In the past several years, there are two main theories which were used to depict the reaction kinetics and the mechanism of the coupling reaction at polymer–polymer interface: one is diffusion-controlled (DC) models suggested by Fredrickson and coworkers [53, 63, 64], and the other is reaction-controlled (RC) models established by O'shaughnessy and coworkers [54, 55, 65–67]. For simplicity, it was assumed in both models that the degree of the polymerization (N) of two polymers is the same, a flat interface was formed between them under quiescent conditions, and reactive groups were located at the chain end of two polymers. The increase of the number of copolymer chains per unit area (Σ) with the increasing reaction time (t) can be expressed as dΣ/dt = k(t)nAnB, where k(t) is the reaction rate, nA and nB are the content of the reactive chains with function groups A and B, respectively.
For DC reaction, the formation of the copolymers at the interface is controlled by the mass transfer of the functionalized polymer chains. The reaction rate constant (k) was described by :
where D, Rg, and N are the diffusion coefficient, radius of gyration, and degree of polymerization of a reactive chain, respectively. Sv is the ration of the interfacial area to total volume fraction. The factor 2RgSV/ln N in Eq. 1 can be interpreted as the volume fraction of interfacial region of an inhomogeneous system that is accessible to A–B coupling. Reduction of this accessible volume fraction by the factor of 1/ln N ∼ 1/ln τ1 is associated with the subdiffusive Rouse dynamics on the time scale that local equilibrium is achieved inside the interface. It can be obtained that k scales with molecular weight in the present Rouse regime as k ∼ 1/ln N for unentangled chains (N〈Ne〉). When the polymer chains exceed the entanglement threshold (N ≥ Ne), the reputation time of chain τ ∼ N3 according to tube model of Gennes, resultantly, k ∼ 1 (N ln N).
On the basis of the previous theoretical framework, Fredrickson and Milner  proposed that there are three time regimes to characterize the growth of the copolymers. The initial growth rate of the copolymers is described by:
where σ(t) is the number of copolymer chains per area of interface that have been formed in the time interval t after initiating the reaction. ρ0 is the concentration of the reactive chains with function groups A and B. τ0 is a reference time whose definition depends on the state of entanglement.
In the early stages of the reaction (τ ≤ t ≤ τρ), when the density of reactive chains in the interfacial region can be considered to be the same as the value in the bulk, coupling reaction is controlled by the quasilocal reaction rate coefficient.
In the intermediate regime (τρ ≤ t ≤ τσ), a depletion hole of reactants in the interfacial region is formed with the progression of reaction,
Therefore, the reaction is controlled by center-of-mass diffusion of the reactive chains to fill the depletion hole.
Finally, the reaction time corresponds to τ ≥ τσ, the potential barrier arising from previously formed copolymers at the interface suppressed the reaction in the last regime, therefore, the diffusion of the functionalize chains across the barrier plays a very important role in the further coupling reaction at the interface. The copolymer grows slowly as
Therefore, the predicted time dependence is a linear growth of σ(t) in the initial time regime, a t1/2 growth in the intermediate regime, and a (ln t)1/2 growth in the last regime, respectively. In most circumstances, only the crossover from center-of-mass diffusion-limited behavior (t1/2) to saturation behavior [(ln t)1/2] can be observed as expected. Furthermore, Müller  investigated the first two regimes using Mote Carlo simulations and found that the simulant reaction rate agreed with the theoretical prediction for the intermediate regime of the DC reaction, however, the simulation indicated a higher reaction rate than predicted by the theory.
According to the Fredrickson's model, if the interfacial coupling reaction was diffusion-controlled, the functionalized chains located in the vicinity of the interface would first participate into the reaction. As the reaction going on, more functionalized chains need to diffuse from the bulk to the interface, then a concentration gradient of the functionalized chains between the interface region and the bulk would be established. Clarke et al. [69, 70] monitored the physical adsorption progress of deuterated polystyrene end-functionalized with carboxylic acid (dPS-COOH) from PS matrix with various molecular weight to a special oxide layer of a silicon substrate, which has immobile reactive sites. It was shown that depletion zones in the concentration profile of dPS-COOH were up to four times longer than necessary for the chains to diffuse from surface to interface, indicating the progress could not be depicted simply in the term of diffusion model. Norton et al.  investigated the similar interface composed of dPS-COOH chains and functionalized epoxy network, and also found that the extent of the reaction could be explained more than DC model.
