D ec 2 02 0 Critical behavior of charged AdS black holes surrounded by quintessence via an alternative phase space

Considering the variable cosmological constant in the extended phase space has a significant background in the black hole physics. It was shown that the thermodynamic behavior of charged AdS black hole surrounded by the quintessence in the extended phase space is similar to the van der Waals fluid. In this paper, we indicate that such a black hole admits the same criticality and van der Waals like behavior in the non-extended phase space. In other words, we keep the cosmological constant as a fixed parameter, and instead, we consider the normalization factor as a thermodynamic variable. We show that there is a first-order small/large black hole phase transition which is analogous to the liquid/gas phase transition in fluids. We introduce a new picture of the equation of state and then we calculate the corresponding critical quantities. Moreover, we obtain the critical exponents and show that they are the same values as the van der Waals system. Finally, we study the photon sphere and the shadow observed by a distant observer and investigate how the shadow radius may be affected by the variation of black hole parameters. We also investigate the relations between shadow radius and phase transitions and calculate the critical shadow radius where the black hole undergoes a second-order phase transition.


I. INTRODUCTION
Undoubtedly, the black hole is one of the most fascinating and mysterious subjects in the world of physics as well as mathematics. Black hole was one of the interesting predictions of general relativity which is confirmed by observational data [1]. The observational evidence of massive objects and detection of the gravitational waves open a new window in modern mathematical physics and data analysis. On the other hand, the discovery of a profound connection between the laws of black hole mechanics with the corresponding laws of ordinary thermodynamic systems has been one of the remarkable achievements of theoretical physics [2,3]. In other words, the consideration of a black hole as a thermodynamic system with a physical temperature and an entropy opened up new avenues in studying their microscopic structure.
In the past two decades, the study of black hole thermodynamics in an anti-de Sitter (AdS) space attracted significant attention. Strictly speaking, the investigation of thermodynamic properties of black holes in such a spacetime provides a deep insight to understand the quantum nature of gravity [4,5]. In particular, the phase transition of AdS black holes has gained a lot of attention due to the AdS/CFT correspondence in recent years. The pioneering work in this regard was realized by Hawking and Page who proved the existence of a certain phase transition (so called Hawking-Page) between thermal radiation and Schwarzschild-AdS black hole [6]. Afterward, our understanding of phase transition has been extended by studying in more complicated backgrounds [7,8]. Among conducted efforts, thermodynamics of charged black holes in the background of an asymptotically AdS spacetime is of particular interest, due to a complete analogy between them and the van der Waals liquid-gas system. Such an analogy will be more precise by considering the cosmological constant as a dynamical pressure and its conjugate quantity as a thermodynamic volume in the extended phase space [9].
It is worthwhile to mention that the conducted investigations in this regard are based on the first law and Smarr relation by comparing black hole mechanics with ordinary thermodynamic systems not CFT point of view. Despite the interesting achievements of AdS/CFT correspondence such as describing the Hawking radiation mechanism and dual interpretation of Hawking-Page phase transition and so on with fixed Λ, this method is not yet able to provide a suitable picture in the extended thermodynamics. In the context of AdS/CFT correspondence, the cosmological constant is set by N , related to the number of coincident branes (M branes or D branes) on the gravity side. On the field theory side, N is typically the rank of a gauge group of the theory, and as such, it also determines the maximum number of available degrees of freedom. The thermodynamic volume in the bulk gravity theory corresponds to the chemical potential in the boundary field theory which is the conjugate variable of the number of colors [10,11].

