Transition of the bounded polymer layer to a rigid amorphous phase: A computational and DSC study

Polymer 55 (2014) 4658e4670

Contents lists available at ScienceDirect

Polymer journal homepage: www.elsevier.com/locate/polymer

Transition of the bounded polymer layer to a rigid amorphous phase: A computational and DSC study Georgios Kritikos* National Center for Scientific Research “Demokritos”, Institute of Physical Chemistry, Molecular Thermodynamics and Modelling of Materials Laboratory, GR-153 10, Aghia Paraskevi, Attikis, Greece

a r t i c l e i n f o

a b s t r a c t

Article history: Received 21 March 2014 Received in revised form 17 July 2014 Accepted 22 July 2014 Available online 1 August 2014

The study focuses on the transition of the bounded to solid surfaces polymer layer to a rigid amorphous phase (RAP). Based on previous Monte Carlo (MC) simulation studies on bulk polyethylene (PE), we refine our numerical variable density Self Consistent Field (nSCF) method in order the calculated density in bulk to follow the predictions of the MC simulation. The proposed modification of the SanchezeLacombe equation of state allows us to examine thermodynamic aspects of the glass transition. By imposing a glass transition (Tg ¼ 220 K) in the bulk we predict an earlier (during cooling), stronger transition of the bounded layer to a RAP, at ~350 K. Also at short separation distances we record the appearance of undisturbed polymer bridges. Differential scanning calorimetry (DSC) experiments on polydimethylsiloxane (PDMS) and polyamide 6 (PA6) nanocomposites suggest that although the crystallization can be significantly suppressed by the addition of nanoparticles, the RAP layer may locally equilibrate, above Tg, for long experimental times with the melt phase. Our results support the idea that a significant free energy barrier of entropic origin appears due to the RAP formation, below the melting temperature (Tm) and above the Tg. © 2014 Elsevier Ltd. All rights reserved.

Keywords: Rigid amorphous phase Glass transition Polymer crystallization

1. Introduction In simple molecules, liquid mobility and diffusion require a minimum of kinetic energy so that the atoms manage to escape the pairwise (e.g. Lennard-Jones) potential. Under a certain melting temperature (Tm) non polymeric (non glassy) materials solidify. But in glassy polymers as the temperature drops, before the liquid to solid transition, the chains can be trapped due to the caging effect [1,2]. Macromolecules need a minimum of free volume in order to diffuse and efficiently scan the configuration space. Moreover, during the glass transition it is observed a deviation from the Arrhenius behavior of constant activation energy [2]. Molecular simulations of confined, bounded on a surface polymers, by the use of accurate surface potentials (e.g graphite force field), have shown [3e11] that macromolecules above the glass transition undergo restricted dynamics, characterized by longer relaxation times. The dynamics of the bounded polymer layer seems to differ from that of a bulk melt near its glass transition temperature (Tg), [4,5]. Nanoparticles with attractive interaction with the polymers not only

* Tel.: þ30 2106503960. E-mail addresses: [email protected], [email protected]. http://dx.doi.org/10.1016/j.polymer.2014.07.048 0032-3861/© 2014 Elsevier Ltd. All rights reserved.

increase the polymer melt/solid interface but also they increase the overlap region between free and adsorbed chains. Depending on their geometry and dimensions nanofillers may nucleate polymer entanglements around their surfaces [10,11]. Several experimental works [12e21] have supported the idea that the bounded layer is immobilized above the major Tg, in a way that finally it does not participate in the glass transition. For the systems where the calculated heat capacity above the Tg is less than expected, the fraction of the missing melt is well established to be called “rigid amorphous fraction” or “RAF” [12]. Since it is neither a well ordered crystalline phase (CP) nor a mobile/melt phase (MAP), we can also use the term fraction of the rigid amorphous phase or RAP, when referring to this interphase [15]. This immobilized layer is found both around crystalline regions and nanoparticles [12e21]. Results [13,14] indicating a balance between the crystalline regions and the rigid amorphous phase on one side and the constant mobile (liquid) phase on the other side, enrich the questions about the origin of the polymer crystallization and the glass transition. Moreover, dielectric experiments on PDMS/silica nanocomposites indicate that the formed RAP layer (around 3 nm at Tg) has not a constant width but slightly decreases with increasing temperature [17]. Calorimetry experiments in amorphous polymer nanocomposites support the idea that the transition to a RAP layer is not

G. Kritikos / Polymer 55 (2014) 4658e4670

related to a second glass transition [16]. In the same case, the cooperative rearranging regions (CRR) were supposed to be either immobilized or mobile since no intermediate states were found [16]. We mention that the CRR have a size comparable the width of the RAP layer [2,17]. In previous computational studies of the polymer melt/solid interface by the variable density nSCF method [22,23], the volume of the lattice site was calculated based on the polymer density at an arbitrary temperature, while small density fluctuations around this temperature (above Tg) were taken into account by the insertion/ removal of voids. In this way, part of the free volume was captured inside the lattice site. This is considered a good approximation given the fact that the used SanchezeLacombe equation of state [24e26] provides us with an accurate estimation of the temperature dependence of the density, above the glass transition. In the glass transition region the equation of state predicts a smooth densification [23]. However, End Bringing Monte Carlo (EBMC) simulations [9] for a bulk polymer indicate a clear change in the slope of the temperatureedensity curve. Based on the simulation study by Alexiadis O, Mavrantzas VG. et al. [9] we attempt to refine our variable density numerical Self Consistent Field (nSCF) method in order to better fit the atomistic detail of the EBMC method and follow the density predictions for the bulk polyethylene. A direct comparison between the numerical SCF theory and the atomistic simulations has been also presented previously [8]. Here we extend the coupling of the mean field nSCF method with the atomistic EBMC to a broader temperature range focusing on the polymer/ solid interface. In the next section we give details about the theoretical formulation of the improved variable density nSCF method. The lattice is created based on the density of the polymer at the frozen state (~50 K below glass transition/Vogel temperature). Then the liquefaction of part or the whole polymer molecule is done due to the insertion of voids. This approach calculates more accurately the free volume distribution at the polymer/solid interface and allows us to make estimations about possible gradients in the viscosity and the mobility of the chain. Also we report the parameters describing the PE mapping. The fitting to the EBMC temperatureedensity curve allows us to propose a modification to the SanchezeLacombe equation of state. In this way we can shed light on thermodynamic aspects of the glass transition. In the third section we present our results. The choice of a computational study of an accurately simulated macromolecule like PE allows us to reach the global characteristics of the polymer/solid interface of real semicrystalline polymer nanocomposites. Our aim is to investigate the fragility of the transition of the bounded layer to a RAP layer comparing against an imposed glass transition in bulk [9]. The proposed model is supported by experimental results from a DSC study on two (PDMS and PA6) semicrystalline polymer nanocomposites. For these two polymer nanocomposites it has been reported [13,14] that the RAP layer locally equilibrates with the melt phase above Tg. Moreover the immersion in liquid nitrogen technique allows us to examine the influence of the kinetic factor in the polymer solidification (crystallization or RAP formation). Details about the experimental procedure and the used materials are given in the Appendix. Finally we end providing the conclusions of our study.