However, Harton et al. [72, 73] provided direct evidence of depletion hole of the functionalized chains at an interface by studying the coupling reaction between deuterated hydrozy terminated PS (dPS-OH) and methyl methacrylate and methacrylic acide copolymer (PMMA-MAA) at a relatively low temperature of 120°C. In their experiment, dynamic secondary ion mass spectrometry (DSIMS) was used to examine the dPS volume fraction (φdPS) at the interface. The real-space DSIMS depth profiles of the concentration of dPS-OH at the interface from 0 to 100 h were shown in Fig. 2.
Figure 2. Real-space DSIMS depth profiles of the dPS volume fraction (φdPS), including both unreacted dPS-OH within the PS layer and PMMA-g-dPS located at the interface, are shown as a function of sample depth for (a) 0 h, (b) 3 h, (c) 6 h, (d) 12 h, (e) 24 h, and (f) 100 h at 120°C. (Reproduced from Ref.73).
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From Fig. 2a–e, it could be observed that a depletion hole formed and grew to a width with 40 nm apparently from 0 to 24 h, which was a direct evidence of DC coupling reaction at the interface. When the reaction time was up to 100 h, shown in Fig. 2f, the depletion hole disappeared completely and the dPS-OH/PS layer approached diffusion equilibrium. On the basis of the previous experimental results, they concluded that the coupling reaction mechanism at an immiscible polymer–polymer interface was DC, at least in the early stage.
Therefore, the in situ chemical reaction at the interfaces is controlled by DC mechanisms or not depending on the diffusion step of the polymers put in contact and interactions between them. It has been experimentally verified that the diffusion step of polymer chains at interface of two polymers is clearly dependent on the structure of the polymers [74, 75], the relaxation process of chains , the molecular weight and distribution of polymers [77–81], and the composition [82, 83] and compatibility of the interfaces  besides the experimental conditions such as time , temperature [86, 87], and shearing flow [88, 89]. Bousmina et al.  also studied the effect of surface topology and chemistry on the diffusion step by a novel rheometry technique and theoretical analysis that allow determination of diffusion coefficients of both polymer–polymer interfaces and bulk polymers. It was revealed that the diffusion step is strongly dependent on the initial chain ends distribution at the surface before contact.
The RC mechanisms for interfacial reactive compatibilization were theoretically depicted in detail by O'shaughnessy and Sawhney [65, 66]. They considered that the reaction depends on the reactivity of the functional groups and the stage of the reaction. They predicted that the reaction kinetics obeys mean field theory in the reaction between functional groups with weak reactivity (Q « Q*):
where Q, a, and h were the local functional group reactivity, the group size, and the interfacial thickness, respectively.
When the functional groups are sufficiently reactive (Q » Q*), k(t) would be controlled by the diffusion of the reactive chains.
For unentangled melts (N〈Ne)
and for entangled melts (N ≥ Ne)
where R was the coil size, ta was the relaxation time of a single chain unit, and Ne was the critical degree of polymerization for entanglements.
Finally, the crowded copolymer chains formed at the interfaces suppressed k(t) to exponentially small values. Furthermore, O'shaughnessy and Vavylonis  systematically investigated four types of reaction kinetics, through all time regimes, as a function of the reactivity and bulk reactive group densities. Three distinct kinetic sequences which gave rise to different types of reaction kinetics as a function of reaction time were established. Each of these three regions involved a different sequence of reaction kinetics regimes with increasing reaction time. At short time scale, simple mean field (MF) kinetics was applied, the second order rate constant was then independent of time. After this MF phase, there were two possibilities, a direct transition to first-order kinetics may occur. However, if the functional groups were very reactive, then a second-order diffusion regime would onset. In this time regime, it was predicted that Σ ∼ t/(ln t) for unentangled chains and Σ ∼ t/(ln t) during the two t1/4 regimes, whereas Σ ∼ t1/2 for the t1/8 regime for entangled chains. In long time regimes, first-order DC kinetics would be observed, time dependencies during these regimes are Σ ∼ t1/4 for unentangled chains, whereas for entangled systems, in chronological order Σ ∼ t1/4, Σ ∼ t1/8, and Σ ∼ t1/4. When the accumulating copolymer ultimately saturated the interface, first-order DC kinetics with Σ ∼ t1/2 were predicted.