II. THERMODYNAMICS OF CHARGED ADS BLACK HOLE SURROUNDED BY QUINTESSENCE: A BRIEF REVIEW
In this section, we first introduce the thermodynamics charged AdS black hole surrounded by quintessence by reviewing Refs. [46][47][48][49]. The line element of such a black hole is expressed as where where M and Q are the mass and electric charge of the black hole, respectively and ℓ = − 3 Λ is the AdS radius which is related to the cosmological constant. The state parameter ω describes the equation of state p = ωρ where p and ρ are the pressure and energy density of the quintessence, respectively. The normalization factor a is related to the density of quintessence ρ as with [length] −2 dimensions. Solving the equation (f (r = r + ) = 0), one can obtain the total mass of the black hole M as In addition, one can use the Hawking and Bekenstein area law to obtain the entropy as Working in the extended phase space, the cosmological constant and thermodynamic pressure are related to each other with the following relation where the variability of the cosmological constant is associated to the dynamical vacuum energy. It it easy to rewrite Eq. (4) in terms of pressure and entropy as It is obvious that the total mass of black hole plays the role of enthalpy instead of internal energy in the extended phase space. Therefore, regarding the enthalpy representation of the first law of black hole thermodynamics, the intensive parameters conjugate to S, Q, P and a are, respectively, calculated as where T , Φ and V are the temperature, electric potential and thermodynamic volume, respectively and y is the quantity conjugate to the dimensionful factor a. Considering the dimensional analysis, we can obtain the following Smarr relation where confirms that we have to regard a as a thermodynamic quantity. Now, it is straightforward to find that the first law of the black hole is obtained as A. P − V criticality and van der Waals phase transition: Usual method In this section, we briefly review the critical behavior of such black holes in the usual way. Considering the temperature relation, Eq. (8) with the definition of pressure, Eq. (6), one can easily derive the equation of state of the black hole as For the sake of the dependency between r + and specific volume υ [9,14,46], Eq. (14) can be rewritten as The corresponding P − V and T − V diagrams are depicted in Fig. 1. Evidently, the behavior is reminiscent of the van der Waals fluid which confirms the first-order small-large black hole transition for temperatures smaller than the critical temperature. The critical point can be extracted from which results into the following equation for calculating critical volume The critical temperature and pressure is calculated as Equation (17) can be analytically solved for ω = − 2 3 , resulting into the following critical quantities It is worthwhile to mention that the critical volume and pressure are exactly the same as those presented for RN-AdS black hole [9], and only the critical temperature is affected by quintessence dark energy. For a = 0, all these critical quantities reduce to those of the RN-AdS black hole.

III. CRITICAL BEHAVIOR OF THE CHARGED ADS BLACK HOLE SURROUNDED BY QUINTESSENCE: AN ALTERNATIVE APPROACH
Kubiznak and Mann in Ref. [9] showed that charged AdS black holes have a critical behavior similar to van der Waals fluid in the extended phase space. Li also employed such an idea for investigating the critical behavior of the charged AdS black hole surrounded by quintessence [46]. Although the idea of considering the variable cosmological constant has attracted a lot of attention in black hole thermodynamics, it was shown that by keeping the cosmological constant as a fixed parameter and instead considering the square of electric charge as a thermodynamic variable, one can observe such a critical behavior in Q 2 − Ψ plane [35]. The study of phase transition via this alternative approach was made for the charged AdS black hole in the presence of quintessence field in Ref. [50]. Now, we are interested in studying the critical behavior of charged AdS black holes surrounded by quintessence via a new approach by considering both the cosmological constant (Eq. 6) and electric charge as fixed external parameters and allow the normalization factor to vary. Here, we investigate critical behavior of the system for three different cases to find a proper alternative approach. In this subsection, we consider the normalization factor a as a thermodynamic variable and study the critical behavior of the system under its variation. We start by writing the equation of state in the form a(T, y) by using Eq. (8). Inserting Eq. (11) into the relation of temperature, the equation of state is obtained as In order to investigate the critical behavior of the system and compare with the van der Waals fluid, we should plot isotherm diagrams. The corresponding a − y diagram is illustrated in Figs which leads to Studying the heat capacity, one can confirm the criticality behavior mentioned above via the method of reported in Refs. [51,52]. After some manipulations, we find Solving denominator of the heat capacity with respect to a, a new relation for normalization factor (a new ) is obtained which is different from what was obtained in Eq. (20). a new is obtained as follows Evidently, the above relation diverges at ω = − 2 3 . This new relation for normalization factor a has an extremum which exactly coincides with the inflection point of a − y diagram (see dashed lines in Figs. 2(a) and 2(d)). In other words, its extremum is the same critical normalization factor and its proportional y (in which a new is maximum) is y c . By deriving the new normalization factor with respect to y, one can obtain y max as which is the same y c in Eq. (22). It is evident that by inserting Eq. (25) into Eq. (24), one can reach a c (compare a new to a c in Eq. (22)). As we see from Fig. 2(a), for ω > − 2 3 , the new normalization factor has a maximum which matches to the inflection point of a − y diagram. Whereas an opposite behavior can be observed for ω < − 2 3 (see Fig. 2(d)). The phase structure of a thermodynamic system can also be characterized by the Gibbs free energy, The behavior of the Gibbs free energy in term of T is depicted in Figs. 2(b) and 2(e). The existence of swallow-tail shape in G − T diagram indicates that the system has a first order phase transition from small black hole to large black hole. For ω > − 2 3 , system undergoes a first order phase transition for a < a c (see Fig. 2(b)) and T > T c (see Fig. 2(a)). Whereas for ω < − 2 3 , such a phase transition is observed for a > a c and T < T c (see Figs. 2(d) and 2(e)). The coexistence line of two phases of small and large black holes, along which these two phases are in equilibrium, is obtained from Maxwell's equal area law. Figures 2(c) and 2(f) displays the coexistence line of small-large black hole phase transition. The critical point is highlighted by a small circle at the end of the coexistence line.