2. Variable density nSCF method 2.1. Theoretical formulation The used numerical SCF method is based on a lattice (cubic) discretization of the space. The lattice site is chosen according to

4659

the Flory segment volume [27e30] and is calculated by the equation:

31=2

2

7 6 nm Mm  7 lF ¼ 6 5 4 NA rnb lb sin q2b

(1)

where nm is the degree of polymerization, Mm is the monomer molecular weight, r the mass density of the polymer, NА the Avogadro's constant and nb is the number of chemical bonds per chain. The segmental volume (n* ¼ l3F ) is defined by eq. (1) at the temperature where the free volume is zero and only vibrational motion is allowed (~50 K below Tg). For a given temperature T and pressure Pb in the bulk phase, the volume fraction of the chain segments in bulk, FbA , is calculated by the equation [24e26]:

    2  Pb v* 1 FbA þ ln 1  FbA þ c FbA ¼ 0 þ 1 r RT

(2)

where r is the chain length and R is the gas constant. The corresponding FloryeHuggins lattice parameter (c) for the virtual interaction polymer segment/void is:

c¼

T* T

(3)

Also based on the SanchezeLacombe theory we introduce the parameters [24e26]:

T * ¼ Z

and

wAA 2k

P* ¼

RT * v*

(4)

(5)

where wAA is the attractive energy between adjacent segments, Z is the coordination number (for a cubic lattice is 6) and k is the Boltzmann constant. If wAS is the interaction energy between a segment adjacent to the surface and a surface site then we also introduce [25,26] the parameter:

w TS* ¼  AS k

(6)

So for the interaction polymer segment/surface site, the FloryeHuggins parameter is:

ZT *  T * : cS ¼  S T

(7)

The description of the interfacial area is done in a mean field self-consistent approximation, as introduced by Scheutjens and Fleer [22,23,25e32]. In terms of an A-type segment potential:

uA ðzÞ ¼ cAB FВ ðzÞ þ cS þ aðzÞ þ uref A

(8)

which consists of two terms describing the contribution of the interaction of the A-type segment with segments of a different type, a (third) term taking into account the “hard-core” potential that ensures full occupancy of the lattice and a reference potential. In our representation of the interfacial system we use the symbol A for the polymer, the symbol B for the void and the symbol S for the surface. The FВ(z) is the volume fraction of the B-type segment, while the solid surfaces have volume fraction equal to 1 and are placed at the z-boundaries of the lattice. Once the potential uA(z) is

4660

G. Kritikos / Polymer 55 (2014) 4658e4670

known, the probability of finding an A-type segment in layer z of the interfacial system, relative to finding it in the bulk is just the segment weighting factor GA(z), defined as GA ðzÞ ¼ euA ðzÞ=kT . The segments on a macromolecular chain are labeled by an integer number s and then the statistical weight for finding the end of an ssegments long chain in layer z, G(z;s), is defined. The probability of approaching the s-segment by two opposite directions (i.e. starting from the front and the back end) results in two recursion relations. Each recursion relation is solved once we know a proper initial condition. For the forward propagation (i.e. starting from the s ¼ 1 segment) the recursion relation has the following expression [28,29]: bs

G ðZ; sj1Þ ¼ GðZÞ〈G

bs1

ðZ  bs ; s  1j1Þ〉

where 〈Gbs1 ðZ  bs ; s  1j1Þ〉≡ljbs j

P bs1

(9a)

tbs1 bs Gbs1 ðZ  bs ; s  1j1Þ:

Also Gbs is the s segment statistical weight for forward propagation, bs(≡zszs1) is the bond between segment s and segment s1, lbs is the fraction of neighbor sites and tbs1 bs is the bending statistical weight for a pair of successive segments. A similar expression stands for the backward propagation:

Gbsþ1 ðz; sjrÞ ¼ GðzÞ〈Gbsþ2 ðz þ bsþ1 ; s þ 1jrÞ〉

(9b)

where P 〈Gbsþ2 ðz þ bsþ1 ; s þ 1jrÞ〉≡ljbsþ1 j tbsþ1 bsþ2 Gbsþ2 ðz þ bsþ1 ; s þ 1jrÞ. bsþ2

The product of all li gives the number of possible chain configurations on the lattice divided by the configurations in bulk Zr1. It's a measure of the entropy change. Then, by means of a composition law we find the volume fractions (F(z)) for a (in general) polydisperse polymeric system as:

FðzÞ ¼ CGðzÞ1

r X X X s¼1

bs

Gbs ðz; sj1Þtbs bsþ1 Gbsþ1 ðz; sjrÞ

(10)

bsþ1

where C is a normalization constant. In the case of a homopolymer, monodisperse melt coexisting with voids (variable density model), the normalization constant is chosen such that the bulk volume fraction,FbA takes the same value with the prediction of the equation of state (eq. (2)). In our method the polymer stiffness is also taken into account [22,23,27,28,30]. The bending energies are determined from the characteristic ratios of the polymeric chains by matching the meansquare end-to-end distance between the real chain and the equivalent chain used in the calculations, composed of correlated Flory segments [27,28]: F 〈R2 〉 ¼ C∞ nb l2b ¼ C∞ ðr  1Þl2F

(11)

the polymer density and the intramolecular energy are efficiently incorporated. 2.2. Mapping polyethylene The studied PE/solid system is schematically presented in Fig. 1. Exploiting the atomistic EBMC simulation [9] we take into account the temperature dependence of the intramolecular energy. The dependence of the C∞ on the temperature is given by the equation [9]:

C∞ ¼ e2:71:210

3

Τ

(13)

Moreover we have determined the temperature dependence of T* in order to follow closely the density predictions of the EBMC study. The pairwise interaction parameter has a sigmoidal function of the form:

    T * ¼ T2* þ T1*  T2* = 1 þ eðTTg Þ=dT

(14)

where: Tg is the glass transition temperature, dT determines the region where T* becomes temperature dependent, T1* is the value of T* at the Vogel temperature (150 K) and T2* is the value of T* above glass transition as given by the SanchezeLacombe theory [24]. The dT is related to the fragility of the polymeric material. The density in each temperature indicates a mean distance between the segments. Since T* represents (eq. (4)) the interaction energy wAA, which in simulation [9] is described analytically by a Lennard-Jones potential, then the temperature dependence of T* (eq. (14)) could be explained by changes in the mean intersegment distance in a potential vs intersegment spacing curve (inset of Fig. 2). For the polyethylene Tg ¼ 220 K, dT ¼ 20 K, T1* ¼ 649 K [24] and T2* ¼ 924 K. We mention that the fitting (eq. (14)) was done in order to reproduce the MC density results for a PE chain with the same length (r) as the one in Ref. [9]. Then the dependence of the density on r was taken into account according to eq. (2). Also the Tg was taken few degrees lower than the one estimated by the EBMC simulation in order to follow the experimental outcome [13,14,16] that when a significant fraction of RAP is formed above the glass transition, then the Tg tends to reduce/stabilize. Moreover the chosen parameter value for the polymer/surface interaction is TS* ¼ 858 K. The value of TS* is constant in the whole temperature region. The high strength of the surface potential resembles the strong polymer/nanosurface interaction (e.g. hydrogen bond) that is present in the PDMS and PA6 nanocomposites studied by the DSC experiment (section 3.2). Another characteristic of our detailed description of the interface is that the lattice is chosen such that the width of the first layer is close to the width of the square well graphite potential (first adsorption peak) [8]. As the study of the bounded polymer layer by molecular simulations is considered a difficult task [11], the use of

F is the characteristic ratio of the correlated Flory chain, where C∞ which is a chain composed of Flory segments each of size lf. Assuming that the back-folding conformer (V-conformer) is forbidden, the tV statistical weight has a vanishing value and hence the characteristic ratio of the Flory chain is related to the bending statistical weights by

F C∞ ¼1þ

tI 1 ¼ 1 þ eðεL εI Þ=kT 2 2tL

(12)

In the case of non-ideal chains an additional term representing the bending energy has to be added in eq. (8), affecting conformational energies. The inclusion of the bending energies as described above [22,23,27,28,30] allows us to introduce a polymer segment (eq. (1)) shorter than the Kuhn segment. In this way both

Fig. 1. Schematic presentation of the computationally studied PE/solid system. The flat surfaces (plates) extent in the x- and y-directions with dimensions infinitively large in order to neglect the end effects.