Oyama et al. [91, 92] proposed a new reaction model and kinetic equation for interfacial reaction at polymer–polymer interface by discussing the reaction kinetics and mechanisms of interfacial reaction between amorphous PA and polysulfon end-functionalized with phthalic anhydride (PAH), and triazine. They considered that the coupling reaction at polymer–polymer interface was the progression of the formation of a two-dimensional monolayer at the interface between the polymers. The vacant sites available for reaction at interface was defined as the parameter (Σ* − Σ), where Σ corresponds to the areal density of copolymers formed at time t and Σ* is a critical value of the areal density of copolymers which can form the monolayer at the interface. It was demonstrated that the coupling reaction at polymer–polymer interface was similar to the reaction at the gas/solid interface in surface science, and the reaction followed pseudo-first-order kinetics in the parameter (Σ* − Σ) over the whole time scale based on the assumption that the coupling reaction would be terminated at Σ = Σ*. Except for the systems in their experiment, the reaction kinetics in published articles for reactive polymer interfaces such as amorphous styrene-maleic anhydride copolymer/amine-terminated butadiene-acrylonitrile copolymer (SMA/ATBA), SMA/PA11 , and carboxyl end-functional polystyrene/precured epoxy containing excess epoxide groups  were all analyzed by using their model. The results showed that all the experimental data agreed quite well with presudo-first-order kinetics. So, it was suggested that the coupling reaction process between polymers was probably reaction-controlled, and the spatial restriction at the interface played an important role during the coupling reaction.
Schulze et al. [95, 96] measured the interfacial excess (Z*/Rg) at three molecular weights of dPS end-functionalized with an amine group (dPS-NH2-22, −37, and −92) and PMMA end-functionalized with anhydride (PMMA-anh) interface using forward recoil spectrometry (FRES). The measured interface (shown in Fig. 3a) was prepared by floating films containing 8.4 wt% dPS-NH2 and PS onto PMMA-anh layers that had been spun from the solution onto Si wafer. In addition, a special designed trilayer interface geometry shown in Fig. 3b was used to observe independently the diffusion of dPS-NH2 chains through the PS matrix and the reaction with PMMA-anh eventually. The results of the measured growth of Z*/Rg with time for all three molecular weights could be described with polymer chain reactivity, rather than the diffusion of the functionalized chains. Compared with the concentration profiles obtained from these two different designed interface geometries indicated that the dPS-NH2 could diffuse through the PS layer many times before significant reaction could be observed under experimental condition. These results proved that the coupling reaction rate at PS-NH2/PMMA-anh interface was not limited by the diffusion of the functionalized polymer chains.
Later, Yin et al. [97–99] also probed the interfacial reaction between PMMA-anh and PS-NH2 with high and low molecular weight by size exclusion chromatography with a UV detector (SEC-UV). Reactive polymers were labeled with anthracene to increase the sensitivity. It was found that the conversion of PS-NH2 was faster when the reactive chains were shorter. It likely showed that the reaction kinetics was DC controlled because the diffusion coefficient depends on the chain length. However, when the dependence of the interfacial roughness on the molecular weight of the reactive chains was observed by AFM and TEM, it was found that the interfacial modification was slower than the diffusion of the reactive chains. The same conclusion was obtained by them as well as Schulze et al. that the interfacial reaction between PS-NH2 and PMMA-anh was not controlled by diffusion but rather by the reaction kinetics of the anhydride and amine functional groups.
