �vi and vBi � ÿfvi. Furthermore,one can obtain from (4) the change of the localparticle concentration cA;B � dNA;B=dV:

cA;B0

cA;B�

��det 1� 2"A;B� �

q�5�

Note that we assume that monomers of both typeshave the same size. This consideration simpli®esthe conversion between concentration and volumefraction, but it has no relevant in¯uence on thephysical behaviour of the microphase separation.Using (4), the ratio can be expressed as

ln det 1� 2"A;Bÿ �� 6 ln 1� !� � �

2Pi

vA;Bi;i

1� !

ÿ

Pi;l

vA;Bi;l vA;Bl;i

1� !� �2

�6�

up to the second order in the relative deformationgradients vA;Bi;l . Without any relative deformation

cA;B

cA;B0

� 1

1� !� �3 � �

The deformation value � can used as a controlparameter, which describes the action of the exter-nal pressure. Furthermore (6), suggests that thetrace of the relative deformation gradient is one ofthe possible order parameters for the description ofthe microphase separation of IPNs. In the subse-quent calculations we introduce a normalised ver-sion of this order parameter

� � f 1ÿ f� ��13

Xi

vi;i �7�

To understand this choice we obtain from (5) and(6)

cA;B � cA;B0 1� !� �ÿ3 1ÿPi

vA;Bi;i

1� !

264375

up to ®rst order terms in the deformation gradi-ents. Thus, the local concentration of both com-ponents become with c0 � cA0 � cB0 and cA0 � fc0and cB0 � �1ÿ f �c0

cA � c0 1� !� �ÿd fÿf 1ÿ f� �P

i

vi;i

1� !

264375

� c0�� fÿ ��

cB � c0 1� !� �ÿd 1ÿ f�f 1ÿ f� �P

i

vi;i

1� !

264375

� c0� 1ÿ f� ��

Hence, the local composition ratio (volume frac-tion) of components A and B are

'A � cA

cA � cB� fÿ ��r� and

'B � cB

cA � cB� 1ÿ f� ��r�

respectively. Thus, the local order parameter ��r�can be interpreted as the ®rst order displacement ofthe local composition ratio.

AVERAGES AND CORRELATIONS OF THEDISPLACEMENT FIELD

The topological disorder IPNs shows no regularordered structures in the homogeneous and also in



the inhomogeneous phase in contrast to block-copolymers. But compositional ¯uctuations ininhomogeneous phases are partially ®xed over thetotal observation time, whereas ¯uctuations in thehomogeneous phase relax during a characteristictime scale.The relative displacement ®eld is always a sto-

chastic vector ®eld with a spatially homogeneousand isotropic distribution function. Therefore wealways obtain

vi�r�h i � 0 �8�

and as a consequence

vi;j�r� � � vi;j;k�r�

� � . . . � 0

Because of the spatial homogeneity and isotropy ofthe probability distribution function the generaltwo±point correlation of the relative displacement®eld v�r� is

vi�r�vj�r0� � � A jrÿ r0j� ��ij

� B jrÿ r0j� � �ri ÿ r0i��rj ÿ r0j�jrÿ r0j2

Ordinary Fourier transformation

vi�r� �Xq

vi�q� exp iqrf g

vi�q� � 1

V

�d3rvi�r� exp ÿiqrf g

leads to the equivalent representation

vi�q�vj�q0� � � ��jqj��ij � ��jqj� qiqj

q2

� ��q�q0 �9�

In general, the relative displacement ®eld can beused as a 3±dimensional order parameter, whichdescribes the microphase separation in IPNs. In thesubsequent investigations we introduce anotheruseful order parameter.The homogeneous phase is characterised by a

time and space dependent relative displacement®eld v�r; t� � �v�r; t�, i.e. the ¯uctuations of vdecays after a ®nite time interval. The inhomoge-neous phase consists also on a ¯uctuation part�v�r; t� and a static part vs�r�, which suggests apartially frozen in state of the IPN-structure. Thus,the microphase separation transition can be inter-preted also as an transition from an ergodic stateinto a partially nonergodic state. Using

v�r; t� � vs�r� � �v�r; t� �10�

(9) becomes

vi�q�vj�q0� � � �s�jqj��ij � �s�jqj� qiqj

q2

� ��q�q0

� �vi�q��vj�q0� �

Note that the cross correlations hvs;i�q��vj�q0�i canbe neglected without any problems. Clearly, thecorrelations of the ¯uctuation part can be writtenalso in the general form

