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A self-interacting partially directed walk subject to a force

We consider a directed walk model of a homopolymer (in two dimensions) which is self-interacting and can undergo a collapse transition, subject to an applied tensile force. We review and interpret all the results already in the literature concerning the case where this force is in the preferred direction of the walk. We consider the force extension curves at different temperatures as well as the critical-force temperature curve. We demonstrate that this model can be analysed rigorously for all key quantities of interest even when there may not be explicit expressions for these quantities available. We show which of the techniques available can be extended to the full model, where the force has components in the preferred direction and the direction perpendicular to this. Whilst the solution of the generating function is available, its analysis is far more complicated and not all the rigorous techniques are available. However, many results can be extracted including the location of the critical point which gives the general critical-force temperature curve. Lastly, we generalise the model to a three-dimensional analogue and show that several key properties can be analysed if the force is restricted to the plane of preferred directions.


Abstract.
We consider a directed walk model of a homopolymer (in two dimensions) which is self-interacting and can undergo a collapse transition, subject to an applied tensile force. We review and interpret all the results already in the literature concerning the case where this force is in the preferred direction of the walk. We consider the force extension curves at different temperatures as well as the critical-force temperature curve. We demonstrate that this model can be analysed rigorously for all key quantities of interest even when there may not be explicit expressions for these quantities available. We show which of the techniques available can be extended to the full model, where the force has components in the preferred direction and the direction perpendicular to this. Whilst the solution of the generating function is available, its analysis is far more complicated and not all the rigorous techniques are available. However, many results can be extracted including the location of the critical point which gives the general critical-force temperature curve. Lastly, we generalise the model to a three-dimensional analogue and show that several key properties can be analysed if the force is restricted to the plane of preferred directions.

Introduction
The development of atomic force microscopy and optical tweezers has allowed experimentalists to micro-manipulate individual polymer molecules (e.g. Bemis et al. (1999), Haupt et al. (2002), Gunari et al. (2007)), and this has led to a considerable body of theoretical work describing the response of a polymer to an applied force (e.g. Halperin and Zhulina (1991), Cooke and Williams (2003), Rosa et al. (2003)). Several situations have been investigated, including pulling a polymer off a surface at which it is adsorbed, pulling a polymer from a preferred solvent to a less preferred solvent, pulling a copolymer which is localized at an interface between two immiscible liquids, and pulling a collapsed polymer (in a poor solvent) to an extended form. In this paper we shall be concerned with the latter problem.
The stress-strain curve of a linear polymer in a poor solvent, being pulled in an AFM experiment, has been measured by several groups (Haupt et al. 2002, Gunari et al. 2007). The force-extension curve shows a characteristic plateau. For forces below this critical value the polymer will be in a collapsed state, while for forces above the critical value the polymer will be stretched. The plateau region would seem to indicate a firstorder phase transition. Indeed Grassberger and Hsu (2002) have studied self-avoiding walks with nearest-neighbour attraction and an applied force at low temperatures (poor solvents): they predicted a first-order phase transition in three dimensions. On the other hand, they see no sign of a first-order transition in two dimensions.
A well-studied exactly solved model of poor solvent polymers is the self-interacting partially directed self-avoiding walk model (IPDSAW). We begin our discussion by noting that a partially directed self-avoiding walk on the square lattice without selfinteraction is intrinsically anisotropic with a preferred direction so that the polymer's size scales proportionally to its length, and one perpendicular to this so that the polymer's size in this direction scales sub-linearly. As we shall consider the square lattice with the polymer oriented one way, the preferred direction will be the horizontal direction and the other direction will be the vertical direction.
In the absence of an applied force the critical point of this model was found by Binder et al. (1990), and is expected to model the polymer collapse transition (or θpoint). The exact solution of the generating function was found by Brak et al. (1992) and its singularity structure was rigorously elucidated. Prellberg et al (1993) used recurrence relations to generate very long series to estimate the exponents and the scaling function for the phase transition. A second-order phase transition similar to the θ-point was found. The tricritical nature of this transition was described by Owczarek et al. (1993) based upon small parameter expansions and calculations of a related version of the model (see below).
Various calculations were made by Owczarek et al. (1993) that are worth noting. Firstly a semi-continuous version of the model was solved explicitly, building on works by Lauritzen (1968 and1970) on a related model, and showed the same tricritical nature (identical exponents) as the lattice model (from the small parameter expansion and the numerical work of Prellberg et al. (1993)). A later calculation by Prellberg (1995) on the asymptotics of the generating function of the lattice model of staircase polygons enumerated by perimeter and area implicitly demonstrated that the scaling function and the exponents were also the same as in the fully discrete model: the q-Bessel functions involved are the same in the two lattice models. This was made explicit recently by Owczarek and Prellberg (2007). The low temperature scaling of the partition function was found by Owczarek (1993) in the semi-continuous model, and once again this was the same as found by Prellberg et al (1993) numerically for the fully discrete case. We argue below that the uniform asymptotic expansion given by Owczarek and Prellberg (2007) implies that they are indeed similar.
Secondly, Owczarek et al. (1993) generalised the fully discrete model to include a parameter that counts the horizontal span of the walks. The generating function was found by generalising the approach of Brak et al. (1992) for the no-force case. This added parameter is equivalent to considering a force applied in the preferred direction. The semi-continuous model also contained this parameter. In the semi-continuous model it was clear that this parameter did not change the nature of the transition. In the discrete model it was also implicit that the collapse transition was unchanged by this parameter (and confirmed by the work of Prellberg (1995)), and so unchanged by such an applied force. The connection to an applied force was however not made explicit in the paper. Rosa et al. (2003) studied the model numerically with the connection made explicit, and added the consideration of the end-to-end distance scaling function for the discrete model, corroborating the unchanged tricritical nature of the transition when an applied force in the preferred direction is added. They also plotted the critical-forceagainst-temperature curve for the discrete and semi-continuous models. Again the work of Owczarek and Prellberg (2007) makes this conclusion explicit.
Thirdly, in the appendix of Owczarek et al. (1993) a further generalisation of the model was considered and the full generating function of this further generalisation evaluated exactly. In this generalisation the parameter for vertical steps was replaced by two parameters: one for steps in the positive vertical direction and one for steps in the negative vertical direction. No analysis of this generating function was attempted. We note that a recent paper by Kumar and Giri (2007) considered pulling in both directions though their focus was on finite size effects calculated numerically rather than thermodynamic transitions and exact results.
In this paper we make explicit the connection to applying a force in the vertical (non-preferred) direction of the generalisation discussed above. However we go further and solve for the generating function along the surface in the parameter space which should include the transition point. Hence we find an explicit expression for the phase transition point and so are able to plot exactly the critical force against temperature curve. We also observe that one of the tricritical exponents is unchanged when this nonpreferred force is applied. This indicates, though does not prove, that the transition may remain second-order even in the case of this type of force. This is a little unexpected as the force must change the end-to-end scaling of the high temperature phase when pulling in the vertical direction from sub-linear to linear, which it does not when applied in the horizontal direction.
We also derive functional equations for the generating function and show how the solution of this general class of problem can be streamlined with this approach.
Before we explain our work on the vertical pulling problem we summarise all the known results on the horizontal pulling problem making explicit the results in terms of the applied force. Moreover, we apply the rigorous techniques of Brak et al. (1992), which had only been applied to the case of no applied force, to the horizontal pulling problem. In addition we discuss the behaviour of the force-extension curves based upon the exact results. Essentially we bring together all the known results and extend them as necessary for the case of horizontal pulling of a partially directed polymer in two dimensions.
At the end of this paper we consider a three-dimensional analogue of the model and show it has similar behaviour to its two-dimensional counterparts.