The activation energy of the reaction is another clue, which can be used to judge whether the coupling reaction kinetics at polymer interface are controlled by diffusion or reactivity. It is believed that the DC reactions usually have low activation energy (<30 kJ/mol) because it is a physical process, whereas the RC mechanisms have higher activation energy attributed to a chemical process. Oyama and Inoue  estimated the activation energy for the reaction at SMA/ATBA interface was 120 kJ/mol, indicating that the coupling reaction at this interface was reaction-controlled. Kim et al.  investigated the effect of surface functionalization on the interfacial adhesion between PP and PA6. The functional groups of PP surface modified by low-energy ion-beam irradiation method are concentrated on a very shallow surface layer (less than 70 Å), which makes the diffusion of reactive chains, can be neglected significantly. The interfacial reaction kinetics in this system can be considered as reaction-controlled. The calculated activation energy was 97.8 kJ/mol, which was also in favor of reaction-controlled mechanisms. Kim et al. [101, 102] studied the effect of molecular weight of dPS-NH2 and reaction temperature on the reaction kinetics at PS/poly(2-vinylpyridine) (P2VP) interface. It was experimentally showed that the activation energy of the reaction was 156 ± 7 kJ/mol, and the value was not affected by the molecular weight of dPS-NH2. Jiao  measured the activation energy of the reaction between dPS-NH2 and PSMA interface, and the value was 207 kJ/mol. For the reaction between the same functional groups, the experimental values of the activation energy were not the same, but all the experimental results suggested that the interfacial reaction kinetics was controlled by the reactivity of the functional groups.
Interfacial Adhesion (Gc )
When two immiscible polymer chains are connected by the covalent bond of copolymers formed at the interface, the interfacial adhesion between two immiscible phases will be enhanced. In the case of planar interfaces enhanced via reactive compatibilization, the interfacial adhesion is characterized by the fracture toughness (Gc) of that. It has been shown that an asymmetric double cantilever beam (ADCB) test is a reliable test for measuring Gc of polymer–polymer interfaces [104–106]. It involves propagating a crack along the interface between two polymer beams using a wedge. The specific configuration of the test is shown in Fig. 4. The wedge, typically a razor blade, is inserted at the interface to induce a crack. The elasticity of the two polymer beams enhances the stress at the crack tip, causing the crack to be advanced some distance ahead of the wedge. Adhesion acts to suppress crack propagation, counteracting the elasticity. Therefore, the distance of the crack advances ahead of the wedge is an indication of the adhesion. Generally, constant rate mode is introduced, in the method, the razor blade of width Δ is driven at a constant rate of 3 μm/s, and the crack length (a) at the interface is continuously monitored. Gc is then calculated according to the following equation [107, 108]
where Ei and hi denote the Young's modulus and the thickness of material i, Δ is the thickness of the razor blade, a is the crack length ahead of the blade, Ci is the correctional factor for material i, which is expressed as follows:
On the basis of the above measured method and other analysis techniques, the effect of reaction conditions and the bulk properties of polymers on the Gc of model reactive compatibilization interfaces were extensively studied, including reaction time and temperature, the crystalline structure near the interface, molecular weight, and architecture of the formed copolymers, as well as the relationship between Gc and the areal density of copolymers at the interface (Σ).
Effect of Reaction Time and Temperature on Gc
Theoretically, when the reaction temperature is higher than the glass translation temperature of two polymers, interfacial reaction can occur since the polymer chains near the interface can diffuse each other. But very long time is needed to form effective adhesion at low reaction temperature. When the reaction temperature is above the melt temperature of both polymers, Gc will increase rapidly within first several minutes. To control the variation in Gc over wide order of magnitude as a function of suitable reaction time, the reaction temperature is usually chosen between the melt temperatures of two polymers near the interface.