�vi�q��vj�q0� � � �̂�jqj��ij � �̂�jqj� qiqj

q2

� ��q�q0

Thus, the nonergodic part of an IPN in the homo-geneous phase is characterized by �s � �s � 0,whereas the nonergodic part of each inhomoge-neous regime is controlled by a characteristiclength � ' Qÿ1, which has an order of magnitudeof the averaged size of microphases. For the fol-lowing investigations, it is su�cient to assume asimple �±like structure of �s and �s

�s�jqj� � �04�Q2

��jqj ÿQ� and

�s�jqj� � �04�Q2

��jqj ÿQ�

We remark, that any other reasonable functionalstructure of �s and �s can be used. The correlationof the longitudinal relative displacement ®eldvsk�q� � vs�q�q=jqj can now be written as

vsk�q�vsk�q0�D E

� ÿ�0 � �04�Q2V

��qÿQ��q�q0

whereas the transverse relative displacement ®eldvs?�q� � vs�q� ÿ qvsk=jqj becomes

vs?�q�vs?�q0� � � ÿ �0

2�Q2V��qÿQ��q�q0

From this point of view, we obtain after Fouriertransformation

vsk�r�� 2� �

� ÿ��0 � �0� and vs?�r�ÿ �2D E

� 2�0

Finally, the auto±correlation functions of thedeformation gradients are



Xi;j

vsi;i�r�vsj;j�r�* +

�Xi;j

vsi;j�r�vsj;i�r�* +

� Q2 vsk�r�� 2� � �11�

and

Xi;j

vsi;j�r�vsi;j�r�* +

� Q2

�vsk�r�� 2� �

� vs?�r�ÿ �2D E� �12�

THE FREE ENERGY OF IPNS

Order parameter and random phase approximationTo continue our investigation the free energy

should be expressed in terms of the order parame-ters. As shown below, the normalised trace of therelative deformation gradient � (7) is such a rea-sonable order parameter value. In general we have� � �s�r� � ��r; t�. In terms of the usual randomphase approximation all thermodynamic ¯uctua-tions ��r; t� are neglected, only the static (ornonergodic) ¯uctuations �s�r� are considered.Because of the distribution function of IPNs areinvariant against rotation and translation, the localstatic ¯uctuation is not representative of the wholesystem. we use therefore the averaged value� � ��h�2s �r�i

pas an order parameter of the whole

system.This order parameter takes into consideration

only the longitudinal part of the relative deforma-tion ®eld. Note, that because of (11) h�2s i corre-sponds to hv2ki. Therefore, the transverse part of thenonergodic relative displacement ®eld vs?�r� formsa second order parameter, given byv? �

��h�v?�r��2ip

which can be used as theremaining observable in a complete set of orderparameters.Finally, the inverse characteristic length Q � �ÿ1

is a third observable, which controls the free energyof the IPN and consequently, one must consideralso this value for a complete set of order par-ameters.

Elastic free energy of subnetworksIn the usual rubber elasticity theory, the elastic

free energy density has been expressed as30,33±37

fel � kBT��

2�2x � �2y � �2z ÿ 2b ln��x�y�z�h i

�13�

where � is the e�ective crosslink number density inthe reference state and �i are the elongation ratiosalong the three principal axes of deformation. Asregards the coe�cient b is controversial and severaltheories30,34±37 predict di�erent values, b � 0; 1;2Fÿ1 (F: functionality of the crosslinks). Thequadratic part of (13) accounts for the deformationenergy of the network chains, whereas the loga-rithmic part describes the compressibility of thesubnetwork. Using