Model and definitions
Consider the square lattice and a self-avoiding walk that has one end fixed at the origin on that lattice. Now restrict the configurations considered to self-avoiding walks such that starting at the origin only steps in the (1, 0), (0, 1) and (0, −1) directions are permitted: such a walk is known as a partially directed self-avoiding walk (PDSAW). For convenience, we consider walks that have their first step in the horizontal direction. Let the total number of steps in the walk be n. We label the vertices of the walk i = 0, 1, 2, . . . , n. Let the number of horizontal steps be n x and the number of vertical steps be n y . To define our model we will need finer definitions, so let us define n y + to be the number of (0, 1) steps (positive vertical steps) and n y − to be the number of (0, −1) steps (negative vertical steps). If the walk starts at the origin let the position of the other end-point be (s x , s y ) so that the span in the horizontal direction is s x and the span of the walk in the vertical direction is s y . We therefore have n = n x + n y = n x + n y + + n y − (2.1) and s x = n x s y = n y + − n y − . (2.2) An example configuration along with the associated variables of our model is illustrated in figure 1.
To define our model we add various energies and hence Boltzmann weights to the walk. First, any two occupied sites of the walk not adjacent in the walk though adjacent Figure 1. An example of a partially directed walk (the bold black path) of length n = 21 with n x = 8, n y+ = 8, and n y− = 5 and having six nearest-neighbour 'contacts' (shown as intertwined (red) curves) so m = 6. The horizontal span is s x = 8 while the vertical span is s y = 3. One end is fixed at the origin while forces are applied to the other end (horizontal f x and vertical f y ).
on the lattice are denoted nearest-neighbour contacts or contacts: see figure 1. An energy −J is added for each such contact. We define a Boltzmann weight ω = e βJ associated with these contacts, where β = 1/k B T and T is the absolute temperature. Without loss of generality we shall take the units of energy to be such that J = 1 and therefore ω = e β , except when we discuss pulling polymers without any self-interaction (ω = 1) where we will have J = 0.
An external horizontal force f x pulling at the other end of the walk adds a Boltzmann weight h sx with h = e βfx . An external vertical force f y pulling at the other end of the walk adds a Boltzmann weight v sy with v = e βfy .
The partition function Z n (f x , f y , β) of the model for walks of length n ≥ 1, where for later mathematical convenience the first step of the walk is a horizontal step, is where m(ϕ) is the number of nearest-neighbour contacts in the PDSAW, ϕ. The generating functionĜ(z; h, v, ω) iŝ so z can be considered as a fugacity for the steps of the walk and the generating function as a "generalised partition function". We shall denote the radius of convergence of G(z; h, v, ω) as a function of z as z c (h, v, ω). The mean values of n, m, s x and s y , are given by (2.5) We shall define the number of n-edge partially directed walks with m nearestneighbour contacts, horizontal span s x and vertical span s y as d n (s x , s y , m) so that (2.6) and so b n (s x , m) is the number of n-edge partially directed walks with m nearestneighbour contacts and horizontal span s x . So It is advantageous when working with the generating function to define different variables. Let the generating function G(x, y + , y − , ω) be defined as where the sum over ϕ is over all PDSAW's of length n. Then making the substitutions demonstrates that G(hz, zv, z/v, ω) =Ĝ(z; h, v, ω). (2.12) Hence we have (2.13) It will be useful to define q + = y + ω and q − = y − ω (2.14) and q = √ q + q − , (2.15) and on making the substitutions (2.11) q = ωz . (2.16) We define the reduced limiting free energy as where the existence of the limit can be established by concatention arguments. The radius of convergence z c (h, v, ω) can be related to the free energy κ(f x , f y , β) as (2.18) It will turn out that there is a single phase transition where the free energy is singular as a function of β. We shall denote the phase-transition inverse-temperature as β t ≡ β t (f x , f y ). In general a superscript of t implies the critical value of a parameter, e.g. the critical horizontal force at a fixed temperature and zero vertical force is f t x ≡ f t x (0, β). However, we use a subscript t to denote the critical values of the fugacities. The value of the radius of convergence ofĜ(z; h, v, ω) as a function of z at ).
When f y = 0 we shall set y = y + = y − and so the generating function we need to consider for the case where there is no vertical stretching force is G(x, y, y, ω), noting that G(hz, z, z, ω) =Ĝ(z; h, 1, ω), wherê (2.19) Note that for f y = 0 we have q = yω , (2.20) which agrees with (2.16) when the substitutions (2.11) are made since y = z when In subsequent calculations it is convenient to define the generating function G r (x, y + , y − , ω) for walks that have r vertical steps immediately after the first horizontal step, where −∞ < r < ∞. It will also be convenient to introduce the generating function G + r (x, y + , y − , ω) = G r (x, y + , y − , ω) for walks that have r ≥ 0 steps after the first horizontal step in the positive y direction and G − r (x, y + , y − , ω) = G −r (x, y + , y − , ω) that have r ≥ 0 steps after the first horizontal step in the negative y direction. Clearly, In the functional equation section (section 7) we need Note the asymmetry of the summation index in the final two definitions. Note that and In this section we will begin discussing the case of horizontal pulling force where f y = 0.