Critical exponents
Critical exponents describe the behavior of physical quantities near the critical point. For fixed dimensionality and range of interactions, the critical exponents are independent of the details of a physical system, and therefore, one may regard them quasi-universal. Now, our aim is to calculate the critical exponents in this new approach. To do so, we first introduce the following useful relations where the critical exponents α , β, γ and δ describe the behavior of specific heat C y , the order parameter η, the isothermal compressibility κ T and behavior on the critical isotherm T = T c , respectively. To find the critical exponent, we define the below dimensionless quantities Since the critical exponents are studied near the critical point, we can write the reduced variables in the following form First, we rewrite the entropy in terms of T and y as, which is independent of temperature. So, we find that and hence α = 0. By using Eq. (28), one can expand Eq. (20) near the critical point as where and Differentiating Eq. (34) with respect to ν for a fixed t, we get Now, using the fact that the normalization factor remains constant during the phase transition and employing the Maxwell's area law, we have the following two equations: where ν s and ν l denote the event horizon of small and large black holes, respectively. Equation (36) has a unique non-trivial solution given by According to the table I, the argument under the square root function is always positive. From Eq. (37), one can find that Now, we can differentiate Eq. (34) to calculate the critical exponent γ as Finally, the shape of the critical isotherm t = 0 is given by The obtained results show that the critical exponents in this new approach (with fixed Λ and variable a) are exactly the same as those obtained in [46] (with Λ variable and fixed a) and coincide with the van der Waals fluid system [9]. Since phase structure of the thermodynamic system changes at ω = − 2 3 , one cannot observe a unique behavior in this approach.

B.
Critical behaviour of the black hole via approach II In order to have a unified phase behavior, we consider a ′ = −a as a thermodynamic variable to check whether the thermodynamic system has a unique phase structure. In this case, conjugate quantity to a ′ is as The equation state a ′ (T, x) is given by The behavior a ′ as a function of x is depicted in Fig. 3 which shows that the type of phase transition is van der Waals like. To calculate critical values, we use the properties of inflection point which leads to Taking a look at Figs. 3(a) and 3(b), we see that for ω < − 2 3 , the van der Waals like phase transition occurs for T < T c and a ′ < a ′ c . Whereas for ω > − 2 3 , such a phase transition is possible for T > T c and a ′ > a ′ c (see Figs. 3(d) and 3(e)). Calculating heat capacity and solving its denominator with respect to a ′ , we obtain a ′ new which is observable in Figs. 3(a) and 3(d). For ω < − 2 3 , the function a ′ new has a maximum point which coincides to a ′ c (see dashed line in Fig. 3(a)), while vice versa happens for ω > − 2 3 (see dashed line in Fig. 3(d)). Now, we turn to calculate the critical exponents in this new phase space approach. Since the entropy only depends on x, we find that α = 0. To calculate the other critical exponents, we define the reduced thermodynamic variables as and we write the reduced variables in the form Expanding Eq. (42) near the critical point, one finds where (47) which is the same as Eq. (32), and the only difference between these two equations is B i coefficients given by Our numerical analysis showed that the coefficients B 2 and B 3 are very small and can be considered zero as in the previous case. The obtained critical coefficients β, γ and δ are the same as those presented in the previous subsection and we do not write them here to avoid repetition. Although the obtained critical exponents in this approach coincide with those obtained for van der Waals fluid, similar to the previous case, phase structure of the thermodynamic system does not have a unique behavior. To observe a unified phase behavior, we investigate another state of the normalization factor in the next subsection. In order to observe a van der Waals like phase transition with unified behavior, it is better to define a new parameter dependent on the normalization factor