G. Kritikos / Polymer 55 (2014) 4658e4670

Fig. 2. Density as a function of the temperature for the bulk and the bounded polymer on the flat surface (first lattice layer). In the inset we present the temperature dependence of the negative T* (eq. (14)).

the proposed methodology allows us to overcome significant local equilibration difficulties. The rest parameters describing the PE macromolecule are: lF ¼ 4.45 Å calculated by eq. (1) for r ¼ 0.91 g/cm3 at 150 K [9], lb ¼ 1.54 Å, qb ¼ 114 [9], Mm ¼ 28 g/mol and nm/nb ¼ 0.5. The number of monomers per Flory segment is given by the equation [27,28]:

 3

nmpF ¼ NA rlf Mm

(15)

and is calculated as 1.7 monomers per Flory segment. The pressure was kept constant at 1 bar. In our study we have used chains of 60 Flory segments (of molecular weight 2.9 kg/mol) in a monodisperse system. At 450 K the end to end distance of an undisturbed Flory F ¼3.58) on the lattice is 64.6 Å (R ¼ 26.4 Å). In segment chain (C∞ g case of curved nanosurfaces or surfaces with different adsorption strength we mention the radius of the nanoparticle (Rnp) and the TS* (cS). 3. Results and discussion 3.1. NSCF study of a bounded PE Our analysis constitutes a SCF projection of the MC predictions onto the solid/melt PE interface. Although the cooling of the bulk polyethylene by the efficient EBMC simulation could be considered as a reversible procedure (at least at the temperature region above the glass transition) we will take it as a succession of quasiequilibrium, long living metastable states. As it will be shown experimentally in the next subsection, this assumption is valid in case of high nanofiller contents and short intersurface (separation) distances. In these cases the bounded PE layer could be treated in local equilibrium with the bulk non-crystallized PE, in the temperature region from Tm to just above Tg. So in the same temperature region our nSCF results for the bounded layer should be considered as a safe extension of the EBMC predictions for the bulk polymer. The fitting of the intersegment potential by a sigmoidal function (eq. (14)) in the glass transition region allows us to extract the indicated [9] volume fraction of the voids (free volume) at the

4661

Tg. In this way we are able to make a comparison between the predicted [9] glass transition in bulk and the transition of the bounded layer. We mention that according to the same molecular simulation [9] even the system with the grafted chains had reached equilibrium, exhibiting good agreement with the predictions of the SCF theory. Our computational study of the non crystallized, liquid PE above Tg, could be also valid for the examination of the bounded layer in amorphous polymer nanocomposites [16]. In Fig. 2 we present the temperature dependence of the density in bulk and on the flat surface. The parameter T* describes a mean field pairwise potential for polyethylene, consistent with the EBMC simulation. The sigmoidal function of T*(T), (inset of Fig. 2) indicates a constant value above and below the glass transition and a temperature dependence in the glass transition region. The step increment of T* (from T* ¼ 649 K to T* ¼ 924 K) during cooling seems to be related to an increment in the attractive energy between adjacent segments from ~2 KT just above glass transition (at 300 K) to ~6 KT below the glass transition (at 150 K). We mention that according to the literature [2,33] the polymeric material is expected to deviate from the Arrhenius of constant activation energy in the same region. An important outcome of the imposed step increment in T* is that although the bulk density changes slope at 220 K, the polymer density on the surface (first layer) degreases in a more smooth way (Fig. 2). Let us define the potential energy [Kcal/g] in each layer which is related to the temperature dependence of the T*. It is given by the equation:

UðzÞ ¼ FA ðzÞFB ðzÞ

T * kNA nmpF Mm

(16)

where for the bounded layer on the surface: z ¼ 1 and for the bulk: z ¼ middle layer. In this calculation we take into account only the pairwise contribution to the potential energy. In Fig. 3a it is shown the potential energy in the bulk as a function of the temperature, while in Fig. 3b it is depicted the derivative of the same quantity for the bulk and the first layer. In this way we are able to evaluate the pairwise potential energy contributions to the heat capacity (Fig. 3b). The obtained values for the bulk are (Fig. 3b): Cp ¼ 0.12  104 Kcal/gK (just above the Tg) and Cp ¼ 0.09  104 Kcal/gK (just below the Tg). The deviation from the predictions of the simulation [9] for the configurational Cp should be mainly attributed to the fact that we have not taken into account the intramolecular contributions. The important outcome of our analysis is that although it is predicted a clear heat capacity step for the bulk polymer, on the contrary for the bounded polymer the behavior is not the same implying a gradual densification [23]. We mention that the presented (Fig. 3b) heat capacity values for the bulk, below Tg, should be compared with the DSC results of an annealing experiment (experimental time of order days). According to the Williams, Landel and Ferry (WLF) equation (pg. 338 of Ref. [33]) the dependence of the viscosity (n) on the free volume is given as:

   n 1 1 ¼ exp B  n0 f f0

(17)

where f is the fraction of the free volume (in our method is the same with the volume fraction of the voids, FВ(z)), f0 is the value of the free volume at the Tg, n0 is the value of the viscosity at the Tg and B is an empirical constant of order unit [33]. In Fig. 4 we present the temperature dependence of the viscosity in the bulk and on the surface. Fig. 4a depicts a sudden increase (during cooling) of the viscosity in the bulk that takes place at the glass transition. It is a behavior connected with the EBMC simulation [9]. However the

4662

G. Kritikos / Polymer 55 (2014) 4658e4670

Fig. 3. We present: a) the intermolecular potential energy (eq. (16)) of the bulk polymer and b) the contribution of the intermolecular energy to the heat capacity (derivative of eq. (16)), for the first layer (on surface) and the middle layer (bulk) as a function of the temperature.

imposed glass transition in bulk predicts a less sudden immobilization on the surface (Fig. 4b) [23]. The fragility [2] of the bounded layer is almost twice lower (stronger) than the fragility of the bulk (Fig. 4). The transition of the bounded adsorbed layer to a rigid amorphous phase (RAP) is located at ~350 K (Fig. 4b). This temperature is just below the melting point of PE [9]. The total density profile at various temperatures is shown in Fig. 5. A small depletion layer at high temperatures (around 450 K), at ~15 Å from the surface can be discerned. This tendency vanishes at lower temperatures where the volume fraction of voids is reduced. The density fluctuations are more intense in case of curved surfaces (especially for Rnp ¼ 25 Å), [23]. This picture is also consistent with results from molecular simulations of PE near a flat graphite surface [6,8]. The origin of the observed behavior should be attributed to the tendency of the mobile, liquid tails to move away from the denser loop layer [23]. In this way beyond the depletion layer the chains succeed bulk conformations and dynamics [6,23]. In Fig. 6 we depict the structure of the adsorbed layer. The volume fraction profiles of the tails and the free chains indicate that the density fluctuations could be also related to a difficulty in full overlapping between the shorter tails and the free,

Fig. 4. Temperature dependence of the viscosity for the cases of: a) the bulk polymer and b) the bounded polymer layer (first lattice layer). The calculations are based on the WLF equation [33]. In case (a) the derivative at the value 1 of the reduced temperature (Tg/T) is 669, while in the inset of the case (b) the derivative at the same value is 372. In the second case the Arrhenius plot in the inset is based on the transition temperature (Ttr z 350 K) of the bounded layer to a rigid amorphous phase. At this temperature the free volume on the surface is the same with the free volume of the bulk, at the Tg.

non adsorbed chains [23,34]. As shown in Fig. 7 the overlap region between free and adsorbed chains increases as temperature drops. This is considered as a combination of the decrease in the density (and the chain mobility) but also due to the increase in the chain stiffness (see eqs. 12 and 13). It has been observed (not shown) that by keeping constant all other parameters (in the same temperature) while increasing the chain stiffness (referring to amorphous polymers) the overlap region increases while the adsorbed amount reduces. We point that at ~350 K the overlap region contains an immobilized polymer layer (trains [31]), a bounded polymer layer (loops and tails) and free chains. But the dynamics are supposed to be determined by the existence or not of entanglements between the bounded and the free chains. We mention that the molecular weight of the PE chains in the monodisperse melt is above the critical value of the entanglement molecular weight [35]. Moreover cooling, as shown in Fig. 8, causes a decrease in the adsorbed

G. Kritikos / Polymer 55 (2014) 4658e4670

4663

Fig. 5. Total density profile in various temperatures at infinite intersurface distance.

amount. This is explained by the difference between the density in the bulk and the density on the surface. The results are consistent with the density profiles shown in Fig. 5. The overlap region can give us a measure of the entanglements between free and adsorbed chains. Especially in a non reversible separation of the two flat surfaces the existence of this region should justify an increase in the applied pressure [22]. On the other hand the measure of the interfacial tension in the polymer/solid interface is given by the equation [25,26,32]:

  2ðg  gS ÞaS 1 ex X FB ðzÞ q þ ¼ 1 ln r A kT FbB Z   2  X FA ðzÞ〈FA ðzÞ〉  FbB þc

(18)

z

where g is the surface tension of the pure polymer, gS is the surface tension of the S phase (if the S phase is a gas then gS ¼ 0 [25,26]), aS P is the area per site and qex ½FA ðzÞ  FbA  is the excess volume A ¼ z

Fig. 6. Structure of the bounded polymer at the solid/polymer interface, at 450 K and infinite intersurface distance. We present the volume fraction profile of the total adsorbed chains (black circles), the volume fraction of the tails (squares), the volume fraction of the loops (stars) and the volume fraction of the free chains (white circles).