No self-interactions (ω = 1)
Let us begin by considering the case of no interactions and no force so that J = f x = f y = 0. It is easy to see that lim n→∞ n −1 logb n = log(1 + √ 2). Let us consider the case of no vertical force so that f y = 0 and v = 1. As discussed in the previous section we are interested in the generating functionĜ(z; h, 1, ω) = G(hz, z, z, ω). As defined above this generating function converges when z < z c (h, 1, ω). Recall that the free energy κ(f x , 0, β) is given by κ = − log z c (assuming that the limit defining the free energy exists, which will be proved in a later section).
When ω = 1 (which corresponds to turning off the vertex-vertex interaction and hence to good solvent conditions),Ĝ(z; 1, h, 1) satisfies the equation From this we can readily calculate the ratio s x / n and take the thermodynamic limit by letting z → z c (h, 1, 1) where In figure 2 is a plot of lim z→zc s x / n against βf x . Note that the stress-strain curve is qualitatively the same as that found experimentally for polystyrene in a good solvent (toluene) (Gunari et al. 2007, figure 1).
For ω = 1 the situation is more difficult and we derive some results about the generating function in the following sections.

Convexity and continuity for ω = 1
In this section we establish the existence of the thermodynamic limit for the canonical problem, and prove convexity and continuity. This implies continuity of the phase boundary. It will be convenient to consider a subset of the walks counted byĜ(z; h, 1, ω). Let a n (s x , m) be the number of partially directed walks with n edges, m contacts and x-span equal to s x which satisfy the additional constraint that their last step is in the east direction. (That is, both the first and last steps are east steps.) These walks can be concatenated by identifying the last vertex of one walk with the first vertex of the other walk which yields the inequality Defining the partition function this implies the super-multiplicative inequality Since A n (h, ω) ≤ 3 n ω n h n , it follows that n −1 log A n (h, ω) is bounded above for h, ω < ∞.
Hence the super-multiplicative inequality above implies the existence of the limit for h, ω < ∞. We now recall the definition of the partition function To prove that κ(f x , β) is a convex function of β and βf x we note that Hölder's inequality implies that This shows that n −1 log B n (h, ω) is a convex function of β and βf x . (It is convex as a surface, not just separately convex in each variable.) Since the limit of a sequence of convex functions (when it exists) is a convex function, κ(f x , 0, β) is a convex function of β and βf x . Hence κ(f x , 0, β) is continuous and is differentiable almost everywhere. Since κ(f x , 0, β) = − log z c (h, 1, ω), the singularity surface z = z c (h, 1, ω) is also continuous and differentiable almost everywhere.