Boucher et al. [109, 110] investigated Gc of PP/PA6 interface annealed at the temperature between 185°C and 223°C as a function of the annealing time, and 5 wt% anhydride grafted PP was blended with PP for reactive compatibilization. As shown in Fig. 5, Gc increased with the annealing time and then reached a level of saturation for all annealing temperature. Many other researchers obtained similar experimental results in other experimental systems [111–113]. When the annealing temperature was a litter higher than the melt temperature of PA6, i.e., 223°C, the rate and the level of saturation of Gc increased dramatically. Laurens et al.  also studied the effect of annealing temperature on Gc of PP/PA6 by using functionalized PP (PPf) with different molecular weight as the compatibilizer. It was found that annealing the samples above the PA6 melting temperature increased strongly the adhesion in samples containing high molecular weight PPf, while such an increase was not observed in other samples. Therefore, they believed that a sufficient molecular weight of functionalized polymer chains should be a necessary condition to obtain a dramatic increase in interfacial adhesion at high reaction temperature.
However, some other researchers observed a maximum in the saturation Gc at a certain reaction temperature when they studied the similar experimental systems. Seo and Ninh  found Gc of PP/PA6 compatibilized with 1 wt% maleic anhydride grafted PP showed a maximum around 220°C and then decreased at higher annealing temperature. Cho and Li  also observed an unexpected maximum in the saturation Gc of PP and amorphous PA6 interface at around 170°C, which was ascribed to a different interfacial morphology around this temperature. Kim et al.  obtained a maximum saturation value of Gc at 200°C when they studied the effect of surface functionalization by low-energy ion-beam irradiation on the enhancement of interfacial adhesion between PP and PA6. Since the PP molecules were all in melt state, the influence of the PP crystallization on the Gc could be eliminated. They argued that the appearance of a maximum at 200°C was attributed to the change of fracture mechanisms at the interface.
Effect of Functionalized Chains Content on Gc
In general, 5 wt% functional polymer is enough for reactive compatibilization in immiscible polymer systems prepared by melt blending, because new reactive interface can be formed continually due to shear effect. But the reactive interface with model planar geometry is constant, whether more functionalized polymer or not need to be added for reactive compatibilization in this system. Cho and Li  investigated the effect of maleic anhydride grafted PP (mPP) content on the interfacial fracture toughness of amorphous PA6/PP at the same bonding temperature. It was shown that Gc could not be enhanced when the amount of mPP was lower than 0.5 wt%, even at a bonding time of 2 h. As the mPP content increased, Gc increased obviously which was ascribed to the in situ formation of copolymers at the interface. When the mPP concentration reached 3 wt%, Gc became slightly leveled off. Gc of the samples as a function of bonding time for 3 and 5 wt% mPP were nearly overlapped, indicating 3 wt% mPP were enough to compatibilize this system effectively.
Koriyama et al.  examined the interfacial thickness and adhesion of the reactive polysulfon (PSU) and amorphous PA6 interfaces as a function of MAH group content in PSU. It was observed that the interfacial thickness was almost independent of the MAH group content, especially when the MAH group content was more than 0.56 wt%. According to their results, the interfacial adhesion increased with increasing interfacial thickness following the relationship, Gc ∼ λ2. So, they believed that MAH group content of 0.2 wt% was sufficient to form a thick interface, thus, efficiency interfacial reinforcement.