Fig. 7. Overlap region between free and adsorbed chains as measured by the integral of the product of the normalized (no voids) volume fractions of the free and the adsorbed chains [23].

fraction. In Fig. 9 we present the dependence of the interfacial tension on the temperature in case of infinite distance between the opposite surfaces (the opposite adsorbed layers/brushes do not interact). The parameter (cS) for the interaction polymer/surface takes the value 10 at 450 K and the value 28 at 150 K. This interaction determines the adsorbed amount (Fig. 8) that justifies negative values of the interfacial tension (or positive values of the adhesion tension [26]). In case of a pronounced depletion layer in the density profile, near the flat surface, then eq. (18) would give positive values, indicating desorption (pg. 179 of Ref. [32]). In the inset of Fig. 9 we also present the cost for the creation of a melt PE/ solid PE interface. It is accompanied by a depletion layer in the density profile. As the two opposite flat surfaces approach each other it is possible to observe the formation of polymer “bridges”. Chains can be adsorbed by both ends from different surfaces. In case of semicrystalline polymer nanocomposites [13,14] this disturbance could justify an additional difficulty in polymer diffusion that could affect

Fig. 8. Adsorbed amount (

P ½rðzÞ  rb ) as a function of temperature. z

4664

G. Kritikos / Polymer 55 (2014) 4658e4670

Fig. 9. a) Surface tension of the solid/polymer system (TS* ¼ 858 K) at various temperatures. In the inset (b) we present the tension of the solid PE/melt PE interface, where cS ¼ 0 for all temperatures. The axes of the inset graph have the same units with main graph.



Fig. 11. Bond order parameter of the chain bridges (Si ¼ 〈 3 cos2 Q  1 =2〉) in the middle layer of the lattice (see Fig. 10). The reference axis is perpendicular to the surface and Q is the angle between each bond and the reference axis. Zero value of Si represents the undisturbed configurations, while 0.5 (1.0) the parallel (vertical) to surface alignment.

crystallization. In Fig. 10 we observe that for TS* ¼ 858 K the “bridges” appear at distances below 2.5Rg, close to the end to end distance of the undisturbed chain. This picture does not change notably even if we choose TS* ¼ 7608 K (~-100 KT at 450 K). The reason for this behavior is that by increasing the enthalpy gain for the formation of more elongated ‘bridges’ we also increase the enthalpy barrier for the desorption form the one surface. The formation of loops and less disturbed tails surfacing above the loop layer [23] is thermodynamically more favorable than the formation of loops and stretched bridges, at least at a temperature region around the Tm. In Fig. 11 we follow the bond order parameter of the bridges in the middle region (middle layer). We observe that when the bridges constitute a significant fraction of the chains in the middle layer, then the free chains have vanished but the bond order parameter of the bridges is zero. The whole behavior doesn't

where b a is the surface area, F is the exerted force and D is the separation distance. In Fig. 12 we show the force per area (MPa) for three cases of adsorbing surfaces, at 450 K. We observe that as the surfaces come closer, at distances where the opposite adsorbed layers (Fig. 6) start to compress (D < 3Rg), then low negative pressure values are detected. Especially at distances below the end

Fig. 10. Volume fraction (normalized) in the middle layer of the total adsorbed chains, the chains adsorbed by both ends from different surfaces (bridges) and the non adsorbed (free) chains as a function of the intersurface distance.

Fig. 12. Force per unit surface at 450 K for various cS. For cS ¼ 100 the highest pressure value is 35 MPa.

change significantly by varying the curvature of the adsorbing solid surfaces. The negative derivative of the interfacial tension gives us the pressure when the opposite surfaces approach each other reversibly [22,26]:

F d ½g  gS  ¼ 2 b dD a

(19)

G. Kritikos / Polymer 55 (2014) 4658e4670

4665

to end distance (64.6 Å) random conformations are cut and more flattened chains obtain higher probability/volume fraction. The tails obtain such configurations in order to allow bulk dynamics in the middle region (Figs. 10 and 11). So the approaching surfaces force the chains to align parallel to the adsorptive surfaces (bond order parameter on the surface layer ~ -0.4). This causes a slight increase in the adhesion tension. Moreover the appearance, at short separation distances, of polymer bridges (Figs. 10 and 11) should additionally justify the negative pressure [26]. For the case of cS ¼ 2 (TS* ¼ 258 K), where at infinite surface separation, due to entropic reasons, the adsorbed amount is ~0 mg/m2, the approaching surfaces increase significantly the contact between the polymer and the solid surface justifying only negative/attractive forces. For the case of cS ¼ 100 (TS* ¼ 7608 K) the early (~90 Å) attractive forces are more pronounced. But at the distance of Rg (26.4 Å) the pressure starts to increase reaching the value of ~35 MPa at 0.5Rg. The case of cS ¼ 10 (TS* ¼ 858 K) depicts an intermediate behavior, while the increased stresses at the distance of 0.5Rg reach the absolute value of 6 MPa. At distances shorter than 0.5Rg the chains in all cases leave the intermediate gap and the pressure drops to zero. At the lower temperature of 350 K and cS ¼ 10 the pressure profile depicts (not shown) a similar behavior with the case of T ¼ 450 K (Fig. 12). The minimum value at Rg is ~ -2 MPa and the maximum value at 0.5Rg is ~4 MPa. The whole pressure profile (Fig. 12) depicts a resemblance with the experimentally observed behavior of confined polymers in poor solvent [36]. We mention that in case of polymer nanocomposites with spherical nanoparticles, as the nanofiller content [37] increases, the adsorbed layer is assumed to be disturbed by a uniform shell wall and the end effects could be neglected. Also we assume that at short separation distances (>Rg) the density in the middle of the interparticle distance could be described by eq. (2). In Fig. 13 we present the bond order parameter of the total adsorbed layer in various curvatures. It is observed that in case of nanoparticles of Rnp ¼ 25 Å the adsorbed layer, following the disturbance of the tails [23], does not depict a high bond order parameter but is described by the conformations of a star polymer [11,23,34,38]. The probability distribution of the free ends revealed [23] that although in planar surfaces the ends tend to lay in the same z layer, achieving undisturbed conformations (random walk) in the xy-plane, on the other hand on the spherical lattice the chain ends segregate in different layers (spherical shells). In this way the

chains obtain again end to end distances closer to the unperturbed value, while the average height of the adsorbed layer slightly increases [23]. Using the block version of the variable density nSCF method [22,30] we are able to distinguish the “V” Kuhn segment conformations (Fig. 14) that belong to adsorbed chains and distribute the chain to the inner layers (closer to the surface) and the “V” segment conformations that belong to free chains and distribute the chain to the outer layers. The product of the probability of these segment conformations should be proportional to the probability of an entanglement [35] between free and adsorbed chains. These entanglements are expected to live longer time. The integral of the product in the whole overlap region [22] indicates that in case of a planar geometry these entanglements (Fig. 14) have a twice higher probability than in the case of more star-like conformations of Rnp ¼ 25 Å (see Scheme 5 in Ref. [23]). Moreover for these highly curved surfaces the increasing chain stiffness by reducing temperature (eq. (13)), slightly affects the stretching of the nearly random walk chain (not shown). The same effect as the temperature decreases should be considered for the middle region of the bridges (Fig. 11). So, high filler contents of well dispersed, adsorbing, spherical nanoparticles having a radius of a few nanometers constitute a limit of a wide surface were macromolecules densify, releasing free volume, while they cause a less disturbed overlap region between free and adsorbed chains. The high confinement of a chain above the Tg causes a force of entropic origin (positive pressure values in Fig. 12). When the caging affect [1,2] takes place and since the formed 3-D wall by the neighbor chains is not a solid one but has the ability to rearrange (Fig. 15), the exerted force could justify the creation of cooperative rearranging regions [2]. The same effect is responsible for the formation of the bilayer structure in the bimodal brushes [23]. Even an anisotropic stretching could be favorable if part of the macromolecules enjoy an area with higher free volume, succeeding an overall lower reduction in the entropy [23,29]. Few degrees above the glass transition temperature a dynamic situation is established where local breakthroughs of molecular mobility could be organized by density fluctuations [2]. The dynamic heterogeneity grows as the temperature approaches Tg [38]. In our variable density nSCF study we have taken a close to Lennard-Jones curve (eq. (14)) for the pairwise energy (T*) as a function of the mean intersegment/ interatomic distance in each temperature. We have assumed that in case of a non polymeric, non glassy material only the interatomic distances where the pairwise energy is almost independent from the temperature can be occupied. These are the lowest pairwise energy value of the solid phase (Fig. 15a) and the tail of the Lennard-Jones function at high temperatures (Fig. 15c). But according to our nSCF analysis of the EBMC temperature-density