The generating function for ω = 1
While the calculation of the generating function for ω = 1 appears in Owczarek et al. (1993) (see section 4 of that paper) we feel that it is worth summarizing and presenting in a slightly different way. In order to consider the case ω = 1 it was necessary to generalize a method originally due to Temperley (1956) and used for the case h = 1 by Brak et al. (1992). For convenience let the partial generating functionĜ r (z; h, 1, ω) in the case where v = 1 (no vertical force) be denoted as and g 0 =Ĝ 0 (z; h, 1, ω). Now the generating function required isĜ(z; h, 1, ω) which is the sum over all r aŝ (3.14) Noting thatĜ −r (z; h, 1, ω) =Ĝ r (z; h, 1, ω), we have the partial generating functions g r satisfying the relations See figure 3. These results imply that g r satisfies the recurrence Figure 3. Walks counted by g r , with r > 0, either contain a single horizontal bond (and so are an ⌋ or an ⌉) or can be constructed by appending an ⌋ or ⌉ configuration of bonds to a walk counted by g k . Summing over all these possibilities gives equation (3.16).
( 3.22) The quadratic equation (3.20) has the two solutions λ 1 = z and λ 2 = q = ωz and the general solution for g r , r > 0 is Arguments similar to those in Brak et al. (1992) show that A 2 = 0 and A 1 can be determined by noting that and where a = hz 2 (2 + hz − ωhz) and b = hz 2 (1 + hz + ω − ωhz). These simultaneous equations giveĜ which can be written as ( 3.28) The solution given by Owczarek et al. (1993) was written as .
( 3.29) where H(y, q, t) = H(y, q, qt) H(y, q, t) (3.30) and The two forms can be seen to be the same when one notes that and that ( 3.33) The implications for the singularity diagram of this expression forĜ(z; h, 1, ω), and the phase transitions, will be considered in the next section.
3.4. Solution on the special surface z = 1/ω Before we proceed to discuss the singularity diagram let us consider the solution on the special surface defined by q = 1 since the solution described by (3.28) is singular when z = 1/ω though the generating function is finite for large enough ω.
The recurrence for the partial generating functions becomes and now the simpler Ansatz of g r = Cµ r can be used to find (3.35) When h = 1 we recover the previously calculated result of Brak et al. (1992).

The phase diagram for a horizontal force
In order to analyse the phase transition structure we need to analyse the form of the singularity diagram as a function of the ω and h parameters. We derive a functional equation for a slight generalization of the quantity g r . We introduce a parameter t and define the function 1 is equal to zg(tq; q, h, ω) evaluated at t = 1 while g (1) 0 = g(1; q, h, ω). The function g(t; q, h, ω) satisfies the functional equation g(t; q, h, ω) + g(q 2 t; q, h, ω)/ω = (1 + 1/ω + (1/ω − 1)hq 2 t/ω)g(qt; q, h, ω).
(4.2) DefiningĤ (t; q, h, ω) = g(t; q, h, ω)/g(tq; q, h, ω) (4.3) givesĤ (t; q, h, ω) which leads to a continued fraction representation forĤ(t; q, h, ω). Note that and therefore (4.6) The functionĤ(q, h, ω) is singular on the hyperbola q = ωz = 1 for all values of h and the only other singularities in G come from the poles corresponding to the denominator of G being zero, i.e. determined by the solution of the equation (1 + hz + ω − ωhz)Ĥ(q, h, ω) − 2 = 0. This line of poles intersects the hyperbola at a point determined by solving the equation (1 + hz + ω − ωhz)Ĥ(1, h, ω) − 2 = 0, whereĤ(1, h, ω) is determined by the quadratic equation which comes from (4.4) on setting q = t = 1. The variables ω and h are related through the equation which implies that the critical value of h is given by Let us consider q c (h, 1, ω) = z c (h, 1, ω)ω. We have that 0 < q c < 1 for 1 ≤ ω < ω t (h, 1). We have argued that there is a pole in the generating function which means that there is a zero of D = 1/(1 + G). That is, the function We also have analyticity of H for 0 < q < 1, shown by the ratio test. When ω = 1 there is a simple pole in the generating function. We have and so that ∂D(q, 1, h) ∂q < 0 (4.13) at q = q c (h, 1, 1). The derivative is negative for all 0 < q < 1. We now prove that there is a simple pole for all 1 ≤ ω < ω t (h, 1). For this, we compute (4.14) Observe that H(y, q, t) is related to the generating function G sc (x, y, q) of staircase polygons enumerated by width, height, and area. Equations (4.6) and (4.9) of Prellberg and Brak (1995) imply that It follows that H(q/ω, q, hq 2 (ω − 1)/ω 2 ) is a power-series in q with non-negative coefficients. Therefore, its derivative with respect to q is positive. Hence ∂D(q, ω, h) ∂q < 0 (4.16) and therefore there is a simple pole in the generating function at all values of 1 ≤ ω < ω t (h, 1). Now, using a version of the implicit function theorem that gives analyticity (see chapter 6 of Krantz and Parks 2002) it follows that q c (h, 1, ω) is an analytic function of both ω and h.

(4.19)
To evaluate the continued fraction C efficiently we note that if C m is the truncation of C at order m then C m can be written as a rational function whose numerator and denominator depend on m and are determined by recurrences. This gives a convenient way to evaluate C to the required accuracy. In figure 4 we show the phase boundaries in the (ω, z)-plane for several values of h.