Effect of Molecular Architecture of Functionalized Polymer on Gc
When the miscibility between functionalized polymer and the matrix is really poor, it will induce a lack of entanglements or cocrystallization between the in situ formed copolymers and the matrix. Therefore, it is not efficient to transfer stress across the interface during fracture test. On the contrary, when the functionalized polymer has a good compatibility with the matrix, the block of the copolymers can entangle or cocrystallize with the matrix chains well, leading to an effective reinforcement of interface by developing a plastic deformation zones at the craze tip. The molecular weight of the functionalized polymer also influences the entanglement between the copolymers and the matrix near the interfaces. Laurens  synthesized a series of succinic anhydride functionalized PP with different molecular architectures (isotatic-PP, isotatic-PP with 5 wt% PE, syndiotactic-PP, metallocene-PP) and discussed the role of the architecture of PP block of the in situ formed copolymers on the interfacial fracture toughness enhancement for isotactic-PP/PA6 interface. It was found that the kinetics of fracture toughness enhancement of samples annealing at the same conditions was strongly influenced by the miscibility of functionalized PP (PPf) with the matrix PP. When the PPf was isotatic-PP, Gc of i-PP/PA6 interface increased rapidly and then reached saturation value with annealing time due to its good compatibility with the matrix. When the partially compatible PPf (PE-PPf) was incorporated, the extent of the increase in Gc became less obvious and a plateau was not always observed. In addition, the molecular weight of PPf also influenced Gc, which was shown that the maximum Gc was larger for the high molecular weight PPf. They proposed that long copolymer chains were more efficient than short copolymer ones to reinforce the interface, which was attributed to the fact that long copolymer chains can link several lamellae in the vicinity of the interface. Eastwood and Dadmun  compared the ability of a series of styrene and methyl methacrylate copolymers with varying architectures to compatibilize PS/PMMA interface. It was demonstrated that the ability of reinforcement of the interface decreased in the order pentablock 〉 triblock 〉 diblock 〉 heptablocks 〉 random copolymers, and the multiblock copolymers except heptablocks provided good interfacial adhesion, which was attributed to multiple interface crossings between blocks of monomer and homopolymers. To form adequate entanglements with homopolymer, a critical molecular weight was required for block length. This was exactly the reason that why the heptablock copolymers give the relatively weak interface.
Besides the aforementioned factors, The enhanced adhesion between immiscible polymer interfaces through in situ reactive compatibilization depends on many other parameters, such as cooling rate , chains orientation , and crystalline structure at the interface , especially for semicrystalline polymer interfaces.
Relationship Between Gc and Σ
It is evident that the covalent bond connection at the interface will build up with the increasing Σ, thus, Gc increases as the progress of interfacial reaction. The changes of Gc are strongly depended on the fracture mechanisms at the interface. To understand which mechanisms are responsible for the interface reinforcement, it is useful to examine the relationship between Gc and Σ. In the case of glassy polymer interfaces [122, 123], two different relationships between Gc and Σ have been observed: when the fracture mechanisms of the interface are simple chain scission or chain pull-out without any plastic deformation, Gc is found to vary as Σ linearly; when crazing and plastics deformation are initiated at the interface, then Gc is observed to be scaled with Σ2. For semicrystalline polymer interfaces, the relationship between Gc and Σ is more complicated because the crystalline structure at the interface has profound effects [124–127] on the plastic deformation properties of the polymers. Boucher et al.  first established a theoretical model to correlate Gc with Σ at semicrystalline PA6/PP interfaces. By the combination of the results of ADCB test and XPS analysis, they were surprised to obtain that all date plotted on log–log scale in agreement Gc ∝ Σ2.03 ± 0.18 scaling, which was similar to that observed for glassy polymer interface in the crazing regime predicted by Brown .
But in further work by Boucher , the relationship between Gc and Σ was not fit to the above model when PP/PA6 interface was annealed a litter above the melt temperature of PA6, especially when high molecular weight PPf was incorporated as compatibilizer. As shown in Fig. 6, when the annealing temperature was below 220°C, it was likely that the relation between Gc and Σ was not influenced by the conditions for preparing the samples, and the interfacial fracture mechanisms were similar for all values of Σ. However, it was not right in the case of the samples prepared at a temperature of 223°C. According to Boucher's further investigation, the change in fracture mechanisms for sample with high molecular weight made at 223°C was related to the fact that the β-form PP crystalline structure was developed at the interface. It implied that the interfacial fracture mechanisms were not only influenced by the copolymers formed but also by the crystalline structure near the interfaces. Lauren [114, 118] reported that the relationships between Gc and Σ were not consistent with Boucher's model in some experimental systems. He proposed that the relationship between Gc and Σ depends not only on annealing conditions but also on the molecular weight of functionalized polymers and the miscibility between the functionalized polymer and the matrix. The structure of the in situ formed copolymers and the crystalline and orientation behavior near the interfaces should also be considered. Because of the complicated influencing factors, a consistent and universal model has not been established to describe the relationship between Gc and Σ for reactive compatibilized semicrystalline polymer interface.