with [length] −2 dimensions. Taking a look at Eq. (3) and the equation of state p = ωρ, we notice that the pressure of quintessence can be considered to this purpose. By doing so, total mass (4) can be rewritten in term of p as The temperature and conjugate quantity to p are as First law and Smarr relation are given, respectively, as To calculate the equation of state p(T, χ), we use Eq. (50) and rewrite it in the following form where B = (3ω 2 ) 1 3 . The p − χ and T − χ diagrams are shown in Fig. 4. The behaviors observed in these diagrams point out that the critical behavior is in complete analogy with the van der Waals liquid-gas system. It should be noted that the oscillating part ( ∂p ∂χ > 0) of the isotherm diagram indicates instability region which is physically not acceptable. In fact, in order to observe an acceptable behavior, the pressure of the quintessence should be a decreasing function of χ. In any place that such a principle is violated, the black hole is unstable and may undergo a phase transition in that region. In order to see whether the pressure of the quintessence is a decreasing/increasing function of its conjugate quantity, we calculate its first order derivation with respect to χ dp dχ if this expression is negative, the black hole admits the mentioned principle and that region is physically accessible, while positivity of this expression means that a phase transition takes place in that region. It is worthwhile to mention that places where the signature of dp dχ changes are where the pressure of the quintessence acquires an extremum. To express such a possibility, we have plotted diagrams in Fig. 5.
To get more information about the phase transition, we investigate G − T diagram. The Gibbs free energy in the canonical ensemble G = M − T S can be calculated as where r + is related to χ which is a function of T and p through Eq. (53). The phase transitions of a system can be categorized by their orders which are characterized by the discontinuity in n th derivatives of the Gibbs free energy. For example, in a first-order phase transition, G is a continuous function but its first derivative (the entropy or volume) changes abruptly whereas in the second-order one, both G and its first derivative are continuous and the heat capacity (the second derivative of G) is a discontinuous function. Formation of the swallow-tail shape in G − T diagram (continuous line of Fig. 6a) represents a first-order phase transition in the system. The phase transition point is located at the cross point in the G − T diagram, where small black hole (SBH) and large black hole (LBH) exist simultaneously (see Fig. 6a). Figure 6b displays the coexistence line of small/large BH phase transition. The critical which leads to As we see, the obtained critical temperature is independent of quintessence field matching exactly with that of presented in [9] (with Λ variable) for Reissner-Nordström black holes. However, χ c and p c depend on the state parameter, as expected. Figure 7 shows how Q and ω affect the critical pressure and temperature in the coexistence curve. Regarding the effect of ω on the critical pressure, it is evident that increasing the state parameter from −1 to − 1 3 makes the decreasing of the critical pressure. We can also see from Fig. 7b that state parameter ω reduces the large black hole region. As for electric charge, both critical pressure and temperature are decreasing functions of Q. To calculate new relation for the pressure of quintessence, we obtain the heat capacity as Solving denominator of the heat capacity with respect to p, the new pressure is determined as follows This new pressure has a maximum point which exactly coincides with the inflection point of p− χ diagram (see dashed line in Fig. 4a). The maximum pressure is the same critical pressure and its proportional χ is χ c . Inserting Eq. (59) into the relation of temperature Eq. (50), one finds a new relation for the temperature which is independent of pressure the existence of maximum in the obtained relation is representing the critical temperature (see dashed line in Fig.  4b). By deriving the new pressure or temperature with respect to χ, one can obtain χ Max as which is the same χ c . Inserting Eq. (61) into Eqs. (60) and (59), one gets Evidently, T Max and P Max are the same critical temperature and pressure in Eq. (57).