Extension under a fixed horizontal force
We shall now consider the average extension of the polymer as a function of the applied force at different temperatures. From section 3 we know that κ(f x , 0, β) is a continuous function of β and f x . Moreover for any fixed f x the free energy κ(f x , 0, β) is analytic for 0 < β < ∞ except at at most one point which we label β t (f x ) when it exists. This transition point is defined (see Owczarek and Prellberg 2007) via there is a single solution. Let the solution for this 'critical force' be labelled f t x and so by (5.1) it is given by the function which is the result (11) of Rosa et al. (2003). In figure 5 we plot the critical force f t x against β −1 . We note that for β ≤ β t (0, 0) ≈ 1.218 (equivalently T t ≈ 0.8205) there is no positive critical force, and that for β > β t (0, 0) there is a single critical force f t x .  Figure 5. The temperature dependence of the critical force for horizontal pulling in the n → ∞ limit. When the force is less than the critical force the walk is compact and when it is greater than the critical force it is expanded.
At fixed force and below the critical temperature (which depends on the force), we have z c (h, 1, ω) = 1/ω and so For 0 < β ≤ β t (f x , 0) we know that z c (h, 1, ω) is a strictly decreasing function of β, and so κ(f x , 0, β) is a strictly increasing function of β. Near the transition we know from Owczarek and Prellberg (2007) (see the discussion after equation (3.9)) that and similarly for some constants C and D.
For 0 < β < β t (f x , 0), the singularity in z closest to the origin inĜ(z; h, 1, ω) is a simple pole. From this we deduce that Z n (f x , 0, β) = e κ(fx,0,β)n+O(1) for 0 < β < β t (f x , 0) . (5.6) In the semi-continuous model ) the following was derived for the scaling of the partition function at low temperatures Z n (f x , 0, β) = e βn+κs(fx,0,β)n 1/2 +O(1) for β t (f x , 0) < β , where κ s (f x , 0, β) is a negative non-constant analytic function of β and f x , and also we have that lim β→β t κ s (f x , 0, β t ) = 0. Both the discrete model (given explicitly by Owczarek and Prellberg (2007)) and the semi-continuous model (given by Owczarek et al. (1993)) have generating functions that have uniform asymptotics with the same algebraic structure, given by a ratio of Airy functions. In particular the results in Owczarek et al. (1993) depend on the scaling of the locations of the poles in the generating function which has the same scaling in both cases. Without making the explicit calculation one can surmise that equation (5.7) also holds for the discrete model. For completeness we note that at β = β t (f x , 0) there is an algebraic singularity in the generating function (see Owczarek and Prellberg (2007)). Assuming the conditions of Darboux's theorem hold gives us One could calculate the average extension in the generalised ensemble simply by differentiating the generating function with respect to the force. In the discrete model this gives a complicated expression in terms of q-Bessel functions and its derivatives. However as the equivalence of the thermodynamic limits of the canonical ensemble and generalised ensemble is limited to 0 < β ≤ β t it is simpler to use the results above to deduce the behaviour of the average extension in the canonical ensemble directly. In the canonical ensemble the average extension s x n (f x , f y , β) is defined by Assuming the conditions for differentiating the asymptotic expansion term-by-term hold the result (5.6) implies that (5.10) so that the thermodynamic limit extension per unit length Moreover the scaling near β t in equation (5.5) as given by Owczarek and Prellberg (2007) implies that One can calculate this quantity from the generalised ensemble in the semi-continuous model (see equation (3.46) in Owczarek et al 1993) and analyse it in a more straightforward manner than in the discrete case (since it involve Bessel functions rather than q-Bessel functions), and it shows the same behaviour.
On the other hand the result (5.7) (once again assuming differentiability of the asymptotic expansion) implies that and so that We have calculated the stress-strain curves by computing the free energy from the boundary of convergence (section 4) and then numerically differentiating. Using these results and the above arguments we can make the following comments. In all cases the function of the average extension per unit length S At high temperatures where β < β t (0, 0) there is no critical force and the average extension per unit length S x (0, 0, β) > 0 at zero force. At such fixed β the function S x (f x , 0, β) is an analytic function of f x for f x ≥ 0 and is strictly increasing with increasing force. There is a unit horizontal asymptote, approached from below for large forces. There is a finite nonzero slope ∂Sx(fx,0,β) ∂fx at f x = 0. The qualitative behaviour is the same as that shown in figure 2 for the the ω = 1 case.
At the critical temperature for zero force β = β t (0, 0) the average extension per unit length S x (0, 0, β) = 0 at zero force. Again the function S x (f x , 0, β t (0, 0)) is an analytic function of f x for f x > 0, is strictly increasing with increasing force, and has a unit horizontal asymptote, approached from below for large forces. Now however the slope ∂Sx(fx,0,β t (0,0)) ∂fx diverges when f x → 0 + as f −1/2 x . This is shown in figure 6. At low temperatures where β > β t (0, 0) (so there is a critical force f t x > 0) there are two regimes. For 0 ≤ f x ≤ f t x the average extension per unit length S x (f x , 0, β) = 0 regardless of the force. However for f t x < f x the function S x (f x , 0, β) is an analytic function of f x , is strictly increasing with increasing force, and has a unit horizontal asymptote, approached from below for large forces. Again the slope ∂Sx(fx,0,β) ∂fx diverges when f x → (f t x ) + as (f x − f t x ) −1/2 . This is illustrated in figure 6. 6. The full model: Pulling in both directions

No self-interactions (ω = 1)
As in subsection 3.1 which describes the solution when there is no vertical pulling (and no self-interactions) one can write down a simple functional equation for the generating function G(x, y + , y − , 1) as and setting x = hz, y + = zv and y − = z/v we get 1 +Ĝ(z; h, v, 1) = 1 So the answer is a simple rational function, and the critical z (that is z c (h, v, 1)) is the root of a cubic. For h = v = 1 this simplifies as expected.
If we set f x = 0 then 1 +Ĝ(z; 1, v, 1) = 1 (6.4) and from this we can readily calculate the ratio s y / n and take the thermodynamic limit by letting z → z c (1, v, 1). In figure 7 is a plot of lim z→zc s y / n against βf y .