�i � dl0idli�

��1� 2"�i�

qwhere "�i� are the diagonal terms of the deforma-tion tensor represented in terms of its principalaxes, one obtains immediately

�2x � �2y � �2z � 3� 2Xi

"ii and

�x�y�z ��det�1� 2"�

pIn the sense of a random phase approximation, weneglect all contributions of the thermodynamic¯uctuations. The averaged nonergodic part of theelastic free energy density of the subnetworkdepends only on the three order parameters dis-cussed above. We obtain with vA;B � A;Bv� A � 1ÿ f and B � ÿf � and (4), (7), (11) and(12)

2P

i"ii

� � 3 �1� !�2 ÿ 1h i

� 2�1� !� A;B Pivi;i

�� A;B�2P

l;ivl;ivl;ih i

� 3 �ÿ23 ÿ 1

h i� A;Bÿ �2P

l;i

�vsl;iv

sl;i

�� vl;i�vl;ih i�� 3 �ÿ

23 ÿ 1

h i� A;Bÿ �2

Q2

�vsk�r�� 2� �

� vs?�r�ÿ �2D E�

� 3 �ÿ23 ÿ 1

h i� A;Bÿ �2hDP

ivsi;i�r�

�Pjvsj;j�r�

E�Q2 vs?�r�

ÿ �2D Ei� 3 �ÿ

23 ÿ 1

h i��ÿ

23� A;B�2�2

f 2�1ÿ f �2 � A;Bÿ �2

Q2v2?

�14�

In view of the random phase approximation thecontribution of the thermodynamic ¯uctuationh�vl; �vl;ii was neglected. Furthermore, we obtain



ln det�1� 2"�h i � ÿ2 ln �ÿ A;Bÿ �2

�2

f 2�1ÿ f �2

ÿ�23 A;Bÿ �2

Q2v2?

�15�

Inserting (14) and (15) into (13) one ®nds theaveraged free energy of one subnetwork. Thesuperposition of the free energy of both subnet-works yields the elastic part of the averaged freeenergy density. Supposing that both networks havedi�erent e�ective crosslink number densities �A

and �B we obtain

2 Felh ikBT��0V0

� 3�ÿ23 ÿ 3� 2b ln �

� gÿ 2fg� f 2

f 2�1ÿ f �2��2 �ÿ

23 � b

� �� f 2�1ÿ f �2Q2v2? 1� b�ÿ

23

� �� 16�

with �0 � �A � �B and g � �A=�0 and the volume V0

of the non deformed state (note that � � V0=V�.

Density gradientsThe thermodynamic composition ¯uctuations

can be taken into account by the introduction of anadditional gradient term. This gradient contribu-tion to the free energy density can be written interms of the local concentrations of component Aand B, respectively38

fgrad � a0rcAÿ �2cA

� rcB� �2

cB

!

(a0 is a molecular constant, which is mainly deter-mined by the persistence length of the networkchains). For su�ciently small deformation gradi-ents (which control the monomer concentrations),one obtains up to second order

Fgrad � a0cA � cB

f �1ÿ f � �r��2 � a0

c0�

f �1ÿ f � �r��2

The averaging procedure leads under considerationof the random phase approximation arguments tothe result

�r��2D E

� f 2�1ÿ f �2�23

nrP

ivsi;i

ÿ �2D E� rP

i�vi;i

ÿ �2D Eo� Q2f 2�1ÿ f �2�2

3P

ivsi;i

ÿ �2D E� Q2�2

Thus

Fgrad

� � a0c0V0Q

2�2

f �1ÿ f � �17�

Monomer interactionThe e�ective interaction energy between both

components is in the lowest order a simple bilinearform in the particle concentrations cA and cB,which is proportional to the probability of a directcontact between both components, i.e.

fint � �cAcB � � cA � cBÿ �2

'A'B

� �c20�2 f �1ÿ f � � ��1ÿ 2f � ÿ �2� � �18�

The interaction parameter � can be transformed asfollows. The interaction free energy per monomercan be obtain from fint;mon � Fint�cA � cB�ÿ1, or inunits of kBT : fint;mon=kBT � �co�'A'B=kBT. There-fore, the proportionality constant � � �c0�=kBT isthe well known temperature-dependent Flory±Huggins parameter, which is a measure of thee�ective repulsion (� > 0) or attraction (� < 0)between the A and B units. Thus, the averagedinteraction energy can be expressed as