Behavior near critical point
Let us now compute the critical exponents for the black hole system. We start with the behaviour of the entropy and rewrite it in terms of T and χ as, which is independent of temperature. So, we find that and hence α = 0. Expanding around the critical point and defining ̺ = p pc , the equation of state (53) is rewritten as where Differentiating Eq. (66) with respect to ϕ for a fixed t < 0 , we get Applying Maxwell's area law with the fact that during the phase transition the pressure remains constant, leads to where ϕ s and ϕ l denote the event horizon of small and large black holes, respectively. Equation (69) has a unique non-trivial solution given by hence one can find that Now, we can differentiate Eq. (66) to calculate the critical exponent γ as Finally, the shape of the critical isotherm t = 0 is given by The obtained results show that the critical exponents in this new approach are exactly coincident with the van der Waals fluid system and one can observe a unique phase behavior as well.

IV. PHOTON SPHERE AND SHADOW
The image of a supermassive black hole in the galaxy M 87, a dark part which is surrounded by a bright ring, was direct support of the Einstein's general relativity and the existence of the black hole in our universe [53]. The image of the black hole gives us the information regarding its jets and matter accretion. The black hole shadow is one of the useful tools for a better understanding of the fundamental properties of the black hole and comparing alternative theories with general relativity. The gravitational field near the black hole's event horizon is so strong that can affect light paths and causes spherical light rings. The shadow of a black hole is caused by gravitational light deflection.
It is worthwhile to mention that in preliminary studies in the context of black hole shadow, the black hole was assumed to be eternal, i.e, the spacetime was assumed to be time independent. So, a static or stationary observer could see a time-independent shadow. But, modern observational results have shown that our universe is expanding with acceleration. This reveals the fact that shadow depends on time. Although for the black hole candidates at the center of the Milky Way galaxy and at the centers of nearby galaxies the effect of the cosmological expansion is negligible, for galaxies at a larger distance the influence on the diameter of the shadow is significant [54]. One method to explain the amazing accelerating expansion is to introduce dark energy which makes up about 70 percent of the universe. Cosmological constant and quintessence are two well-known kinds of dark energy models. Recently, the role of the cosmological constant in gravitational lensing has been the subject of focused studies [55][56][57][58][59]. The black hole shadow arises as a result of gravitational lensing in a strong gravity regime. So, one can inspect the effect of the cosmological constant on the shadow of black holes [60][61][62].
It should be noted that although the expansion of the universe was based on a positive cosmological constant, some pieces of evidence show that it can be associated with a negative cosmological constant. As we know, an interesting approach to examine the accelerated cosmic expansion and study properties of dark energy is through observational Hubble constant data which has gained significant attention in recent years [63][64][65]. The Hubble constant, H(z), is measured as a function of cosmological redshift. The investigation of H(z) behavior at low redshift data showed that the dark energy density has a negative minimum for certain redshift ranges which can be simply modeled through a negative cosmological constant [66]. The other reason to consider a negative cosmological constant is the concept of stability of the accelerating universe. In Ref. [67], authors analyzed the possibility of de Sitter expanding spacetime with a constant internal space and demonstrated that de Sitter solution would be stable just in the presence of the negative cosmological constant. The other interesting reason is through supernova data. Although there is strong observational evidence from high-redshift supernova that the expansion of the Universe is accelerating due to a positive cosmological constant, the supernova data themselves derive a negative mass density in the Universe [68,69]. Several galaxy cluster observations appear to have inferred the presence of a negative mass in cluster environments. In Ref. [70] was shown that a negative mass density can be equivalent to a negative cosmological constant. In fact, the introduction of negative masses can lead to an Anti de Sitter space. This would correspond to one of the most researched areas of string theory, the AdS/CFT correspondence. Now, we would like to investigate how black hole parameters affect the shadow radius of the corresponding black hole. To do so, we employ the Hamilton-Jacobi method for a photon in the black hole spacetime. The Hamilton-Jacobi equation is expressed as [71,72] ∂S ∂σ where S and σ are the Jacobi action and affine parameter along the geodesics, respectively. The Hamiltonian of the photon moving in the static spherically symmetric spacetime is Due to the spherically symmetric property of the black hole, one can consider a photon motion on the equatorial plane with θ = π 2 . So, Eq. (75) reduces to Taking into account the fact that the Hamiltonian does not depend explicitly on the coordinates t and φ, one can define where constants E and L are, respectively, the energy and angular momentum of the photon. Using the Hamiltonian formalism, the equations of motion are obtained aṡ The effective potential of the photon is obtained aṡ Fig. 8 depicts the behavior of the photon's effective