Interfacial Morphological Development by Reaction
Some investigation has shown that the copolymers generated at the interface can form polymeric surfactants, thus, decrease the interfacial tension between the incompatible polymer phases and result in a significant diminution in the particle size of dispersed phase [128–131]. Many studies have found that the final morphology of polymer blends with an in situ reactive compatibilizer depends on the bulk properties of the components, blend composition, the amount and the molecular weight of the formed copolymer, viscosity ratio between different phases, and the processing parameters during melt blend. Among them, the shear force is a very important factor because it can induce droplet breakup and bring continual new reactive interfaces. For flat polymer interface system, discussed in this article, there is no shear effect during reaction compatibilization. How does the morphology develop in this kind of experimental condition is another interesting topic.
From the theoretical studies [63, 65] concerning the kinetics of coupling reaction at a melt polymer interface, a free energy barrier (μ*/kBT) to the coupling reaction will develop due to the entropy loss involved in stretching the “brush” of grafted chains as the ratio of interface excess of the grafting chain to its radius of gyration (Z*/Rg) increases. The buildup of such a barrier during the reaction will suppress the coupling reaction and ultimately limit the Z*/Rg achievable within experimental time. O'shaughnessy and Sawhney  asserted that the reaction rates will slow down to near zero before the reaction can form sufficient copolymers at the melt polymer interface to effectively diminish the interfacial tension. However, Jiao et al.  demonstrated that it is possible to make interfacial tension at polymer interface decrease significantly by reactive compatibilization for short functionalized chains, and the decreases of the interfacial tension is large enough to cause polymer interface instability, which leads to interfacial roughness and eventually the formation of microemulsions. A theoretical model correlated to −Δγ/γ0 and Z*/Rg was developed to predict the transition of the interface from flat to corrugated, where the interfacial tension becomes to zero.
On the basis of the self consistent mean field calculations of Shull , the decrease in interfacial tension (−Δγ) due to the increased interfacial excess of copolymer chains can be written in the following equation:
where ρ0 is the monomer number density, a is the statistical segment length or polymer, α is a function of Z*/Rg tabulated by Shull for the limit where the unreactive matrix chains are much longer than the end-functional chains.
For the high molecular weight polymers without graft copolymers, the initial interfacial tension (γ0) is given by the Helfand and Tagami results ,
where χ is the Flory–Huggins interaction parameter.
Figure 7. RMS roughness of the PS/PSMA interface after washing with cyclohexane measured by SFM as a function of Z/Rg. N = 55 (□) and N = 270 (○). (Reproduced from Refs.132 and135).
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Some recent experimental results [98, 102, 136] also proved that the decrease of interfacial tension due to reactive compatibilization can result in the interfacial roughening or interfacial emulsification at a planar polymer interfaces in the absence of flow. The development of morphology is closely related to the reaction conditions, the concentration, and molecular weight of reactive functionalized polymer.
Lyu et al. [137, 138] examined the morphological development at layered reactive PS-NH2/PMMA-anh interface as a function of annealing time by AFM and TEM observation. According to TEM results, the interface between PS and PMMA was flat before annealing, shown in Fig. 8a. The interface became quite rough after annealing the sample for only 20 min, and some parts of the PS domain appeared to have pinched off at the interface and moved into PMMA phase, as shown in Fig. 8b. When the annealing time increased to 1 h, the interfacial roughness increased further and the magnitude of the width of the roughening zone was about 0.5 μm shown in Fig. 8c. The interfacial roughening during annealing was also observed by AFM. It was observed that the magnitude of the width of the roughening zone increased from 0 to 0.2 μm when the sample was annealed for 1 h.
Figure 8. Representative morphologies of PS-NH2/PMMA-ah interface after static reaction of (a) 0, (b) 20, and (c) 60 min. (Reproduced from Ref.137).