Fig. 13. Bond order parameter of only the adsorbed layer for various curvatures of the solid surface at infinite intersurface distance (T ¼ 450 K).

Fig. 14. Block “V” Kuhn segment conformations [22,30] of the adsorbed (black) and the free (gray) chains.

4666

G. Kritikos / Polymer 55 (2014) 4658e4670

Fig. 15. Three cases of local polymer concentration in the temperature region where the pairwise interaction parameter (T*) becomes temperature dependent (eq. (14)): a) case of zero free volume, b) caging effect and c) liquid mobility. White circles represent the neighbor chains of the gray chain. The chains have the density of a solid polymer (~50 K below Tg), while the rest volume is considered as free volume. The average volume fraction of the voids at the Tg was calculated (by the variable density nSCF method) as: fTg ¼ 0.015 [33].

curve, in polymers also intermediate, mean interatomic distances can occur (Fig. 2). In this region the pairwise energy is temperature dependent (inset of Fig. 2). Macromolecules achieve such intermediate mean interatomic distances (density values at the glass transition) probably due to local density fluctuations (Fig. 15) related to the existence of cooperative rearranging regions. We suppose that none of these locally high concentrations justify an exothermic peak in the DSC experiment as the one observed in the case of semicrystallization. 3.2. RAP and crystallization/DSC study It has been reported, in experimental DSC studies on polyamide 6 (PA6)/layered silicates nanocomposites [13] and polydimethylsiloxane (PDMS)/silica nanocomposites [14] that the crystallization in all neat and polymer nanocomposites proceeds until the fractions of the solidified polymer (CP þ RAP) and the liquid MAP reach a certain limit, independent from the filler content. The interpretation of the suppressed degree of crystallinity (or crystallinity (Xc)) was done [13,14] based on traditional kinetic arguments, which have to do with the relation between the molecular relaxation time and the experimental time. With the same arguments (difficulty in chain diffusion) it was explained the reduction [13,14] in the crystallization temperature (Tc). But the computational study provided us with strong indications that the bounded layer may be irreversibly solidified, around the Tc, on the wide nanoparticles surface. It is a result consistent with the experimentally observed reduction in the heat capacity step [13,14]. Moreover in previous work [23,30] it was shown, based on thermodynamic arguments, that the densification on the nanoparticles could be accompanied by an appearance of a thin (few nanometers) depletion layer in the density profile. In this region, next to the adsorbed layer, they can be observed reduced relaxation times and an enchantment in diffusion [6,15]. Also in both semicrystalline polymer nanocomposites [13,14] the crystallinity reduction was not

accompanied by a significant increase in the Tg. On the contrary it was observed a Tg reduction/stabilization as a function of the filler content, which should be related to a decrease in the average viscosity. Looking for the characteristics (kinetic or thermodynamic) of the connection between the total RAP (around both lamellar crystals and nanoparticles) and the CP, we have conducted a supplementary DSC study on the same semicrystalline neat polymers (PDMS, PA6) and their nanocomposites. Details about the used materials, the apparatus and the experimental procedure are given in the Appendix. The cooling was done according to three protocols: (a) fast cooling by immersion of the samples in the liquid nitrogen (over 1000 K/min), (b) standard cooling rate (10 K/min) and (c) annealing at the Tc and the cold crystallization temperature Tcc. Heating was always done in the standard rate of 10 K/min. By varying the cooling rate we were able to examine the influence of the kinetic factor [39e41]. Neat PDMS network exhibits a high degree of crystallinity, with a value (Xc ¼ 0.8, [14]) close to the one of PE. Also it has Tc ¼ 195 K and Tm ¼ 225 K [14]. We point that the addition of the cross-linking agent in the linear PDMS, which is the same as the precursor for the in situ synthesis of the silica, did not cause a significant inhibition in chains' ability to crystallize [14,42]. We also assume that the synthesis of the silica does not allow further cross-linking because in that case we should expect a dramatic increase in the Tg and the Tm. Besides the nSCF study supports the idea that even a strong attractive interaction with the surface (cS), when it is applied to the whole linear chain, will not promote the appearance of more stretched bridges (Figs. 10e11). The addition of silica spherical nanoparticles (diameter ~5 nm) caused a Tg reduction/stabilization to a value almost 5 K below the value (149 K) of the neat PDMS [14,42]. Also strong interactions are developed between PDMS and silica [14,42]. We mention that TEM images [42] of the same nanocomposites indicate that the filler particles are uniformly dispersed. According to the results presented in Ref. [14], standard heating rate prevents the cold crystallization in the standard cooled nanocomposites (PDMS/5.7 wt% silica and PDMS/10 wt% silica). This implies that the crystallization in the standard cooled nanocomposites finds a barrier at Xc values close to: 0.42 and 0.39 respectively. In this way no cold crystallization correction is needed. This was also confirmed by annealing experiments at the Tc (not shown). In higher filler contents (31 wt% and 36 wt%) there is a weak cold crystallization [14]. But annealing experiments at the Tc for these high filler contents although cause a weak crystallinity enhancement, confirm the significant suppression in the crystallization with the addition of silica nanoparticles. We mention that in case of a crystallinity reduction due to high cooling rate (or short experimental time) then a heating rate of the order of 104 K/s is required in order to avoid cold crystallization [39e41]. The heating thermograms after the fast cooling are shown (black lines) in Fig. 16. In both samples of the neat and the nanocomposite (10 wt% silica) PDMS we observe an almost 80% reduction of the Xc compared to the case of standard cooling (gray lines). In the neat PDMS a significant cold crystallization is observed at 176 K during heating of samples imposed on the fast cooling process. But at 10 wt% silica, the cold crystallization is significantly suppressed, broadened and observed later at higher temperature (196 K) during heating, after the fast cooling process. This is a behavior similar to the case when nanofillers act as nucleation agents [39e41]. According to these results (Fig. 16) when the crystallization in the neat PDMS is suppressed due to kinetic reasons imposed by the significantly high cooling rates (over 1000 K/ min), the polymer manages to respond through cold crystallization in a pronounced way [39e41]. Assuming that the significant reduction of the crystallinity in the PDMS/silica nanocomposites is

G. Kritikos / Polymer 55 (2014) 4658e4670

4667

Fig. 16. DSC thermograms for the PDMS and PDMS/silica nanocomposites during heating, after fast (black lines) and standard (gray lines) cooling, in the temperature range from 150 to 240 K.