potential for E = 1 and various L. We consider p = 0.6 to be close to the critical point p c . As we see, there exists a peak of the effective potential which increases with increasing L. Due to the constraintṙ 2 ≥ 0, we expect that the effective potential satisfies V ef f ≤ 0. So, an ingoing photon from infinity with the negative effective potential falls into the black hole inevitably, whereas bounce back if V ef f > 0. An interesting occurrence is related to the critical angular momentum L = L p (V max ef f = 0). In this case, the ingoing photon loses both its radial velocity and acceleration at r = r max completely. But for the sake of its non-vanishing transverse velocity, it can circle the black hole. So, the case of r = r max is called the photon orbit and it is denoted as r = r p . From what was expressed, one can find that the photon orbits are circular and unstable associated to the maximum value of the effective potential. In order to obtain such a maximum value, we use the following conditions, simultaneously the first two conditions determine the critical angular momentum of the photon sphere (L p ) and the photon sphere radius (r p ), respectively. The third condition ensures that the photon orbits are unstable. The radius of the photon orbits is calculated as The orbit equation for the photon is obtained in the following form The turning point of the photon orbit is expressed by the following constraint dr dφ Using Eqs. (76) and (83), one gets Considering a light ray sending from a static observer placed at r 0 and transmitting into the past with an angle Θ with respect to the radial direction, one can write [73,74] cot Θ = √ g rr g φφ dr dφ r=r0 . (85) Hence, the shadow radius of the black hole can be obtained as Imposing the constraint f (r 0 ) = 1 for an observer at spatial infinity, one can calculate the shadow radius as where r p is given in Eq. (81). The apparent shape of a shadow is obtained by using the celestial coordinates α and β which are introduced as the perpendicular distance of the shadow from the axis of symmetry and the apparent perpendicular distance of the shadow from its projection on the equatorial plane, respectively. The coordinates α and β are defined as [75,76] where (r 0 , θ 0 ) are the position coordinates of the observer.
To investigate the effect of electric charge and state parameter ω on the size of the black hole shadow, we have plotted Fig. 9. From this figure, one can find that the size of the circular shape of the black hole shadow shrinks with increasing these two parameters. As we see, the variation of Q has a weaker effect on the shadow size than the state parameter. In Fig. 10, we plot the black hole shadow for different values of the quintessence pressure and temperature. We observe that increasing the pressure (temperature) leads to decreasing (increasing) of the size of the black hole shadow.

A. Relations between shadow radius and phase transitions
Now we are interested in exploring the relations between the shadow radius and phase transitions. According to [73,74,77,78], there is a close connection between black hole shadows and the black hole thermodynamics. Heat capacity is one of the interesting thermodynamic quantities which provides us with information related to thermal stability and phase transition of a thermodynamic system. The signature of the heat capacity determines thermal stability/instability of black holes. The positivity (negativity) of this quantity indicates a black hole is thermally stable (unstable). In addition, the discontinuities in heat capacity could be interpreted as phase transition points. In fact, the phase transition point are where heat capacity diverges. According to Eq. 58, heat capacity can be written as By using the fact that ∂S ∂r+ > 0, the sign of C is directly inducted from ∂T ∂r+ which can be rewritten as ∂T ∂r + = ∂T ∂r s ∂r s ∂r + .
By satisfying the constraint ∂rs ∂r+ > 0 [73,74], one can draw a conclusion that the sign of C is controlled by ∂T ∂rs . To investigate the link between phase transition and black hole shadow, we consider the temperature expression Eq. (50) and the heat capacity Eq. (58), and plot the isobare curves on the T − r s and C − r s in Figs. 11b and 11d. As we see, these curves exhibit similar behaviors as T − χ and C − χ curves in Figs. 11a and 11c. For p > p c , the temperature is only a monotone increasing function of r s without any extremum (see dotted line in Fig. 11b). The heat capacity is also a continuous function for variable r s (see dotted line in Fig. 11d). For the case p < p c , a non-monotonic behavior appears for temperature with one local maximum and one minimum which corresponds to the first-order phase transition. According to the definition of heat capacity, these extrema coincide with divergence points of C. Evidently, a change of signature occurs at these points. Such that it changes from positive to negative at the first divergency and then it becomes positive again at the second one. So, there are three black holes competing thermodynamically. Small black holes which are located between root and smaller divergency are in a stable phase. For an intermediate range of the shadow radius, the black holes are thermodynamically unstable. The region after larger divergency is related to large black holes which are thermally stable. Fig. 12 displays these three phases and the effect of black hole parameters on these regions. As we see the quintessence pressure and state parameter have an effective role on the stability/instability of the black hole. In other words, a stable black hole may exit in its stable state when it is surrounded by quintessence. This reveals the fact that it is logical to consider the normalization factor as a variable quantity.