Solution of the full model
In the appendix of Owczarek et al. (1993) it was shown that if one adds fugacity variables y + for steps in the positive vertical direction and y − for steps in the negative vertical direction then a generalisation of the method for solving the standard problem yields the generating function G(x, y + , y − , ω) as described in section 2. Using q + = y + ω and q − = y − ω it is given by , (6.8) and P [λ] = (λ − 1)(λ − ω). (6.9) Note that q + q − = q 2 , and given that all the parameters are positive we have q = √ q + q − .
This solution is then a clear generalisation of the form for the horizontal pulling model where .
( 6.10) where H(y, q, t) = H(y, q, qt)/H(y, q, t) is given in terms of H(y, q, t) defined by equation (3.31). However the problem of generalising the analysis of Prellberg (1995) is a daunting one. On the other hand we can still make some deductions about the behaviour of the generating function. Since the partition function is positive we know thatĜ(z; h, v, ω) is a strictly increasing function of z for fixed h, v, ω > 0.
Let us consider ω ≥ 1. The functions A ± r and B ± r converge whenever q < 1 and moreover there are singularities when |q| = 1. The solution for the generating function G(x, y + , y − , ω) is given in the next section and one finds that G(x, y + , y − , ω) on this surface is finite when ω > ω t , where ω t is given by the solution of equation (6.27). The only singularities that can occur in the generating function for q < 1 are poles occurring when the denominator of (6.5) is zero. The nature of these poles is unclear but we conjecture that they are simple poles and that they only exist when ω < ω t .
6.3. Solving the full model on the special surface z = 1/ω While expression (6.6) is rather unwieldy, it is possible to find a sub-manifold of parameters for which the solution can be made more explicit. If we restrict to q + q − = q 2 = 1 that is q = ωz = 1 , (6.11) then (6.6) has a singular limit. This is analogous to the fact that in the case of (6.10) one obtains an algebraic function for q = 1. Hence we have q + = v and q − = 1/v. Also, we shall retain the variable x for convenience during the calculations before substituting x = h/ω as required in the final expressions.
Let g ± r = y ± −r G ± r where G ± r is the generating function for walks with r steps in the positive and negative directions respectively, and we have in analogy with the derivation of the full problem g ± r+4 − (ω + 1)(q ± + 1)g ± r+3 + (ω(1 + q ± 2 ) + (ω + 1) 2 q ± )g ± r+2 (6.12) −ωq ± (ω + 1)(q ± + 1)g ± r+1 + ω 2 q ± 2 g ± r = q ± x 2 (ω − 1) 2 g ± r+2 . The characteristic polynomial is (6.13) Despite being a polynomial of degree four, P ± (λ) has sufficient symmetry to allow for simple explicit solutions. The key observation is that if λ is a root of P ± (λ), then so is q ± ω/λ. If we define µ ± = λ + ωq ± λ , (6.14) then µ ± satisfies We can therefore obtain all four roots of P ± (λ) by solving the quadratic equation (6.15) for µ ± , followed by solving the quadratic equation (6.14) for λ. The general solution for (6.12) is a linear combination After some algebra the generating function 1+Ĝ(1/ω; h, v, ω) = G(h/ω, v/ω, 1/vω, ω) can be found to be with t 1 and t 2 the two distinct roots of L(t) = 0 with We note that L(t) is a quartic polynomial in t. If we let s = 1 t + vωt (6.20) then s satisfies and the two solutions of this equation are Now only one of the solutions of (6.20) is applicable, namely which gives us that (6.24) and .
(6.25) Importantly, on substitution into the generating function the denominator can be found to be a polynomial with a factor and no other relevant factors. Hence the generating function is singular on the curve q = 1 when this factor is zero. It can be seen that the algebraic singularity (a square root) occurs in the generating function at the same place by considering the discriminant of (6.13). It can therefore be deduced that the exponent γ u = 1/2 describing the singularity in the generating function approaching the transition point tangentially (as described in Owczarek et al 1993). The exponent is independent of the value of h and v and so of whether there is a horizontal and/or vertical pulling force. We therefore have the location of the critical point as One can rewrite this as When there is no horizontal force, that is h = 1, we therefore find the critical point to be when (ω 2 + 1) 2 = (v + 2 + 1 v )ω 3 , (6.29) which reduces to the known result of Binder et al. (1990) when v = 1. From this equation the critical force temperature plot can be found as In figure 8 we plot the critical force against temperature for vertical pulling. Note that as the temperature approaches the critical value for no pulling force the slope of this curve diverges in contrast to the analogous horizontal pulling curve. Note that when v = 1 in (6.27) we find equation (5.1), that was derived in a previous section.