fast cooling, the thermogram depicts a significant cold crystallization peak at 338 K (Fig. 17b). It seems again that when crystallization during cooling is significantly suppressed due to extremely high cooling rate, then the polymeric material manages through cold crystallization to correct the uncompleted crystallization [39e41]. The obtained crystallinity value is reduced by 50% as compared to standard cooling, possibly due to absence of enough crystallization nuclei [39e41]. Annealing of the neat polymer at Tcc (after quenching) showed that the melting enthalpy is significantly increased, reaching a value almost 10% lower than the one of the standard cooled sample. Thus for the neat PA6, in agreement with the behavior of the neat PDMS, the respective heating thermogram indicates that the kinetic factor affects significantly the observed crystallization and the glass transition. On the other hand in the PA6 nanocomposites, as the CNT content increases the cold crystallization “correction” is significantly suppressed and the peak is broadened [39e41]. The behavior resembles the one in the case of PDMS/10 wt% silica nanocomposites (Fig. 16). Although in the PA6 nanocomposites the nuclei for crystallization are provided by CNTs, for contents higher than 5 wt% no cold crystallization is observed (Fig. 17), [39e41]. Annealing at Tcc in the nanocomposites does not impose a

exclusively justified by traditional kinetic arguments it is interesting that no significant cold crystallization is observed during standard heating of the standard cooled nanocomposites [14]. Furthermore, annealing at Tcc in the fast cooled nanocomposite (PDMS/10 wt% silica) leads to a significant increase in the melting enthalpy. The obtained value is lower (~20%) as compared to the values obtained on standard cooled nanocomposite and far below the neat polymer melting enthalpy. In all, when the crystallinity in the neat and nanocomposite PDMS samples (Fig. 16) is reduced due to the fast cooling rate by almost 80%, then the annealing at Tcc tends to recover at maximum the values obtained on the standard cooled samples. Concerning the glass transition, due to the fact that the DSC apparatus used in these experiments (Perkin Elmer Pyris 6) can cool until 153 K, we were not able to detect the associated endothermic step for higher silica content, since in that case (31 wt % and 36 wt%) the Tg shifts much below this limit (145 K and 144 K respectively [14]). Moreover, annealing experiments at Tc confirm [14] that the MAP fraction has a constant value (~0.1), independent from the filler content, just above the glass transition. Next we deal with the study of the PA6 nanocomposites that contain carbon nanotubes (CNT). The neat PA6 [43] used in our DSC study depicts the same characteristics (Tg ¼ 328 K, Xc ¼ 0.31, Tc ¼ 460 K, Tm ¼ 491 K, etc. [43]) with the one used for the PA6/ layered silicate nanocomposites studied by others [13]. But the addition of CNTs enhanced the crystallization of PA6 [43], while the addition of layered silicate nanofillers [13] caused a significant reduction in the observed crystallinity. Results from SEM and TEM on the used samples [43] indicate that the nanotubes are nicely dispersed as single tubes. It is mentioned that in the interfacial layer again strong substanceeconstraint interactions are present [43]. According to the previous computational study of PE, these strong interactions should be related to a reduction of the free volume, causing an increase in the viscosity (Fig. 4b). We suppose that geometry/dimensions of the nanofillers [10,11,44,45] may also play an important role in the kind of the imposed polymer densification (CP or RAP). Certainly geometry affects the bond order parameter of the adsorbed layer (Fig. 13). In Fig. 17a and b we present the thermograms during heating (always at 10 K/min) of PA6/CNT system after fast cooling (black lines) and standard cooling (gray lines). In case of the neat PA6 after

Fig. 17. DSC thermograms for the PA6 and the PA6/CNT nanocomposites during heating (10 K/min), after fast (black lines) and standard (gray lines) cooling, in the temperature range: (a) 273e520 K and (b) in the glass transition region.

4668

G. Kritikos / Polymer 55 (2014) 4658e4670

significant improvement. Assuming that the polymer should reach in equilibrium Xc ¼ 1.0, higher than the observed crystallinity of the neat polymer, then cold crystallization in the CNT nanocomposites could be enriched further. However, our experimental results do not confirm such assumptions implying that the addition of CNT in PA6 matrix, according to the kinetic approach, helps the first stage of nucleation and inhibits the second stage of crystal growth [39e41,44,45]. Nevertheless we consider the crystal growth as a densification process that releases free volume. It is remarkable that the final degree of crystallinity in the PA6/CNT nanocomposites although increasing (reaching the value ~0.4 for 10 wt% CNT [43]) is less affected by the kinetic factor (Fig. 17a). Annealing thermograms of the PA6/CNT samples at 463 K depict (not shown) that as the filler content increases the crystallinity improvement (compared to standard cooling, DXc/Xc) decreases notably. So in the PA6/CNT nanocomposites even the crystallization growth is affected by the cooling rate only after a certain value of Xc is reached. The same trend is observed for the Tg changes. It seems that as the fraction of the polymer which is crystallized (CP) before Tg is less affected by the cooling rate, then the fraction of the polymer which participates in the glass transition follows a similar dependence (Fig. 17b). It is interesting that the standard cooled PA6/CNT nanocomposites depict a Tg reduction as a function of CNT content, reaching the minimum value of 322 K. They show the same tendency (depression of Tg) and minimum Tg value with the one presented for the PA6/layered silicate nanocomposites [13], where a significant reduction in the crystallinity was observed. Also after the annealing at 463 K (Tc) the normalized heat capacity step, which is a measure of the mobile polymer fraction that undergoes glass transition, seems to fluctuate around the same value of 0.15 ± 0.05 J/gamK (MAP~0.3) presented by Wurm A. et al. [13]. The heat capacity step seems to be independent from the CNT content. We assume that the RAP is only formed around the crystalline regions. Analyzing the quenching behavior of the two studied nanocomposite systems we mention that in the system of PDMS/silica the crystallization is assumed homogeneous in all neat and nanocomposite polymeric materials. For this reason the degree of crystallinity is affected by the fast cooling almost the same (~80% reduction) in both neat and nanocomposite PDMS. But the Tg tends to reduce/stabilize to the value of ~145 K probably because the solidification on the silica (RAP formation) is not affected by the cooling rate. In general DSC experiments in both PDMS/silica and PA6/CNT nanocomposites seem to support the idea that as the amount of the polymer which is solidified (CP þ RAP) around the nanofillers is less affected by the cooling rate, the heat capacity step and the Tg tend to reduce/stabilize. Polymer semicrystallization is still a challenging fundamental problem in polymer physics [12,46e48]. The melting point and the crystal thickness are related through the ThompsoneGibbs equation. The change in the free energy (DG) on melting is given by (pg. 144 of Ref [46]):

DG ¼ DG* þ

n X

Ai si

layer around nanoparticles, modifying in this way the surface free energy (si) of the melt/pure crystal interface. As the lamella crystal thickness is of few nanometers [46,49], well dispersed nanoparticles may substitute crystalline regions on the surface of which the RAP layer is formed during cooling. Concerning the mechanism of RAP formation we suppose that the mobility of the bulk/melt macromolecules forces the adsorb chains on the wide nanoparticles' wall, while the solid phase acts during cooling as an “antenna” of heat absorption [23]. According to these arguments the observed constant fraction of the solidified polymer (CP þ RAP) and the consequent crystallinity suppression, shown in Figs. 16 and 17 could be also interpreted by the cost (Fig. 9) when the surface (Ai) of the melt/solid interface exceeds a certain value. This approach takes into account the cost of the entropy reduction due to the solidification on the nanoparticles. It implies an explanation of entropic, thermodynamic origin [23,50] for the observed free energy barrier that affects the creation of additional crystallization nuclei [13,14]. Annealing DSC experiments of the neat polydimethylsiloxane (PDMS) [14], the neat polyamide 6 (PA6) [43] and other neat semicrystalline polymers have shown that the enthalpy of isothermal crystallization (DН c) was 80e100% of the absorbed heat from Tm to Tc (QTmTc ¼ Cp(TmTc)). According to this observation the significant free energy barrier that restricts further crystallization seems to appear as the ratio of the heat (free volume) released during crystallization/solidification to the heat (free volume) absorbed during cooling from Tm (of the neat polymer) to Tg goes to unity. We assume a linear relationship between the free volume and the temperature [33], (Fig. 2). Moreover the explanation of a gradual solidification during cooling from Tm to Tg (Figs. 2, 3b and 4b) seems reasonable as the densification of the significant RAP fraction (over 20 wt%, [12e14]) is not depicted pronouncedly (as an exothermic peak) in the DSC thermograms (Fig. 16), [13,14,16]. As shown in Fig. 18 at high nanofiller contents, when adsorbing nanoparticles do not act as crystallization agents [13,14], the assumption that a nanocomposite semicrystalline polymeric material could be trapped for a long time (longer than common experimental time) in a metastable state, with a suppressed crystallinity is valid even in standard cooling rates. In our computationally studied PE/solid system, this seems to be related to

(20)

i

where DG*(¼DHTDS) is the surface independent change in the free energy (DН: enthalpy change, DS: entropy change) and si is the specific surface free energy of surface Ai. Since the decoupling of the crystallites from the melt could create an interface that causes a significant free energy barrier (surface free energy) [12], the same could be assumed for the nanoparticles' RAP/melt interface, below a certain temperature (Tm of the neat polymer), where the loops and the tails start becoming one rigid, immobilized amorphous layer (Figs. 2 and 4). Besides crystallites are usually surrounded by a RAP layer that resembles a lot the RAP

Fig. 18. Annealing DSC experiments, at the Tc, for the samples: neat PDMS (black line) and PDMS/31 wt% silica (gray line). The degrees of crystallinity (Xc) are 0.8 ± 0.05 and 0.14 ± 0.05 respectively [14].