Finth i � �kBTc0V0 f �1ÿ f � ÿ �2� �

Mutually entanglementsThe free energy given by the sum of the three

above mentioned contributions gives a goodapproximation of the free energy of an IPN in thehomogeneous regime30,39. But it can be shown by asimple calculation, that there exists a critical tem-perature and therefore a critical �c for which thestatic structure factor becomes divergent at smallwave vectors q, S�q � 0� ! 1 i.e. the long range¯uctuations become divergent. This e�ect is chara-cteristic of a macrophase separation, but it is instrong disagreement with the behaviour of an IPN.The cause for this disagreement is due to the per-manent closed mutual entanglements which areneglected in the theory. These entanglements pre-vent the complete phase separation of both sub-networks and therefore they suppress the longrange composition ¯uctuations. Note that the term`entanglements' has no unique meaning in polymerscience. This term is used often in connection withnonequilibrium (transport) properties of polymermelts. Equilibrium properties of polymer solutionsor melts are not in¯uenced by entanglement e�ects.



This situation is changed in polymer networks. Insuch materials exists a class of entanglements ofnetwork chains which are ®xed because of theirreversibly closed crosslinks (permanent entangle-ments). The quenched topology (as a result of theirreversible preparation or the network or the IPN)is not longer controlled by the parameters of thethermodynamical equilibrium, i.e. a transitionbetween network con®gurations with di�erenttopology is not allowed. These permanent entan-glements of the network can not be neglected by ananalyses of the free energy density of IPNs. How-ever, the entanglements of two or more networkchains of the IPN behave as additional e�ectivecrosslinks between both subnetworks. In principle,one can replaces the action of the mutual entanglednetwork chains of both subnetworks by the actionof virtual chains, which connect these networks.Both the averaged length Le� of these virtualchains and their density are values, which aredetermined by the actual topological structure: Le�

is determined by the averaged number of entangle-ments per network chain and the averaged lengthof the network chains of both subnetworks. Thedensity of the virtual chains is approximately equalto the density of entanglements �ent in the referencestate. However, both values must be determinedseparately for each given topology of the IPN orthey must be calculated from the properties of thesubnetworks by using reasonable considerations.From the view of a phenomenological descriptionthese values can be understood as two new controlparameters of the IPN.The force which is necessary for ®xing both ends

of an e�ective chain in a distance R is given by thescaling

K � kBT

l0

R

Leff

� �

(l0 is the characteristic segment length). A Taylorexpansion of leads always in ®rst order to 3RLÿ1eff

which corresponds to a gaussian behaviour atsmall R. The higher order terms depend on theunderlying model, but in general, only odd termsgives nonvanishing contributions. Usually becomes divergent for RLÿ1eff ! 1. For the follow-ing investigations it is su�cient to consider onlythe ®rst nonlinear contribution, i.e.

K � kBT

l0

3R

Leff��3�

R

Leff

� �3" #

Thus the free energy density of the mutual entan-glements are de®ned by

fent � ��ent

�R0

K�R0�dR0

� kBT��entLeff

2l03

R�r�Leff

� �2

��3�

2

R�r�Leff

� �4" #

where R(r) is the deformation of the e�ectivechains at the position r. Thus, one obtainR�r� � juA�r� ÿ uB�r�j � jv�r�j and the averagedfree energy density (without the thermodynamical¯uctuations) can be rewritten as

2l0 Fenth i2kBT�V�entLeff

��vsk�r��2 � �vs?�r��2D E

L2eff

��3�

6

�vsk�r��2 � �vs?�r��2� �2� �

L4eff

�19�

The higher order moments of the static relativedisplacement ®eld, e.g.h�vsk�r��4i can be approxi-mated by products of second order moments, e.g.h�vsk�r��2ih�vsk�r��2i. But in contrast to the otherthree parts of the free energy (19) contains no gra-dient terms of the relative displacement ®eld. Thus,the nonharmonic term can not be automaticallyneglected, because large displacements of themonomers are not unusual for polymer networks.On the other hand the local gradient terms vi;j canbe assumed to be always su�ciently small.Thus (19) becomes ®nally

2l0 Fenth i3kBTV0�entLeff

��2

Q2f 2�1ÿf �2�23

� v2?