Horizontal pulling
For the case of horizontal pulling we can construct a recursive functional equation for the partial generating function, cf section 2. Let us define g r (x, y, ω) = 2G r (x, y, y, ω) if r ≥ 1 G 0 (x, y, y, ω) if r = 0 (7.1)  column is added to the left of a walk. Let P r be the set of all paths with at least one horizontal step followed by r ≥ 0 first column vertical steps. Let r ′ be the number of second column vertical steps. We can partition P r into the following five disjoint subsets.
Case I: The walk has only one step. This is generated by x.
Case II: These are all walks with only one horizontal step and at least one first column vertical step. These are generated by 2xyp/(1 − yp), the factor of two giving all upwards and all downwards sequences of steps, both generated by yp/(1 − yp).
Case III: These are all paths with at least two horizontal steps and the first column vertical steps (if any) are in the same direction to the second column vertical steps (if any). These are generated by xF (1)/(1 − yp). There is no factor of two for the following reason. The product xF (1)/(1 − yp) corresponds to concatenating a sequence of vertical steps, V → 1/(1 − yp) to and arbitrary path F (1). The generating variable p is set to one in F (p) since the leftmost vertical steps arise from the V sequence. If the first vertical sequence of steps of F (1) are upward, then the V steps are interpreted as also being upward (and hence no contacts are created by the concatenation, thus no ω factor). If the first vertical sequence of steps of F (1) are downward, then the V steps are interpreted as also being downward (and again, no contacts are created by the concatenation).
Case IV: These are all paths with at least two horizontal steps and 0 < r ≤ r ′ and with the further condition that the vertical steps in the first column are in the opposite direction to those in the second column. These are generated by xypF (ωyp)/(1 − yp). The argument for this form is similar to that of Case III, except now contacts are created by the concatenation. The contacts are accounted for as follows. All new contacts occur in the overlap between the first and second columns. Since the p in F (p) tracks the number of vertical steps in the second column, the new contacts can be accounted for by replacing p by ωp. The factor of yp in F (ωyp) is interpreted as giving rise to first r ′ vertical steps in the first column (these can be thought of as 'virtual steps') and the remaining r ′ − r vertical steps are generated by 1/(1 − yp) and are interpreted as being concatenated on to the virtual steps, as illustrated by the fourth term in figure 9.
Case V: These are all paths with at least two horizontal steps and 0 < r ′ ≤ r and with the further condition that vertical steps in the first column are in the opposite direction to those in the second column. These are generated by xωyp [F (1) − F (ωyp)] /(1 − ωyp) which can be shown using inclusion-exclusion. The term xωyp/(1 − ωyp) counts an infinite sequence of vertical bonds and contacts. So the generating function, T 1 = xωypF (1)/(1 − ωyp), counts all the required configurations, but also those with r ′ > r. The contribution of these over-counted configurations is then given by T 2 = xωypF (ωyp)/(1 − ωyp). Hence the contribution of this case is T 1 − T 2 . See figure 10. Combining all five cases together, as illustrated in figure 9, generates all the walks. Thus we get, This functional equation can be solved for F (p) by the method of iteration (Bousquet-Mélou 1996). The case ωy = 1 is much simpler to solve. Letting ωy = 1 gives, Note, to find F (1) from equation (7.4) we cannot put p = 1 because of the denominator 1 − p. To avoid this problem, first collect coefficients of F (p) to give, where the "kernel", K(p), is given by If we can put K(p) = 0 in (7.5) then, since the lefthand side is zero, we can solve for F (1). Note, the condition K(p) = 0 implicitly constrains p, which, since K(p) is quadratic in p, implies that we get two possible functions p ± (x, y). However some care must be taken as this way of finding F (p) assumes F (p)K(p) = 0 which is only the case if lim p→p ± F (p)K(p) = 0. Thus, we first assume this is the case for one of the solutions, say p + and then verify this assumption once F (p) has been explicitly computed. As is readily shown, each of the two assumptions lim p→p + F (p)K(p) = 0 and lim p→p − F (p)K(p) = 0 give rise to different functions, denoted F ± (p). In fact, F (p) is an algebraic function and each of the two functions F ± (p) are the two branches of F (p). The limit lim p→p + F (p)K(p) only vanishes if the correct branch is combined with the correct limit (the other branch has a pole at p + and hence the limit is the non-zero residue of F (p) at p + ). Thus, assuming lim p→p + F + (p)K(p) = 0 and taking the limit as p → p + on both sides of (7.5) gives F + (1). Thus we obtain the branch F + (1) on the line ωy = 1 as or, explicitly 1 + F + (1) = y − 1 x 2 (y − 1) + 2x(y + 1) + y − 1 . (7.8) Using this solution, one then verifies the assumption that lim p→p + F + (p)K(p) = 0. This solution has a square root singularity at x 2 (y − 1) + 2x(y + 1) + y − 1 = 0, (7.9) which gives the same critical value of x, x c as given by equation (5.1) using the transformations x = h/ω and y = 1/ω. Thus we see that, using this method, x c arises via p + and hence from the kernel. The functional equation (7.3) is also closely linked to the recurrence relation (3.16). If we make the substitution (2.21) into (7.3) and equate coefficients of p, then, for r > 0 we get g r = xy r+1 2 + k≥0 g k + r−1 k=0 ω k g k + ω r κ≥0 g k − ω r r−1 k=0 g k (7.10) which is readily put in the same form as (3.16).