For p = p c , the small black hole and the large one merge into one squeezing out the unstable black hole. This can be found as a deflection point in the T − r s plot which forms critical point of the second-order phase transition. Such behavior of temperature is very similar to van der Waals liquid-gas system which goes under a first order phase transition for T < T c and p < p c , and undergoes a second-order phase transition at T = T c and p = p c . Inserting Eqs. (49) and (81) into Eq. (86), and using the critical values (57), one can obtain the critical shadow radius as One can analyze the behavior of the shadow radius before and after the second-order small-large black hole phase transition. To do so, we have depicted the changes of the shadow radius (∆r s = r L s − r S s ) as a function of the reduced pressure (p/p c ) in Fig. 13b. We see that ∆r s and ∆χ have similar behaviors and they both are monotonically decreasing functions of the reduced pressure. They approach to zero at p = p c , where the first-order phase transition becomes a second-order one.

V. CONCLUSION
In this paper, we have studied the analogy of charged AdS black holes surrounded by quintessence with van der Waals fluid system with a new viewpoint, in which we kept the cosmological constant as a constant parameter and instead allow the normalization factor to vary as a thermodynamic quantity. The obtained results showed that although the critical exponents in this alternative approach are exactly coincident with the van der Waals fluid system, a unique phase behavior cannot be observed in such a phase space. In continue, we considered the pressure of quintessence as a thermodynamic variable and studied the phase structure of the system under its variation. Obtaining the equation of state p = p(T, χ) and depicting p − χ and G − T diagram, we found that the corresponding black holes have very similar thermodynamic behavior in this alternative phase space as the van der Waals fluid system. We calculated critical quantities and noticed that the critical temperature in this alternative approach is exactly the same as that obtained in Ref. [9]. We also studied the behavior of physical quantities near the critical point and observed that the corresponding critical exponents coincide with those of the van der Waals fluid. It is worthwhile to mention that regarding our results, the pressure of quintessence is a positive quantity. So, in contrast to what was expressed in previous works, in this context, the energy density of quintessence and normalization factor should be negative.
It is worthwhile to mention that although one can investigate phase transition of a black hole with help of P −V and Q 2 − Ψ planes, it is more logical to take the normalization factor, which indicates the intensity of the quintessence field as a variable quantity. As we know, the cosmological constant does not change which basically has a constant value. In contrast, the quintessence field is a dynamic parameter that changes over time. Consideration of cosmological constant as a thermodynamic pressure shows that the pressure of the system depends only on the space-time parameter, while the pressure of quintessence depends on the black hole parameters as well. Regarding the consideration the square of the electric charge as a thermodynamic quantity, although one can employ this method to explore phase transition, it should be noted that according to [35,36] a first-order phase transition takes place for temperatures above its critical values which is a little different from the behavior of van der Waals fluid and other ordinary phase transitions in everyday systems. In addition, since the electromagnetic repulsion in compressing an electrically charged mass is dramatically greater than the gravitational attraction, it is not expected that black holes with a significant electric charge will be formed in nature. So, the electric charge of a black hole cannot change over the time so much.
Finally, we investigated the photon sphere and the shadow observed by a distant observer. We found that the shadow size shrinks with increasing the electric charge and state parameter ω. We also observed that as the quintessence pressure (temperature) increases, the shadow radius decreases (increases). We also explored the connection between shadow radius and phase transition and found that there exist non-monotonic behaviors of the shadow radius for the pressure below its critical value which corresponds to a first-order phase transition. We showed that such a phase transition will become a second-order one at critical pressure. Studying thermal stability of the system in this point of view, we noticed that the quintessence pressure and state parameter have a significant influence on the stability/instability of the black hole. This revealed the fact that a stable black hole may exit in its stable state if it is surrounded by the quintessence.