G. Kritikos / Polymer 55 (2014) 4658e4670

separation distances shorter than ~3Rg. The pressureeintersurface distance profile (Fig. 12) at these separation distances indicates negative pressure. In these cases the densified bounded layer may locally equilibrate with the melt polymer (MAP), above Tg. The densification on the wide nanoparticles' surface could cause a suppression of the caging effect (negative pressure/low local density [38]). Based on this thermodynamic approach we could add arguments in order to explain the suppression of the cooperative rearranging regions [16] and the reduced/stabilized Tg observed in several nanocomposite polymers [13,14,16]. A decrease in viscosity should be connected with this behavior [51,52]. The same “relief effect” could be responsible for the reduction in the Tc observed in the semicrystalline polymer nanocomposites [13,14]. Although annealing at Tc gives us the impression of equilibrium (Fig. 18), the same experiment below Tg causes a measured enthalpy relaxation, lasting several hours/days [20]. The formation of the RAP seems to justify a reduced enthalpy relaxation [16,19,20]. Our proposed thermodynamic model, first presented [23] for the interpretation of dielectric and DSC results for the PDMS/silica nanocomposites [14], seems to be supported by recent experimental and simulation works [50,53]. 4. Conclusions Based on a previous EBMC study on polyethylene, we have improved the variable density nSCF method. Our computational study managed to capture the transition of the bounded polyethylene layer to a rigid amorphous phase just below the Tm, at ~350 K. This transition was estimated at least twice stronger than the imposed major glass transition at 220 K. The change in the slope of the temperature-density curve at Tg required a sigmoidal step in the mean field pairwise parameter, T*. Our study indicates that even at temperatures around the Tc of the neat polymer, the fraction of the mobile polymer available for crystallization decreases significantly. Especially spherical adsorbing nanoparticles constitute a limit of a wide surface where macromolecules are immobilized, releasing free volume, while they cause less disturbed overlap regions between free and adsorbed chains. The whole behavior seems to be related to a Tg reduction/stabilization. Immersion in liquid nitrogen technique allowed us to obtain DSC thermograms on extremely fast cooled (~1000 K/min) polymeric samples. The experimental finding that a dramatic reduction in the degree of crystallinity due to kinetic reasons (neat PDMS and neat PA6) is compensated by a pronounced cold crystallization, implies that in the case of the PDMS/silica and the PA6/layered silicate nanocomposites [13,14], where negligible cold crystallization effects are detected, the observed suppressed crystallization should not be interpreted exclusively on the basis of traditional kinetic arguments. DSC experiments in PDSM/silica and PA6/CNT nanocomposites support the idea that the solid fraction (CP þ RAP) could be trapped above Tg in local equilibrium with the melt fraction for long (experimental) time. The created solid polymer/melt polymer interface should justify a significant free energy barrier. From a thermodynamic point of view the proposed model could provide an explanation for the experimental observation of a connection between CP þ RAP and MAP. Acknowledgments The author would like to thank the laboratory of Prof. Pissis P. for the technical support and the provision of the materials studied in the DSC experiments. Also the author would like to thank Mrs. Korkoli Efi for her advices.

4669

Appendix Materials Our DSC study is based on polydimethylsiloxane/silica (PDMS/ silica) nanocomposites and on polyamide 6/multiwall carbon nanotubes (PA6/CNT). Both polymers are semicrystalline and have been previously characterized by microscopy, calorimetry and other relative techniques [14,42,43]. Dispersion of nanofillers was recorded to be good and the systems showed homogeneous behaviors, concerning their thermal transitions and structures. In the system PDMS/silica the amorphous spherical silica nanoparticles were in situ synthesized and dispersed, via Sol-Gel technique, in the polymer matrix [14,42]. The diameter of nanoparticles is around 5 nm. The unfilled polymer network was prepared from hydroxylterminated PDMS (18 kg/mol) by end-linking reactions using tetraethoxysilane (TEOS) as cross-linking agent. The experimental densities are 2.65 g/cm3 for silica and 1.62 g/cm3 for PDMS [14]. The PA6/CNT nanocomposites were prepared by melt mixing. A masterbatch of 20 wt% multiwall CNT in PA6 was obtained from Hyperion Catalysis International, Cambridge, MA in pellet form [43]. PA6 nanocomposites contained nanotubes of 10 nm diameter and over 10 mm length. The final concentration of the CNT in the samples varied from 2.5 to 20 wt%. The geometrical characteristics of the nanotubes give a high aspect ratio of about 1000. Differential scanning calorimetry Thermal transitions (melting, crystallization and glass transition) of the samples were investigated in nitrogen atmosphere in the temperature range from 153 to 523 K using a Perkin Elmer Pyris 6 instrument, which was calibrated using Indium sample. We performed three different types of thermal-crystallization treatment (protocols) for the samples: (a) suppressed, (b) standard and (c) annealed crystallization. At first, we performed a heating scan of the sample at a temperature well above melting point (Tm), in order to erase any thermal history [54]. For the suppressed polymer crystallization (a), the sample was immersed quickly in liquid nitrogen (quenching conditions, cooling at ~1000 K/min) and stayed there for several minutes. At the same time the DSC cell was cooled down and stabilized at a temperature (~50 K below Tg) depending on the type of polymer. Then the sample was inserted in the measurement cell, stayed isothermally for 20 min at this low temperature and then heated up to temperatures well above Tm at 10 K/min. In case of the PDMS/silica system, after the samples were immersed in liquid nitrogen they were inserted in the DSC cell, at 153 K and stayed there for 5 min instead of 20 min as in other semicrystalline samples, in order to prevent relaxation. We mention that the lowest temperature allowed by the DSC (Perkin Elmer Pyris 6) instrument was 153 K. Subsequently, for standard crystallization (b), the sample was cooled down at 10 K/min to the lower temperature, stayed there for 2 min and reheated above Tm. In order to enhance crystallization during cooling (c), measurements were carried out also after a 30 min isothermal stay (annealing) of the sample at a temperature between the onset and the peak (Tc) of crystallization during cooling or the onset and the peak (Tcc) of cold crystallization event. After the annealing at Tc, the sample was cooled down, at 10 K/min, to the lower temperature, stayed there for 2 min and reheated well above Tm. In all cases studied in this work, the degree of crystallinity is calculated by melting enthalpy after subtracting cold crystallization enthalpy [54]. The heat of fusion for the 100% crystalline polymer was taken equal to: 37.4 J/g for PDMS [14] and 240.0 J/g for PA6 [43].