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Zhang et al.  studied the relationship between the interfacial roughening process and the extent of coupling reaction at PS/PMMA interfaces. The coupling reaction was controlled by varying the concentration of PS-NH2 in an unreactive PS matrix, while maintaining the same pure PMMA-anh layer. It was obtained that interfacial roughness was strongly influenced by the concentration of PS-NH2 in the PS layer. For 10 wt% PS-NH2 sample, the RMS roughness increased to about 1.4 nm in the first 5 min and then remained the same value within 120 min. When the concentration of PS-NH2 was high, take 75 wt% for example, the RMS roughness increased to above 10 nm dramatically in about 10 min and then increased to above 15 nm very slowly over 120 min. There was an abrupt increase in RMS roughness when the PS-NH2 concentration changed from 25 to 30 wt% within the same annealing time of 1 h at 175°C, just shown in Fig. 9. To explain the development of interfacial roughness, the maximum interfacial coverage (Σ*) was defined. They got a conclusion that when the concentrations of PS-NH2 was low, Σ < Σ*, the interfacial roughness increased with reaction conversion but remained low. When the concentration was increased to 30 wt%, Σ > Σ*, the interfacial roughness increased dramatically, and interfacial emulsification phenomena happened, which was proved by TEM observation of the interfacial morphology.
The length of the functionalized chains, i.e., the molecular weight also influence the interfacial roughness [98, 102]. On the one hand, the short reactive chains can diffuse to interface faster than long ones, on the other hand, when long functionalized chains are introduced, the high molecular weight copolymers formed at the interface do not leave the interface, and suppress the reaction rate to zero, whereas, the copolymers formed by short chains can diffuse across the interface, which is favorable to the reaction progress and interfacial roughness.
When reactive compatibilized polymer blends are prepared in the presence of shear (or elongational) effect, micelles are usually formed in the matrix phase depending strongly on the amount of in situ formed copolymers as well as the molecular structure of those [140–143]. In the Jiao's and Zhang's research, the interfacial roughness both remain nearly constant after the transition point though the extent of reaction increases. They attributed this to the fact that the entire emulsified region (copolymers coated droplets) formed after the transition point was removed by the selective solvent during the preparation of sample for AFM observation. Direct TEM cross-sectional imaging of the interface supported their assumption. Kim and coworkers [144, 145] first reported the formation of microemulsions at reactive interface of PS-mCOOH/PMMA-GMA in the absence of shear when the samples were annealed at 180°C for 17 h. A simple illustration given in Fig. 10 was used to explain the interfacial morphological change with reaction stages and the formation progress of microemulsions in the studied system. Three distinct stages were considered: in Stage I, the functionalized chains reacted each other mainly near the interface and the interface was not roughened. When the in situ formed copolymers were covered at least a single layer at the interface (Stage II), the functionalized chains diffuse into the brush like copolymer layer, the interface became corrugated. At the beginning of the Stage III, the reactant chains diffuse through the brush like copolymer layer, the interface became more corrugated with the process of reaction. When t > tmicroemulsion, the copolymers began to pinch off and then PS chains were encapsulated by PMMA chains, which finally became microemulsions. They verified the hypothetic process by TEM observation .
Figure 10. Schematic describing variations of interfacial morphology development for (PS-mCOOH)/(PMMA-GMA) bilayer at 180°C. (Reproduced from Ref.145).
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They suggested that the microemulsions structure in the planar interface depended on the reaction time and the molecular weight of the functionalized chains. Longer reaction time and lower molecular weight more likely to form microemulsions structure. Since the above experiment was performed in plate rheometer, a very small oscillatory shear force was applied on reactive bilayer sample during reaction, it might be argued that whether the microemulsion formation was induced by this very small shear effect. According to the experimental results of Kim, the morphological development near the interface of the (PS-mCOOH)/(PMMA-GMA) bilayer sample annealing in the presence of oscillatory shearing used in their rheological measurements was essentially the same as that in the absence of oscillatory shearing. Microemulsions were also observed in the bilayer sample annealing at 170°C under quiescent conditions.
In a word, the copolymers formed at the planar interface is expected to decrease the interfacial tension of this polymer interface. Instability and roughening of the interface is observed when the interfacial tension decreases to a critical value. The extent of interfacial roughness depends on the concentration and molecular weight of functionalized chains and reaction conditions. There exists a critical values for Z*/Rg and the concentration of the functionalized chains, where the interfacial tension vanishes and the interfacial morphology changes dramatically from flat to corrugated one.