L2eff

��3�

6

�2

Q2f 2�1ÿf �2�23

� v2?

� �2

L4eff

THERMODYNAMIC EQUILIBRIUM OF IPNS

The averaged total free energy in the random phaseapproximation is then a sum of these four parts,i.e. hFIPNi � hFel � Fgrad � Fint � Fenti Followingclassical Landau theory40, the equilibrium statecorresponds to the minimum of the total freeenergy with respect to the order parameters gradi-ent �, Q and v?:

@FIPN

@�� 0

@FIPN

@Q� 0

@FIPN

@v?� 0



The last equation leads always to the trivial solu-tion v? � 0, i.e. v? is an irrelevant order parameterfor the description of IPN equilibrium states. Thus,we can substitute v? � 0 in hFIPNi and obtain therelevant part of the free energy

FIPN � V0

(�0 � �2�2 � �Q2�2

� kBTV0�entLeff

2l02

X

Q

� �2

��3�

2

X

Q

� �4" #)

with

�0 � kBT�02

3�ÿ23 ÿ 3� 2b ln �

n o� �kBTc0 f �1ÿ f �

�2 � kBT�02

gÿ 2fg� f 2

f 2�1ÿ f �2 �ÿ23 � b

h iÿ �kBTc0

X � �

f �1ÿ f ��13Leff

� � a0c0f �1ÿ f �

�20�

After introducing the reduced order parameter

x � �ÿ13

f �1ÿ f �Leff

�

Q

y � �ÿ16�2l0��

14��f �1ÿ f �Leff

p3kBT�entLeff� �14

Q

one obtain,

FIPNh i � �0 �0�0� By2x2 � y4x2 � x2 ��3�

6x4

� �with

�0 � 3kBT�entLeff

2l0

B � �2�13��

�0�p Leff f �1ÿ f �

�21�

Equilibrium is de®ned by

0 � 1

�0

@FIPN

@x� 2By2x� 2y4x� 2x� 2�3�

3x3

0 � 1

�0

@FIPN

@y� 2Byx2 � 4y3x2

�22�

This system can be decoupled. One obtains afterelimination of B

x2 � 3

�3�y4 ÿ 1ÿ � �23�

and straightforwardly from the second equation in(22)

0 � y2 B� 2y2ÿ �2

y2 � 1ÿ �

y2 ÿ 1ÿ � �24�

Note that each solution of (24) must satisfy thecondition

y � 1

otherwise (23) has no real solution for the orderparameter x.

DISCUSSION AND CONCLUSIONS

There exist two possible solutions of (24)

y � 1 and y ��ÿB

2

r

A simple computation shows, that the ®rst solutionis stable for B � Bc � ÿ2 whereas the secondsolution becomes stable for B � Bc. Thus, the ran-dom phase approximation suggests a second orderphase transition for IPNs at the critical parameterB � Bc. The corresponding �-parameter can bedetermined from (20) and (21):

��

6a0kBTnentl0Leff

s�ÿ

13

f �1ÿ f �� 32

� 1

2n0

gÿ 2fg� f 2

f 2�1ÿ f �2 �ÿ23 � b

h i �25�

with the averaged number of monomers per net-work chain n0 � c0=�0 and the averaged number ofmonomers per entanglement nent � c0=�ent. Fromthis point of view the knowledge of the tempera-ture±dependent �-parameter � � ��T� allows thedetermination of the shift of the critical tempera-ture due to crosslinking and external pressure (thecritical temperature is de®ned by ��Tc� � �c��,see Figure 1.Note that the microphase separation transition

becomes of second order as a result of the pre-averaging over all topological inhomogeneities inthe mean ®eld theory. A more detailed theory,



using a quenched average over the topology of thenetworks and their entanglements, may give con-tributions which lead to a ®rst order phase transi-tion in analogy to the Hartree Fock corrections inblock-copolymers22. However, the general beha-viour, which predicts a microphase separationtransition �� shows only some few irrelevantcorrections to (25) when the contributions over thequenched averaged topology are considered.The in¯uence of the pressure can be separated

into three regimes. The high pressure regime (i.e.� � V0=V!1� is dominated by the second termin (25)