Vertical pulling
Let us defineF Using similar arguments to the derivation of equation (7.3) we obtain the following pair of coupled equations. (7.14) These equations can again be solved by the method of iteration, resulting in complex q-series similar to (6.6).
There is also the simple case occurring when ωy = 1. Using the notation we can write the two functional equations as where the two kernels are given by Thus we choose p such that which is the same equation as the characteristic equation (6.13) with p → λ and y replaced by 1/ω. Thus we see that the kernel that arises from the functional equation approach corresponds to the characteristic equation (6.13) required to solve the Temperley recurrence relations (6.12).

Three-dimensional model
The model can be generalized to a three-dimensional partially directed walk model in which the walk is self-avoiding, can take steps in the +x or +y directions or in the ±z directions. Again it will be convenient to require that the first step is in the +x or +y direction. We shall only consider a force applied in the (x, y)-plane and as such we shall keep track of the span in the x and y directions as well as the number of contacts. If we write c n for the number of these walks with n steps it is easy to see that and we define the generating function Concatenation arguments can be used, as in section 3, to establish the existence of the limiting free energy κ(h x , h y , ω) = lim n→∞ n −1 log Z n (h x , h y , ω). (8.4) Methods exactly analogous to those in section 3 can be used to show that κ(h x , h y , ω) is convex and continuous, and differentiable almost everywhere. At fixed ω, h x and h y G converges if z < z c (h x , h y , ω) = exp[−κ(h x , h y , ω)], and the phase boundary z = z c is continuous.
If we turn off the interactions by setting ω = 1 and write G 0 ≡ G(z; h x , h y , 1) then G 0 satisfies the equation . (8.6) The force-extension curve can be calculated as in section 3 and this has the same general form as for the two-dimensional model, and as found experimentally for good solvent conditions (Gunari et al. 2007).
For attractive interactions (ω > 1) the Temperley approach described in section 3 for the two-dimensional case immediately generalises to this model. Defining partial generating functions as in section 2, the partial generating functions obey the relations g 0 = (h x + h y )z(1 + G) (8.7) and g r = (h x + h y )z r+1 2 + r k=0 (1 + ω k )g k + (1 + ω r ) k>r g k , (8.8) so that g r satisfies the relation g r+1 − (z + q)g r + q r (h x + h y )z(q − z)g r + qzg r−1 = 0 (8.9) with q = ωz. The analysis goes through as in section 3 and one can find the temperature dependence of the critical force. By choosing the relative magnitudes of h x and h y one can change the direction in the (x, y)-plane in which the force is applied. In all cases the force-temperature curve is monotonic and does not show reentrance. Presumably this is because the model does not have extensive ground state entropy. The phase diagram can also be calculated, using the same methods as in section 3, with qualitatively similar results.  Figure 11. The temperature dependence of the critical force for the three-dimensional model in the n → ∞ limit when the force is applied in the x-direction.
If we set h y = 0 we reproduce the results of section 3. We consider two other cases. First we set h x = h and h y = 1. This corresponds to applying a force in the x-direction. We show the critical force-temperature curve in figure 11 and the boundary of convergence in figure 12.
These are qualitatively similar to the corresponding figures for horizontal pulling in two dimensions, though with quantitative differences, of course. In particular the critical force-temperature curve has zero slope in the T → 0 limit and is monotone decreasing. We also set h x = h y = h which corresponds to pulling in the xy-plane but at 45 o to the x-axis. We show the critical force-temperature curve in figure 13 and the boundary of convergence in figure 14.
The critical force now has negative limiting slope in the T → 0 limit but remains monotone decreasing. This is because of the curious feature of this model that the ground state has no (extensive) entropy in the compact state but acquires entropy under the influence of a force. The low temperature behaviour can be understood by the following crude low temperature argument. Think of an n-edge walk at low temperature T under a tensile force f at 45 o to the x-axis. If n − m edges of the walk are in a compact state and the remaining m edges are extended the (extensive) free energy can be written as Setting ǫ = −1 gives a critical force of 1 at T = 0 and the limiting slope df /dT = − log 2 < 0.

Discussion
We have analysed the polymer model of partially directed walks with self-interaction, so as to induce a collapse transition, under the influence of tensile forces on the ends of the polymer. The problem of forces only in the preferred direction of the walk can be elucidated completely. The phase transition which is a second-order transition without any force is unchanged by the presence of such a force. The force extension curves at high temperatures look qualitatively similar to those of AFM experiments (Gunari et al. (2007)). The solution of the full model is more problematic. While the exact solution of the generating function of partition functions can be written down in terms of q-series, these functions are even more complicated than those encountered in the standard model that has only forces in the preferred direction (those are q-Bessel functions). We have been able to solve the model on the important special curve in parameter space which should contain the transition point. This seems to indicate that, again, the transition is unaffected by the force. This is more difficult to understand physically as the force must change the high temperature state of the polymer, unlike a force applied only in the preferred direction. It would therefore be interesting to analyse this full model further.