4670

G. Kritikos / Polymer 55 (2014) 4658e4670

References [1] Goetze W. Complex dynamics of glass-forming liquids: a mode-coupling theory. Oxford: Oxford University Press; 2008. [2] Donth E. The glass transition. Springer; 2001. [3] Mansfield KF, Theodorou DN. Macromolecules 1989;22:3143e52. http://dx. doi.org/10.1021/ma00197a042. [4] Smith GD, Bedrov D, Borodin O. Phys Rev Lett 2003;90:2261031e4. http://dx. doi.org/10.1103/PhysRevLett.90.226103. [5] Vogel M. Macromolecules 2009;42:9498e505. http://dx.doi.org/10.1021/ ma901517z. [6] Harmandaris VA, Daoulas KCh, Mavrantzas VG. Macromolecules 2008;38: 5796e809. http://dx.doi.org/10.1021/ma050177j. [7] Yelash L, Virnau P, Binder K, Paul W. EPL 2012;98:28006. p. 1e5, http://dx.doi. org/10.1209/0295-5075/98/28006. [8] Daoulas KCh, Theodorou DN, Harmandaris VA, Karayiannis NCh, Mavrantzas VG. Macromolecules 2005;38:7134e49. http://dx.doi.org/10. 1021/ma050218b. [9] Alexiadis O, Mavrantzas VG, Khare R, Beckers J, Baljon ARC. Macromolecules 2008;41:987e96. http://dx.doi.org/10.1021/ma071173c. € ger M, Liu WK. Macromolecule 2012;45:2099e112. http://dx.doi.org/ [10] Li Y, Kro 10.1021/ma202289a. [11] Pandey YN, Doxastakis M. J Chem Phys 2012;136:094901e11. http://dx.doi. org/10.1063/1.3689316. [12] Wunderlich B. Prog Polym Sci 2003;28:383e450. http://dx.doi.org/10.1016/ S0079-6700(02)00085-0. [13] Wurm A, Ismail M, Kretzschmar B, Pospiech D, Schick C. Macromolecules 2010;43:1480e7. http://dx.doi.org/10.1021/ma902175r. [14] Klonos P, Panagopoulou A, Bokobza L, Kyritsis A, Peoglos V, Pissis P. Polymer 2010;51:5490e9. http://dx.doi.org/10.1016/j.polymer.2010.09.054. [15] del Rio J, Etxeberria A, Lopez-Rodriguez N, Lizundia E, Sarasua JR. Macromolecules 2010;43:4698e707. http://dx.doi.org/10.1021/ma902247y. [16] Sargsyan A, Tonoyan A, Davtyan S, Schick C. Eur Polym J 2007;43:3113e27. http://dx.doi.org/10.1016/j.eurpolymj.2007.05.011. [17] Fragiadakis D, Pissis P, Bokobza L. J Non-crystalline Solids 2006;352:4969e72. http://dx.doi.org/10.1016/j.jnoncrysol.2006.02.159. [18] Tsagaropoulos G, Eisenberg A. Macromolecules 1995;28:6067e77. http://dx. doi.org/10.1021/ma00122a011. [19] Rittigstein P, Torkelson JM. Polym Sci Part B 2006;44:2935e43. http://dx.doi. org/10.1002/polb.20925. [20] Cangialosi D, Boucher VM, Alegría A, Colmenero J. Polymer 2012;53:1362e72. http://dx.doi.org/10.1016/j.polymer.2012.01.033. [21] Harton SE, Kumar SK, Yang H, Koga T, Hicks K, Lee H, et al. Macromolecules 2010;43:3415e21. http://dx.doi.org/10.1021/ma902484d. [22] Kritikos G, Terzis AF. Polymer 2009;50:5314e25. http://dx.doi.org/10.1016/j. polymer.2009.09.030. [23] Kritikos G, Terzis AF. Eur Polym J 2013;49:613e29. http://dx.doi.org/10.1016/ j.eurpolymj.2012.09.008. [24] Sanchez IC, Lacombe RH. Macromolecules 1978;11:1145e56. http://dx.doi. org/10.1021/ma60066a017. [25] Theodorou DN. Macromolecules 1989;22:4578e89. http://dx.doi.org/10. 1021/ma00202a033. [26] Theodorou DN. Macromolecules 1989;22:4589e97. http://dx.doi.org/10. 1021/ma00202a034.

[27] Fischel LB, Theodorou DN. J Chem Soc Faraday Trans 1995;91:2381e402. http://dx.doi.org/10.1039/FT9959102381. [28] Terzis AF, Theodorou DN, Stroeks A. Macromolecules 2000;33:1385e96. http://dx.doi.org/10.1021/ma991024x. [29] Kritikos G, Terzis AF. Polymer 2005;46:8355e65. http://dx.doi.org/10.1016/j. polymer.2005.06.076. [30] Kritikos G, Terzis AF. Polymer 2007;48:638e51. http://dx.doi.org/10.1016/j. polymer.2006.11.039. [31] Scheutjens JMHM, Fleer GJ. J Phys Chem 1980;84:178e90. http://dx.doi.org/ 10.1021/j100439a011. [32] Fleer GJ, Cohen Stuart MA, Scheutjens JMHM, Cosgrove T, Vincent B. Polymers at interfaces. London: Chapman and Hall; 1998. [33] Rubinstein M, Colby RH. Polymer physics. Oxford University Press; 2003. [34] Harton SE, Kumar SK. J Polym Sci Part B: Polym Phys 2008;46:351e8. http:// dx.doi.org/10.1002/polb.21346. [35] Tzoumanekas C, Theodorou DN. Macromolecules 2006;39:4592e604. http:// dx.doi.org/10.1021/ma0607057. [36] Ingersent K, Klein J, Pincus P. Macromolecules 1986;19:1374e81. http://dx. doi.org/10.1021/ma00159a016. [37] Pelster R, Simon U. Colloid Polym Sci 1999;277:2e14. http://dx.doi.org/10. 1007/s003960050361. [38] Vogiatzis GG, Theodorou DN. Macromolecules 2014;47:387e404. http://dx. doi.org/10.1021/ma402214r. [39] Zhuravlev E, Schmelzer JWP, Wunderlich B, Schick C. Polymer 2011;52: 1983e97. http://dx.doi.org/10.1016/j.polymer.2011.03.013. €tschke P, Androsch R, Schmelzer JWP, Schick C. Eur [40] Zhuravlev E, Wurm A, Po Polym J 2014;52:1e11. http://dx.doi.org/10.1016/j.eurpolymj.2013.12.015. [41] Androsch R, Schick C, Schmelzer JWP. Eur Polym J 2014;53:100e8. http://dx. doi.org/10.1016/j.eurpolymj.2014.01.012. [42] Bokobza L, Diop AL. Express Polym Lett 2010;4:355e63. http://dx.doi.org/10. 3144/expresspolymlett.2010.45. [43] Logakis E, Pandis Ch, Peoglos V, Pissis P, Stergiou C, Pionteck J, et al. J Polym Sci Part B: Polym Phys 2009;47:764e74. http://dx.doi.org/10.1002/polb.21681. [44] Grady BP. J Polym Sci Part B: Polym Phys 2012;50:591e623. http://dx.doi.org/ 10.1002/polb.23052. [45] Ning N, Fu S, Zhang W, Chen F, Wang K, Deng H, et al. Prog Polym Sci 2012;37: 1425e55. http://dx.doi.org/10.1016/j.progpolymsci.2011.12.005. [46] Gedde UW. Polymer physics. London: Chapman and Hall; 1995. [47] Muthukumar M. 1st chapter in progress in understanding of polymer crystallization. Berlin Heidelberg: Springer-Verlag; 2007. [48] Yi P, Locker CR, Rutledge GC. Macromolecules 2013;46:4723e33. http://dx. doi.org/10.1021/ma4004659. [49] Weber CHM, Chiche A, Krausch G, Rosenfeldt S, Ballauff M, Harnau L, et al. Nano Lett 2007;7:2024e9. http://dx.doi.org/10.1021/nl070859f. [50] Vanroy B, Wübbenhorst M, Napolitano S. ACS Macro Lett 2013;2:168e72. http://dx.doi.org/10.1021/mz300641x. [51] Mackay ME, Dao TT, Tuteja A, Ho DL, Van Horn B, Kim HC, et al. Nat Mater 2003;2:762e6. http://dx.doi.org/10.1038/nmat999. [52] Kim K, Miyazaki K, Saito S. J Phys Condens Matter 2011;23:234123. p. 1e9, http://dx.doi.org/10.1088/0953-8984/23/23/234123. [53] Batistakis C, Lyulin AV. Comput Phys Commun 2014;185:1223e9. http://dx. doi.org/10.1016/j.cpc.2013.12.020. [54] Sorai M. Comprehensive handbook of calorimetry and thermal analysis. West Sussex: Wiley; 2004.