��!1� � b

2n0

gÿ 2fg� f 2

f 2�1ÿ f �2

It should be noted, that mutual entanglementsbecomes irrelevant for the high pressure regime.Furthermore, the treatment of the high pressureregime is only an approximation, because the usedlinear theory of elasticity lost their validity at highpressure. However, introducing the asymmetry ofan IPN by using �g � gÿ f we obtain

��1� � b

2n0

�1ÿ 2f ��g� f �1ÿ f �f 2�1ÿ f �2

Each symmetric IPN (�g � 0) has a positive critical�-parameter, de®ned by ��1� � b=�2f �1ÿ f �n0�i.e. only symmetrical IPNs with incompatiblemonomers undergo a microphase separation tran-sition at high temperatures. But it should beremarked, that the critical �-parameter decreaseswith increasing pressure, i.e. there exists the possi-bility that a symmetric IPN in the homogeneousstate becomes inhomogeneous with increasingpressure.

On the other hand, an asymmetric IPN with�g > 0 and low concentration of component B(f ' 1) and an asymmetric IPN with �g < 0 andlow composition of component A (f ' 0), respec-tively can achieve a negative critical �±parameter.Thus, also IPNs with compatible components but astrong asymmetry can undergo a microphaseseparation at high pressure (see also Figure 2). Thecause of this behaviour is the di�erent networkelasticity of both subnetworks, which becomesrelevant at high pressure. Furthermore, the char-acteristic length � of the microphases becomes nearthe phase transition is

� � Qÿ1 � �ÿ162l0a0c0 f �1ÿ f �Leff

3kBT�ent

� �14

' 0

because �!1.Small pressures, corresponding to � � 1, inten-

sify the in¯uence of the ®rst term of (25), i.e. thee�ects of the asymmetry are covered partially bythe ®rst term. Thus, a microphase separation of theIPN occurs only for a su�ciently positive �-para-meter, i.e. for incompatible components A and B.As expected, the mutually entanglements becomesnow important and control now the characteristiclength of the microphases. In this case one canobtain a ®nite characteristic length � if the micro-phase transition

� � Qÿ1 � 2l0a0c0 f �1ÿ f �Leff

3kBT�ent

� �14

which is controlled by the inverse density ofmutually entanglements �ÿ1=4ent .The last regime corresponds to a negative pres-

sure, i.e. �! 0. Such a negative pressure can be

Figure 1 w(�!1) for di�erent asymmetry coe�cients: �g � 0 (fullline), �g � 0:02 (dashed line) �g � 0:04 (dotted line)

Figure 2 w(�) for small values of � for a symmetrical IPN �g � 0with �=1.5, 1.0, 0.3, 0.1, 0.03, 0.01, 0.003, 0.001, 0.0003 and 0.0001.� decreases in the direction of the arrow



simulated by a swelling the IPN, which is discussedin a subsequent paper. Here, we will discuss only thee�ect of a su�ciently small � on the microphasetransition. In general one can observe a strongincrease of �� for small values of � (Figure 2).Surprisingly the second term in (25) becomesimportant again, i.e. symmetry e�ects becomeimportant for the same relations between �g and thecomposition ratio f already discussed for the highpressure case. The origin of this behaviour is againthe di�erent network elasticity of both subnetworks.

ACKNOWLEDGEMENTS

This work has been supported by the NationalScience Foundation, Grant No. DMR 962-8224and the Deutsche Forschungsgemeinschaft, GrantNo. schu 934/1-3.

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Microphase separation transition in uniform deformed interpenetrating polymer networks